Equilibrium Index of Invariant Sets and Global Static Bifurcation for Nonlinear Evolution Equations

TL;DR

This paper develops an equilibrium index for invariant sets in nonlinear evolution equations, providing a new global bifurcation theorem that applies even when the crossing number is even, with applications to boundary value problems.

Contribution

It introduces a novel equilibrium index for invariant sets and establishes a global static bifurcation theorem applicable to cases with even crossing numbers.

Findings

Established an index formula for bifurcating invariant sets.

Proved a global bifurcation theorem valid for even crossing numbers.

Provided an example involving bifurcations in boundary value problems.

Abstract

We introduce the notion of equilibrium index for statically isolated invariant sets of the system $u_t+A u=f_\lambda(u)$ on Banach space $X$ (where $A$ is a sectorial operator with compact resolvent) and present a reduction theorem and an index formula for bifurcating invariant sets near equilibrium points. Then we prove a new global static bifurcation theorem where the crossing number $\mathfrak{m}$ may be even. In particular, in case $\mathfrak{m}=2$, we show that the system undergoes either an attractor/repeller bifurcation, or a global static bifurcation. An illustrating example is also given by considering the bifurcations of the periodic boundary value problem of second-order differential equations.

Figures

Equations

$$u_t+Au=f_{\lambda }(u)$$

$$u_t+Au=f(u)$$

$$F=(aI+A{)}^{-1}(aI+f)$$

$$\text{mbox}Ind(\Phi ,K)=(-1{)}^{{\mathfrak{m}}_1}\text{mbox}Ind({\Phi }^c,K),$$

$$\begin{array}[]{ll}\mbox{Ind}\,(\Phi_{\lambda},K_{\lambda})&=\left\{\begin{array}[]{ll}\chi\left(h(\Phi_{\lambda_{0}},\{0\})\right)-(-1)^{\mathfrak{m}_{1}},&\lambda<\lambda_{0};\\[4.30554pt] \chi\left(h(\Phi_{\lambda_{0}},\{0\})\right)-(-1)^{\mathfrak{m}_{1}+\mathfrak{m}_{2}},&\lambda>\lambda_{0}.\end{array}\right.\\[17.22217pt] &=(-1)^{\mathfrak{m}_{1}}\left\{\begin{array}[]{ll}\chi\left(h(\Phi_{\lambda_{0}}^{c},\{0\})\right)-1,&\lambda<\lambda_{0};\\[4.30554pt] \chi\left(h(\Phi_{\lambda_{0}}^{c},\{0\})\right)-(-1)^{\mathfrak{m}_{2}},&\lambda>\lambda_{0},\end{array}\right.\end{array}$$

$$h({\Phi }_{{\lambda }_0}^c,\{0\})={\Sigma }^0,$$

$$\begin{array}[]{lcl}\mbox{Ind}\,(\Phi_{\lambda},K_{\lambda})&=(-1)^{\mathfrak{m}_{1}}\left\{\begin{array}[]{ll}0,&\mbox{$\mathfrak{m}_{2}$\,=\,even};\\[4.30554pt] 2,&\mbox{$\mathfrak{m}_{2}$\,=\,odd},\end{array}\right.\end{array}$$

$$\chi \left(h({\Phi }_{{\lambda }_0}^c,\{0\})\right)\ne 1\text{mbox}or(-1{)}^{{\mathfrak{m}}_2},$$

$$-u^{\prime \prime }=\lambda u+a(x)u^2+h(x,u),x\in \mathbb{R},$$

$$\sigma (A)={\sigma }_p(A)=\{{\lambda }_n{\}}_{n=0}^{\infty }.$$

$$d(A,B)=\inf \{d(x,y):x\in A,y\in B\}.$$

$$d_{\text{mbox}H}(A,B)=\underset{x\in A}{\sup }d(x,B),{\delta }_{\text{mbox}H}(A,B)=\max \left\{d_{\text{mbox}H}(A,B),d_{\text{mbox}H}(B,A)\right\}$$

$$d_{\text{mbox}H}(A_{\lambda },A_{{\lambda }_0})\to 0,\text{mbox}as\lambda \to {\lambda }_0.$$

$$\begin{array}[]{ll}A\cap C\neq\emptyset\neq B\cap C,\end{array}$$

$$(t,x)\in \mathcal{D}(\Phi )⟺t\in [0,T_x);$$

$$\gamma (t)=\Phi (t-s)\gamma (s),\forall t,s\in J,t\ge s.$$

$$\mathcal{U}(\mathcal{A})=\{x:\underset{t\to \infty }{\lim }d(\Phi (t)x,\mathcal{A})=0\}.$$

$$\begin{array}[]{ll}{\mathcal{A}}^{*}=\{x\in S:\,\,\omega(x)\bigcap{\mathcal{A}}=\emptyset\}.\end{array}$$

$$\Phi ([0,T_x))x\subset N⟹T_x=+\infty .$$

$$\Phi ([0,t])x\subset N⟹\Phi ([0,t])x\subset E;$$

$$\gamma (t)\notin B(\text{mbox}resp.\gamma (t)\in \text{mbox}intB,\text{mbox}resp.\gamma (t)\notin B),\forall t\in (0,\epsilon );$$

$$\gamma (t)\in \text{mbox}intB(\text{mbox}resp.\gamma (t)\notin B,\text{mbox}resp.\gamma (t)\notin B),\forall t\in (-\delta ,t).$$

$$B^{-}=B^e\cup B^b,B^{+}=B^i\cup B^b.$$

$${\Phi }_{{\lambda }_n}(t_n)x_n\to {\Phi }_{\lambda }(t)x\text{mbox}asn\to \infty .$$

$$\Pi (t)(x,\lambda )=({\Phi }_{\lambda }(t)x,\lambda ),(x,\lambda )\in \mathcal{X}:=X\times \Lambda ,t\ge 0.$$

$$u_t+Au=f(u),u=u(t)\in U,$$

$$d(B,\partial U)>0.$$

$$||f(x)-f(y)||\le L||x-y|{|}_{\alpha },\forall x,y\in B.$$

$$\Phi (t)u_0=u(t;u_0),u_0\in U,t\in [0,T_{u_0}).$$

$$\text{mbox}ReZ\le a(resp.\ge a),$$

Full text

**Equilibrium Index of Invariant Sets and Global

Static Bifurcation for Nonlinear

Evolution Equations**

Desheng Li 111Supported by the grant of NSFC (10771159, 11071185)

Department of Mathematics, Tianjin University

Tianjin 300072, China

E-mail: lidsmath@tju.edu.cn

Zhi-Qiang Wang222Corresponding author, supported by the grant of NSFC (11271201)

Department of Mathematics and Statistics

Utah State University Logan, UT 84322

E-mail: zhi-qiang.wang@usu.edu

Abstract. We introduce the notion of equilibrium index for statically isolated invariant sets of the system $u_{t}+Au=f_{\lambda}(u)$ on Banach space $X$ (where $A$ is a sectorial operator with compact resolvent) and present a reduction theorem and an index formula for bifurcating invariant sets near equilibrium points. Then we prove a new global static bifurcation theorem where the crossing number $\mathfrak{m}$ may be even. In particular, in case $\mathfrak{m}=2$, we show that the system undergoes either an attractor/repeller bifurcation, or a global static bifurcation. An illustrating example is also given by considering the bifurcations of the periodic boundary value problem of second-order differential equations.

1 Introduction

In our previous work [17] we studied the dynamic bifurcation of the equation

[TABLE]

on a Banach space $X$, where $\lambda\in\mathbb{R}$ is the bifurcation parameter, $A$ is a sectorial operator on $X$ with compact resolvent, and $f_{\lambda}(u)$ is a continuous mapping from $X^{\alpha}\times\mathbb{R}$ to $X$ for some $0\leq\alpha<1$. Suppose $f_{\lambda}(0)\equiv 0$ for $\lambda\in\mathbb{R},$ and hence $u=0$ is always a trivial equilibrium solution of (1.1). It was shown that if the crossing number at $\lambda=\lambda_{0}$ is nonzero (the “odd-multiplicity condition” is not needed), then the system bifurcates from the equilibrium solution [math] an isolated compact invariant set $K_{\lambda}$ with nontrivial Conley index. Moreover, such a bifurcation enjoys some global features as in the classical Rabinowitz’s global static bifurcation theorem.

Now a natural problem arises: Does the bifurcated invariant set $K_{\lambda}$ contain equilibrium solutions ? For gradient-like systems, this question seems to be somewhat trivial, as any nonempty compact invariant set of such a system necessarily contains an equilibrium. Another particular but important case is the one of the attractor bifurcation, for which Ma and Wang established some index formulas via indices of isolated singular points (see e.g. [22]). When the crossing number is odd, by using Ma and Wang’s index formulas one can assure the existence of equilibrium solutions in $K_{\lambda}$. Here we are interested in the general case. Our strategy is as follows.

First, we introduce the notion of equilibrium index for statically isolated invariant sets for the non-parameterized equation

[TABLE]

on $X$, where $A$ is the same as in (1.1), and $f$ is a locally Lipschitz continuous mapping from $X^{\alpha}$ to $X$ ($0\leq\alpha<1$). Denote $\Phi$ the local semiflow generated by (1.2), and let $K$ be a statically isolated invariant set. We define the equilibrium index $\mbox{Ind}\,(\Phi,K)$ of an isolated invariant set $K$ to be the Leray-Schauder degree $\mbox{deg}\,(I-F,\Omega,0)$, where

[TABLE]

which maps $X^{\alpha}$ into itself, $a$ is a real number such that $\mbox{Re}\,\sigma(aI+A)>0$, and $\Omega$ is a statically isolating neighborhood of $K$ in $X^{\alpha}$. It can be shown that the index $\mbox{Ind}\,(\Phi,K)$ is independent of the choice of the number $a$ and the neighborhood $\Omega$, and hence is well defined. One of the advantages of introducing this notion is that, it allows us to obtain information on equilibrium solutions by directly performing some mathematical analysis on the evolution equation (1.2) without returning back to the corresponding stationary equation and putting it into a proper form so that one can apply the Leray-Schauder degree.

Secondly, we prove a reduction theorem for equilibrium index defined above by using the geometric theory of evolution equations, which allows us to compute the index $\mbox{Ind}\,(\Phi,K)$ of an invariant set $K$ near an equilibrium point $e$ by restring the system $\Phi$ on the local center manifold of $e$. More specifically, let $e$ be an equilibrium of $\Phi$. We show that there is a neighborhood $U$ of $e$ such that for any statically isolated invariant set $K$ of $\Phi$ in $U$, it holds that

[TABLE]

where $\mathfrak{m}_{1}$ is the dimension of the local unstable manifold of $e$, and $\Phi^{c}$ is the restriction of $\Phi$ on the local center manifold. Since invariant manifolds are actually of a pure dynamical nature, the above result indicates that there are inherent connections between static and dynamic objects, which fact was recognized as early as in the work of Chow and Hale [6].

Thirdly, based on the reduction theorem and a result on the relation between topological degree and Conley index for finite dimensional systems given in Rybakowski [34], we establish an index formula for the bifurcating invariant set of (1.1). Denote $\Phi_{\lambda}$ the local semiflow generated by (1.1), and let $\Phi^{c}_{\lambda}$ be the restriction of $\Phi_{\lambda}$ on the local center manifold of [math]. Suppose $\lambda=\lambda_{0}$ is a dynamic bifurcation value, and let $K_{\lambda}$ be the isolated invariant set bifurcated from the trivial equilibrium [math]. We prove that

[TABLE]

where $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$ denote, respectively, the dimensions of the local unstable manifold and the local center manifold of [math] at $\lambda=\lambda_{0}$ ($\mathfrak{m}_{2}$ is precisely the crossing number), and $\chi(h(\Phi,K))$ denotes the Euler number of the Conley index $h(\Phi,K)$. Due to the degeneracy of the equilibrium [math] at $\lambda=\lambda_{0}$, the computation of the index $\mbox{Ind}\,(\Phi_{\lambda_{0}},\{0\})$ via topological degree seems to be of little hope. However, in some cases computing the Conley indices $h(\Phi_{\lambda_{0}},\{0\})$ and $h(\Phi^{c}_{\lambda_{0}},\{0\})$ may be of practical sense. For instance, in the particular case of attractor bifurcation, [math] is an attractor of the $\Phi^{c}_{\lambda_{0}}$. Hence one easily deduces that

[TABLE]

where $\Sigma^{0}$ denotes the homotopy type of the pointed [math]-dimensional sphere (the space consisting of precisely two distinct points with one being the base point). Consequently $\chi\left(h(\Phi_{\lambda_{0}}^{c},\{0\})\right)=1$. Thus for $\lambda>\lambda_{0}$, by (1.3) we see that

[TABLE]

which, in our situation, recovers an index formula in the theory of attractor bifurcation given be Ma and Wang; see e.g. [21, Theorem 6.1]. Another important example is the one where $\mathfrak{m}_{2}=2$ and the equilibrium [math] fails to be an attractor of $\Phi_{\lambda_{0}}^{c}$; see Section 8 for details.

Fourthly, we are interested in the global static bifurcation of (1.1). A well-known result in this line is the famous Rabinowitz’s Global Bifurcation Theorem. However, it requires the crossing number to be odd (“crossing odd-multiplicity” condition). If one drops this condition then situations become very complicated. To the authors’ knowledge, even if for gradient systems the global static bifurcation still remains an open problem. To deal with this problem without assuming the “crossing odd-multiplicity” condition, Ma and Wang [20] proved some new local and global static bifurcation theorems by using higher-order nondegenerate singularities of nonlinearities. In this present work, motivated by the index formula (1.3) we give a new global static bifurcation theorem. Roughly speaking, we show that if $\lambda_{0}$ is a bifurcation value and

[TABLE]

then the equation (1.1) bifurcates from the trivial stationary solution $(0,\lambda_{0})$ a connected branch $\Gamma$ of stationary solutions. (We call $(e,\lambda)$ a stationary solution of (1.1) if $e$ is an equilibrium point of $\Phi_{\lambda}$.) $\Gamma$ enjoys some global features as in the classical Rabinowitz’s Global Bifurcation Theorem. It is worth noticing that if the crossing number $\mathfrak{m}_{2}$ is odd, then the condition (1.5) is automatically satisfied.

Finally, we pay some special attention to the case where $\mathfrak{m}_{2}=2$, i.e., there are exactly two eigenvalues (with multiplicity) crossing the imaginary axis. Let $(0,\lambda_{0})$ be a bifurcation point. Suppose $S_{0}=\{0\}$ is dynamically isolated with respect to the flow $\Phi_{\lambda_{0}}$. We prove that the system either undergoes an attractor/repeller bifurcation (a generalized Hopf bifurcation), or bifurcates from $(0,\lambda_{0})$ a connected global bifurcation branch of stationary solutions. As an illustrating example, we consider the bifurcation of the periodic boundary value problem of the second-order differential equation

[TABLE]

where $h(x,u)=0(|u|^{3})$ at $u=0$. Let $A$ be the differential operator $-\frac{d^{2}}{dx^{2}}$ associated with periodic boundary condition. Then

[TABLE]

The first eigenvalue $\lambda_{0}$ is simple with a corresponding constant eigenfunction, and all the others are of multiplicity $2$. For each $\lambda_{k}$ ($k\geq 1$), we show under appropriate conditions that either there is a two-sided neighborhood $I$ of $\lambda_{k}$ such that for each $\lambda\in I\setminus\{0\}$, the problem has at least two distinct nontrivial solutions, or it bifurcates from $(0,\lambda_{k})$ a global bifurcation branch enjoying the properties in the Rabinowitz’s Global Bifurcation Theorem.

This paper is organised as follows. In Section 2 we make some preliminaries, and in Section 3 we introduce the notion of equilibrium index and discuss its basic properties. In Section 4 we establish a reduction theorem for equilibrium index. In Section 5 we give an index formula for the bifurcating invariant sets of (1.1). Section 6 is devoted to the global static bifurcation of (1.1), in which we prove a global static bifurcation theorem without the “odd-multiplicity condition”. Section 7 consists of some discussions on the special case where the crossing number $\mathfrak{m}_{2}=2$. In Section 8 we give an illustrating example by considering the periodic boundary value problem of (1.6).

2 Preliminaries

This section is concerned with some preliminaries.

2.1 Basic topological notions and facts

Let $X$ be a metric space with metric $d(\cdot,\cdot)$. For convenience we will always identify a singleton $\{x\}$ with the point $x$ for any $x\in X$.

Let $A$ be a nonempty subset of $X$. The closure, interior and boundary of $A$ are denoted, respectively, by $\overline{A}$, int$\,A$ and $\partial A$. A subset $U$ of $X$ is called a neighborhood of $A$ if $\overline{A}\subset\mbox{int}\,U$. The $\varepsilon$-neighborhood of $A$ in $X$, noted by $\mbox{B}_{X}(A,\varepsilon)$, is defined to be the set $\{y\in X:\,d(y,A)<\varepsilon\}.$

Let $A$ and $B$ be two nonempty subsets of $X$. If $A\subset B$ then we will use the notations int${}_{B}\,A$ and $\partial_{B}A$ to denote the interior and boundary of $A$ in $B$, respectively. The verification of the following basic facts is straightforward.

Lemma 2.1

$\partial_{B}(A\cap B)\subset\partial A$. If $A\subset\mbox{\em int}B$ then we also have $\partial_{B}A=\partial A$.

The distance $d(A,B)$ between $A$ and $B$ is defined as

[TABLE]

The Hausdorff semi-distance and distance of $A$ and $B$ are defined as

[TABLE]

respectively. We also assign $d_{\tiny\mbox{H}}(\emptyset,B)=0$.

Let $A_{\lambda}$ ($\lambda\in\Lambda$) be a family of nonempty subsets of $X$, where $\Lambda$ is a metric space. We say that $A_{\lambda}$ is upper semicontinuous in $\lambda$ at $\lambda_{0}\in\Lambda$, this means

[TABLE]

The following two lemmas will play important roles in our discussion.

Lemma 2.2

[28]** Let $X$ be a compact metric space, and let $A$, $B$ be two disjoint closed subsets of $X$. Then either there is a subcontinuum $C$ of $X$ such that

[TABLE]

or $X=X_{A}\cup X_{B}$, where $X_{A}$ and $X_{B}$ are disjoint compact subsets of $X$ containing $A$ and $B$, respectively.

Lemma 2.3

[5]** (pp. 41) Let $X$ be a compact metric space. Denote by ${\mathscr{K}}(X)$ the family of compact subsets of $X$ which is equipped with the Hausdorff metric $\delta_{\mbox{\tiny H}}(\cdot,\cdot)$. Then ${\mathscr{K}}(X)$ is a compact metric space.

2.2 Some fundamental dynamical notions

In this subsection we collect some fundamental dynamical notions for the reader’s convenience.

Let $X$ be a metric space.

Definition 2.4

A local semiflow $\Phi=\Phi(t,x)$ on $X$ is a continuous mapping from an open set $\mathcal{D}(\Phi)\subset\mathbb{R}^{+}\times X$ to $X$ that enjoys the following properties:

1. (1)

for each $x\in X$, there exists $0<T_{x}\leq\infty$ such that

[TABLE] 2. (2)

$\Phi(0,\cdot)=id_{X}$, and $\Phi(t+s,x)=\Phi(t,\Phi(s,x))$ for all $x\in X$ and $t,s\in\mathbb{R}^{+}$ with $t+s\leq T_{x}$ .

Assume that there has been given a local semiflow $\Phi$ on $X$. As usual we will write $\Phi(t,x)=\Phi(t)x$.

Let $J$ be an interval. A trajectory (or solution) of $\Phi$ on $J$ is a continuous mapping $\gamma:J\rightarrow X$ such that

[TABLE]

A trajectory $\gamma$ on $J=\mathbb{R}$ is called a full trajectory.

A set $S\subset X$ is said to be positively invariant (resp. invariant), if $\Phi(t)S\subset S$ (resp. $\Phi(t)S=S$) for all $t\geq 0$.

A compact invariant set ${\mathcal{A}}\subset X$ is said to be an attractor of $\Phi$, if it attracts a neighborhood $U$ of itself, namely, $\lim_{t\rightarrow\infty}d_{H}\left(\Phi(t)U,{\mathcal{A}}\right)=0.$

The attraction basin of an attractor ${\mathcal{A}}$, denoted by ${\mathscr{U}}({\mathcal{A}})$, is defined as

[TABLE]

As in [13, Proposition 3.4], one can easily verify that ${\mathscr{U}}({\mathcal{A}})$ is open.

Let $S$ be a compact invariant set. Then the restriction $\Phi_{S}$ is a semiflow on $S$. A compact set ${\mathcal{A}}\subset S$ is said to be an attractor of $\Phi$ in $S$, this means that ${\mathcal{A}}$ is an attractor of $\Phi_{S}$ in $S$.

Given an attractor ${\mathcal{A}}$ of $\Phi$ in $S$, define

[TABLE]

${\mathcal{A}}^{*}$ is called the *repeller *of $\Phi$ in $S$ dual to ${\mathcal{A}}$, and $({\mathcal{A}},{\mathcal{A}}^{*})$ is called an attractor-repeller pair in $S$.

A subset $N$ of $X$ is said to be admissible [34], if for any sequences $x_{n}\in N$ and $t_{n}\rightarrow+\infty$ with $\Phi([0,t_{n}])x_{n}\subset N$, the sequence $\Phi(t_{n})x_{n}$ has a convergent subsequence. $N$ is said to be strongly admissible [34] if in addition, $\Phi$ does not explode in $N$, namely,

[TABLE]

Definition 2.5

$\Phi$* is said to be asymptotically compact on $X$, if each bounded set $B\subset X$ is strongly admissible.*

Remark 2.6

If $\Phi$ is asymptotically compact, then one easily verifies that each bounded invariant set of $\Phi$ is necessarily precompact.

2.3 Conley index

From now on we assume that $X$ is a complete metric space. Although we don’t require $X$ to be complete in the definition of a local semiflow, completeness of the phase space always plays a crucial role in establishing a dynamical systems theory.

Let $\Phi$ be a local semiflow on $X$. Since we are working in an infinite dimensional space, in the remaining part of this section, we also assume that

1. (AC)

$\Phi$ is asymptotically compact.

A compact invariant set $S$ (we allow $S=\emptyset$) of $\Phi$ is said to be isolated, if there exists a neighborhood $N$ of $S$ such that $S$ is the maximal invariant set in $\overline{N}$. Correspondingly, $N$ is called an isolating neighborhood of $S$.

Let there be given an isolated invariant set $S$. A pair of bounded closed subsets $(N,E)$ is called an index pair of $S$, if (1) $N\setminus E$ is an isolating neighborhood of $S$; (2) $E$ is $N$-invariant, i.e., for any $x\in E$ and $t\geq 0$,

[TABLE]

(3) $E$ is an exit set of $N$. Namely, for any $x\in N$, if $\Phi(t_{1})x\not\in N$ for some $t_{1}>0$, then there exists $0\leq t_{0}\leq t_{1}$ such that $\Phi(t_{0})x\in E.$

Let $B\subset X$ be a closed domain. A point $x\in\partial B$ is called a strict egress (resp. strict ingress, bounce-off) point of $B$, if for every trajectory $\gamma:[-\tau,s]\rightarrow X$ with $\gamma(0)=x$ (where $\tau\geq 0$, and $s>0$),

1. (1)

there exists $0<\varepsilon<s$ such that

[TABLE] 2. (2)

if $\tau>0$ then there exists $0<\delta<\tau$ such that

[TABLE]

Denote $B^{e}$ (resp. $B^{i}$, $B^{b}$) the set of all strict egress (resp. strict ingress, bounce-off) points of the closed set $B$, and set

[TABLE]

A bounded closed domain $B$ is called an isolating block [34], if $B^{-}$ is closed and $\partial B=B^{i}\cup B^{-}.$ For an isolating block $B$, we infer from [34] that $(B,B^{-})$ is an index pair of the maximal compact invariant set $S$ (possibly empty) in $B$.

Definition 2.7

(homotopy index) Let $(N,E)$ be an index pair of $S$. The homotopy Conley index of $S$ is defined to be the homotopy type $[(N/E,[E])]$ of the pointed space $(N/E,[E])$, denoted by $h(\Phi,S)$.

Remark 2.8

Denote $H_{*}$ and $H^{*}$ the singular homology and cohomology theories with coefficient group $\mathbb{Z}$, respectively. Applying $H_{*}$ and $H^{*}$ to $h(\Phi,S)$ one immediately obtains the homology and cohomology Conley index $CH_{*}(\Phi,S)$ and $CH^{*}(\Phi,S)$, respectively.

An important property of the Conley index is the homotopy invariance of the index. Here we state a result in this line for the reader’s convenience, which is actually a particular case of [34, Chap. I, Theorem 12.2].

Let $\Phi_{\lambda}$ ($\lambda\in\Lambda$) be a family of asymptotically compact local semiflows on $X$, where $\Lambda$ is a metric space. We say that $\Phi_{\lambda}$ depends on $\lambda$ continuously, if $\Phi_{\lambda}(t)x$ is defined at $(t,x,\lambda)$, then for any sequence $(t_{n},x_{n},\lambda_{n})$ converging to $(t,x,\lambda)$, $\Phi_{\lambda_{n}}(t_{n})x_{n}$ is also defined for all $n$ sufficiently large, furthermore,

[TABLE]

Suppose the family $\Phi_{\lambda}$ ($\lambda\in\Lambda$) depends on $\lambda$ continuously. Set

[TABLE]

Then $\Pi$ is a local semiflow on the product space ${\mathscr{X}}$. For convenience, we call $\Pi$ the skew-product flow of the family.

We say that $\Phi_{\lambda}$ is $\lambda$-locally uniformly asymptotically compact ($\lambda$-l.u.a.c. in short), if the skew-product flow $\Pi$ is asymptotically compact. It is easy to see that if $\Phi_{\lambda}$ is $\lambda$-l.u.a.c., then the parameter space $\Lambda$ is necessarily locally compact.

Suppose for each $\lambda\in\Lambda$, $\Phi_{\lambda}$ has an isolated invariant set $S_{\lambda}$. We call the pair $(\Phi_{\lambda},S_{\lambda})$ a dynamic continuation on $\Lambda$, if for every $\lambda_{0}\in\Lambda$, there is a neighborhood $W$ of $\lambda_{0}$ in $\Lambda$ and a set $N\subset X$ such that $N$ is an isolating neighborhood of $S_{\lambda}$ for all $\lambda\in W$. In the case where $\Phi_{\lambda}$ is $\lambda$-l.u.a.c., one can easily verify that a dynamic continuation $(\Phi_{\lambda},S_{\lambda})$ is ${\mathcal{S}}$-continuous in the terminology of [34, Chap. I, Def. 12.1]. Hence by [34, Chap. I, Theorem 12.2] we have

Theorem 2.9

Assume that $\Phi_{\lambda}$ is $\lambda$-l.u.a.c. Let $(\Phi_{\lambda},S_{\lambda})$ be a dynamic continuation on $\Lambda$. Then $h(\Phi_{\lambda},S_{\lambda})$ is constant for $\lambda$ in any component of $\Lambda$.

3 Equilibrium Index of Statically Isolated Sets

Let $X$ be a Banach space with norm $\|\cdot\|$, and $A$ be a sectorial operator on $X$ with compact resolvent. Denote $X^{\beta}$ ($\beta\in\mathbb{R}$) the fractional powers of $X$ induced by $A$ equipped with the usual norm $\|\cdot\|_{\alpha}$ (see [10, Chap. 1] for details). Consider the equation

[TABLE]

where $U$ is an open subset of $X^{\alpha}$ for some fixed $\alpha\in[0,1)$. Our main purpose in this section is to introduce the notion of equilibrium index for (3.1).

For convenience in statement, we will write $B\Subset U$, meaning that $B$ is a bounded subset of $U$ with

[TABLE]

We always assume $f:U\rightarrow X$ is locally Lipschitz continuous, namely,

1. (LC)

for any $B\Subset U$, there exists $L>0$ such that

[TABLE]

Under the above assumption, it is well known that the initial value problem of the equation is well-posed. That is, for each $u_{0}\in X^{\alpha}$ the problem has a unique solution $u(t)=u(t;u_{0})$ with $u(0)=u_{0}$ defined on a maximal existence interval $[0,T_{u_{0}})$ with $u(0)=u_{0}$; see e.g. [10, Chap. 3]. Define a local semiflow on $U$ as

[TABLE]

$\Phi$ is usually called the local semiflow generated by (3.1).

Remark 3.1

Note that in general the domain $U$ may not be complete. However, one easily understands that the dynamical systems theory reviewed in Section 2 applies to $\Phi$ quite well on any closed domain $N\Subset U$.

3.1 Definition of the equilibrium index

Given $K\subset U$, denote ${\mathscr{E}}_{\Phi}(K)$ the set of equilibrium points of $\Phi$ in $K$.

Definition 3.2

A set $S\Subset U$ is called statically isolated if it has a neighborhood $N\Subset U$ such that ${\mathscr{E}}_{\Phi}(\overline{N})={\mathscr{E}}_{\Phi}(S).$ Correspondingly $N$ is called a statically isolating neighborhood of $S$.

For convenience in statement, given $Z\subset\mathbb{C}$ and $a\in\mathbb{R}$, we write

[TABLE]

meaning that $\mbox{Re}\,z\leq a\,\,(resp.\,\,\geq a)$ for all $z\in Z.$

Take a number $a\in\mathbb{R}$ with $\mbox{Re}\,\sigma(aI+A)\geq\delta>0$. Let

[TABLE]

Then $F(B)$ is pre-compact in $X^{\alpha}$ for any $B\Subset U$.

Definition 3.3

Let $S\Subset U$ be statically isolated with $N\Subset U$ being an isolating neighborhood. The equilibrium index of $S$, denoted by $\mbox{\em Ind}\,(\Phi,S)$, is defined as

[TABLE]

where $\deg\,(\cdot,\cdot,\cdot)$ denotes the Leray-Schauder degree.

It is trivial to verify that the definition of $\mbox{Ind}\,(\Phi,S)$ is independent of the choice of $N$ and the number $a$.

A basic property of the equilibrium index is that it is invariant under transformations. Specifically, let $Y$ be another Banach space. Suppose there is an isomorphism $T:X\rightarrow Y$ (a bounded linear bijection with bounded inverse). Consider the following equation:

[TABLE]

where $B=TAT^{-1}$, and $g=TfT^{-1}.$ Denote $\Psi$ the local semiflow generated by (3.2). Then $\Psi$ and $\Phi$ are conjugate, namely, $\Psi T=T\Phi$. We have

Proposition 3.4

Let $S\Subset U$ be a statically isolated set of $\Phi$. Then $TS\Subset V$ is a statically isolated set of $\Psi$; furthermore,

[TABLE]

Proof. Take a number $a>0$ so that $\mbox{Re}\,\sigma(A_{a})\geq\delta>0$. Let $f_{a}=aI+f.$ Then (3.2) can be written as

[TABLE]

where

[TABLE]

We have

[TABLE]

Pick an (statically) isolating neighborhood $N$ of $S$. Then $M=TN\Subset V$ is an isolating neighborhood of $TS$. Hence by definition we have

[TABLE]

which completes the proof of the result. $\Box$

3.2 Continuation property

Let $X$, $U$ and $A$ be as above. Consider the equation with parameter $\lambda\in\Lambda$:

[TABLE]

where $\Lambda$ is a connected compact metric space. Instead of (LC) we assume that

1. (UL)

for any $B\Subset U$ and compact subset $\Lambda_{0}$ of $\Lambda$, there exists $L>0$ such that

[TABLE]

Denote $\Phi_{\lambda}$ the local semiflow generated by (3.6) on $U$. It is easy to verify that $\Phi_{\lambda}$ depends on $\lambda$ continuously.

Let $\Pi$ be skew-product flow of the family $\Phi_{\lambda}$ ($\lambda\in\Lambda$) on ${\mathcal{U}}=U\times\Lambda$. Clearly $x$ is an equilibrium of $\Phi_{\lambda}$ iff  $(x,\lambda)$ is an equilibrium of $\Pi$.

The following result is a simple consequence of the continuation property of the Leray-Schauder degree.

Theorem 3.5

Let ${\mathcal{S}}$ be a statically isolated set of the skew-product flow $\Pi$ in ${\mathcal{U}}$. Set $S_{\lambda}:={\mathcal{S}}[\lambda]$. Then $\mbox{\em Ind}\,(\Phi_{\lambda},S_{\lambda})$ remains constant for $\lambda\in\Lambda$.

Proof. Since ${\mathcal{S}}$ is a statically isolated set of $\Pi$, by definition ${\mathcal{S}}$ has a bounded closed isolating neighborhood ${\mathcal{N}}\Subset{\mathcal{U}}$. Note that $d({\mathcal{N}},\partial_{\mathscr{X}}{\mathcal{U}}):=d>0$, where $\partial_{\mathscr{X}}{\mathcal{U}}$ denotes the boundary of ${\mathcal{U}}$ in ${\mathscr{X}}:=X^{\alpha}\times\Lambda$. It is trivial to check that $d({\mathcal{N}},\partial_{\mathscr{X}}{\mathcal{U}})\leq d({\mathcal{N}}[\lambda],\partial U)$. Therefore we deduce that

[TABLE]

Hence we can pick a bounded closed set $B\Subset U$ such that ${\mathcal{N}}[\lambda]\subset B$ for all $\lambda\in\Lambda$.

Using local Lipschitz continuity of $f_{\lambda}$ one easily verifies that \begin{array}[]{ll}\bigcup_{\lambda\in\Lambda}f_{\lambda}\left(B\right)\end{array} is bounded in $X$. Consequently $M:=\bigcup_{\lambda\in\Lambda}(aI+f_{\lambda})(B)$ is a bounded set in $X$. Let

[TABLE]

Then the set \begin{array}[]{ll}\bigcup_{\lambda\in\Lambda}F_{\lambda}(B)=(aI+A)^{-1}M\end{array} is bounded in $D(A)=X^{1}$. As $A$ has compact resolvent, the embedding $X^{1}\hookrightarrow X^{\alpha}$ is compact. Therefore $\bigcup_{\lambda\in\Lambda}F_{\lambda}(B)$ is precompact in $X^{\alpha}$. Now by virtue of continuation property of the Leray-Schauder degree we immediately concludes that

[TABLE]

on $\Lambda$. The proof of the theorem is complete. $\Box$

Suppose $\Phi_{\lambda}$ has a statically isolated invariant set $S_{\lambda}$ for each $\lambda\in\Lambda$ (we allow $S_{\lambda}=\emptyset$). We say that $(\Phi_{\lambda},S_{\lambda})$ is a static continuation on $\Lambda$, if for each $\lambda_{0}\in\Lambda$, there exist $N\Subset U$ and a neighborhood $W$ of $\lambda_{0}$ in $\Lambda$ such that $N$ is a statically isolating neighborhood of $S_{\lambda}$ for all $\lambda\in W$.

Theorem 3.6

Let $(\Phi_{\lambda},S_{\lambda})$ be a static continuation on $\Lambda$. Then $\mbox{\em Ind}\,(\Phi_{\lambda},S_{\lambda})$ remains constant on $\Lambda$. In particular, if $\Lambda=[0,1]$ then $\mbox{\em Ind}\,(\Phi_{0},S_{0})=\mbox{\em Ind}\,(\Phi_{1},S_{1}).$

Proof. To prove Theorem 3.6, it suffices to check that for any $\lambda_{0}\in\Lambda$, there is a neighborhood ${\mathcal{O}}$ of $\lambda_{0}$ in $\Lambda$ such that

[TABLE]

Let $\lambda_{0}\in\Lambda$. Then by the definition of a static continuation, one can find a neighborhood $N\Subset U$ of $S_{\lambda_{0}}$ and a neighborhood $W$ of $\lambda_{0}$ in $\Lambda$ such that $N$ is a statically isolating neighborhood of $S_{\lambda}$ for all $\lambda\in W$. Pick a connected compact neighborhood ${\mathcal{O}}$ of $\lambda_{0}$ with ${\mathcal{O}}\subset W$. One easily sees that ${\mathcal{S}}:=\bigcup_{\lambda\in{\mathcal{O}}}S_{\lambda}\times\{\lambda\}$ is a static isolated set of the skew-product flow of $\Phi_{\lambda}$ ($\lambda\in{\mathcal{O}}$). Applying Theorem 3.5 we immediately conclude the validity of (3.5). $\Box$

3.3 The finite dimensional case

Now we assume $X$ is an $m$-dimensional Banach space. Consider the ODE system:

[TABLE]

where $f:U\rightarrow X$ is locally Lipschitz. Denote $\Phi$ the semiflow generated by (3.6).

Proposition 3.7

Let $S$ be a statically isolated set of $\Phi$. Then

[TABLE]

for any isolating neighborhood $N$ of $S$.

If $S$ is an isolated invariant set of $\Phi$, we also have

[TABLE]

where $\chi(h(\Phi,S))=\Sigma_{q=0}^{\infty}\,(-1)^{q}\beta_{q}(h(\Phi,S))$ is the Euler number of the Conley index $h(\Phi,S)$.

Proof. We may assume $X=\mathbb{R}^{m}$. Take an arbitrary $m\times m$-matrix $A$ and rewrite (3.6) as

[TABLE]

where $\widetilde{f}(x)=Ax+f(x)$. Pick a number $a>0$ so that $\mbox{Re}\,\sigma(aI+A)>0$. Then

[TABLE]

Note that $\mbox{Re}\,\sigma(aI+A)>0$ implies $\mbox{det}\,(aI+A)>0$. Further by the definition of the Brouwer degree it is easy to deduce that

[TABLE]

Hence

[TABLE]

Now assume that $S$ is an isolated invariant set of $\Phi$. Pick an (dynamically) isolating neighborhood $N$ of $S$. Then we infer from Reineck [33] (see also [32, Theorem 3.2]) that $S$ can be continued in $N$ to an isolated invariant set $K$ of a flow $G$ generated by

[TABLE]

where $g$ is a Morse-Smale gradient vector field on a neighborhood $V$ of $K$ with $V\subset N$, and $g(x)=f(x)$ on $\mathbb{R}^{m}\setminus N$. Thus by Rybakowski [34, Chap. III,Theorem 3.8] (see also Chang [3, Chap. II, Theorems 3.1-3.3]) one concludes that

[TABLE]

The proof is complete. $\Box$

Remark 3.8

In [32] the Euler number $\chi(h(\Phi,S))$ is defined to be the Poincaré index of $S$ (see [32, Section 3]).

4 Invariant Manifolds of Nonlinear Equations

For completeness and the reader’s convenience, in this section we briefly recall some results on local invariant manifolds of the following nonlinear evolution equation with parameter $\lambda\in\mathbb{R}$:

[TABLE]

where $A$ is a sectorial operator in $X$ with compact resolvent, and $f_{\lambda}(x)$ is a continuous mapping from $X^{\alpha}\times\mathbb{R}$ to $X$ for some $\alpha\in[0,1)$.

Without loss of generality, assume that $f_{\lambda}(0)\equiv 0$ for $\lambda\in\mathbb{R},$ hence $u=0$ is always an equilibrium solution of (4.1).

Assume that $f_{\lambda}(x)$ is locally Lipschitz in $x$; furthermore, the local Lipschitz continuity assumption (UL) in Subsection 3.2 is fulfilled. Denote $\Phi_{\lambda}$ the local semiflow generated by (4.1). Then we know that $\Phi_{\lambda}$ depends on $\lambda$ continuously. Also, by very standard argument (see e.g.  [34, Chap. I, Theorem 4.4]), it can be shown that $\Phi_{\lambda}$ is $\lambda$-l.u.a.c. on $X^{\alpha}$.

4.1 Fundamental assumptions and notations

Let $L_{\lambda}=A-f_{\lambda}^{\prime}(0)$, and write

[TABLE]

Then (4.1) reads

[TABLE]

Suppose there exists $\nu>0$ such that the following hypotheses (H1)-(H3) are fulfilled for every $\lambda\in J_{0}=[\lambda_{0}-\nu,\,\lambda_{0}+\nu]$:

1. (H1)

The spectral $\sigma(L_{\lambda})$ has a decomposition $\sigma(L_{\lambda})=\sigma_{\lambda}^{1}\cup\sigma_{\lambda}^{2}\cup\sigma_{\lambda}^{3}$ such that

[TABLE]

for some $\delta>0$ (independent of $\lambda\in J_{0}$). 2. (H2)

The space $X$ has a decomposition $X=X^{1}_{\lambda}\oplus X^{2}_{\lambda}\oplus X^{3}_{\lambda}\,,$ corresponding to the decomposition of $\sigma(L_{\lambda})$ in (H1), such that $X^{i}_{\lambda}$ $(i=1,2,3)$ are $L_{\lambda}$-invariant subspaces of $X$. Moreover,

[TABLE] 3. (H3)

The projection operators

[TABLE]

are continuous in $\lambda$.

Remark 4.1

The above assumptions implies that there is a family $T_{\lambda}$ ($\lambda\in J_{0}$) of isomorphisms on $X$ depending continuously on $\lambda$ with $T_{\lambda_{0}}=I$, such that

[TABLE]

see Appendix A for the proof. It is trivial to verify that $T_{\lambda}P_{\lambda}^{i}=P_{\lambda_{0}}^{i}$ for all $\lambda\in J_{0}.$

Let

[TABLE]

and denote $P_{\lambda}^{ij}$ the projection from $X$ to $X_{\lambda}^{ij}$. Then $P_{\lambda}^{ij}=P_{\lambda}^{i}+P_{\lambda}^{j}$. We infer from Remark 4.1 that

[TABLE]

We will rewrite $X^{\alpha}=E$. Let

[TABLE]

Then

[TABLE]

Because $X^{1}_{\lambda}$ and $X^{2}_{\lambda}$ are finite dimensional, we actually have

[TABLE]

For notational simplicity, hereafter we write

[TABLE]

[TABLE]

Denote $\mbox{B}^{i}_{r}$ the ball in $E^{i}$ centered at [math] with radius $r$, and set

[TABLE]

Lemma 4.2

For any neighborhood ${\mathcal{U}}$ of [math] in $E$, there exists $r>0$ such that

[TABLE]

Proof. Since $P^{i}\,{\mathcal{U}}$ is a neighborhood of [math] in $E^{i}$, we can pick an $\varepsilon>0$ sufficiently small so that $\overline{}\mbox{B}^{i}_{\varepsilon}\subset P^{i}{\mathcal{U}}$ for $i=1,2$. Further by using compactness of $\overline{}\mbox{B}^{i}_{\varepsilon}$ ($i=1,2$) one can easily verify that there exists $\varepsilon^{\prime}>0$ such that $\overline{}\mbox{B}^{1}_{\varepsilon}\oplus\overline{}\mbox{B}^{2}_{\varepsilon}\oplus\overline{}\mbox{B}^{3}_{\varepsilon^{\prime}}\subset{\mathcal{U}}$, from which (4.7) immediately follows. $\Box$

4.2 Local invariant manifolds

Let us recall briefly some fundamental results on local invariant manifolds.

Let $T_{\lambda}$ be the isomorphism on $X$ given in Remark 4.1. Setting $v=T_{\lambda}u$, system (4.2) is transformed into an equivalent one:

[TABLE]

where

[TABLE]

It is easy to verify (or see [17, Section 3]) that $\sigma(B_{\lambda})=\sigma(L_{\lambda}).$

Denote $\Psi_{\lambda}=\Psi_{\lambda}(t)$ the local semiflow generated by (4.8).

Since $X^{i}_{\lambda}$ ($i=1,2,3$) are $L_{\lambda}$-invariant, by (4.5) we find that $X^{i}:=X^{i}_{\lambda_{0}}$ ($i=1,2,3$) are $B_{\lambda}$-invariant for all $\lambda\in J_{0}$. Thus by (4.3) we deduce that

[TABLE]

where $B^{i}_{\lambda}=B_{\lambda}|_{X^{i}}$ is the restriction of $B_{\lambda}$ on $X^{i}$. Now the same arguments as in the proofs of Lemmas 3.3 and 3.4 in [17] apply to prove the following theorem.

Theorem 4.3

There exist open convex neighborhoods $V_{i}$ of [math] (independent of $\lambda\in J_{0}$) in $E^{i}$  $(i=1,2,3)$ such that the following assertions hold.

1. $(1)$

There is a continuous mapping $\psi_{\lambda}(x)$ from $(V_{1}\oplus V_{2})\times J_{0}$ to $V_{3}$ which is differentiable in $x$ with $\psi_{\lambda}^{\prime}(x)$ being continuous in $(x,\lambda)$ and $\psi_{\lambda}^{\prime}(0)=0$ such that for each $\lambda\in J_{0}$, the set $M^{12}_{\lambda}=\{x+\psi_{\lambda}(x):\,\,x\in V_{1}\oplus V_{2}\}$ is a local invariant manifold of system (4.8). 2. $(2)$

There is a continuous mapping $\rho_{\lambda}(x_{2})$ from $V_{2}\times J_{0}$ to $V_{1}$ which is differentiable in $x_{2}$ with $\rho_{\lambda}^{\prime}(x_{2})$ being continuous in $(x_{2},\lambda)$ and $\rho_{\lambda}^{\prime}(0)=0$ such that for each $\lambda\in J_{0}$, the set $M^{2}_{\lambda}=\{x_{2}+\zeta_{\lambda}(x_{2}):\,\,x_{2}\in V_{2}\}$ is a local invariant manifold of (4.8), where

[TABLE] 3. $(3)$

$S\subset V:=V_{1}\oplus V_{2}\oplus V_{3}$* is an isolated invariant set of the semiflow $\Psi_{\lambda}$ of (4.8) iff * $S\subset M^{12}_{\lambda}$ ( resp. $M_{\lambda}^{2}$ ) and is an isolated invariant set of $\Psi_{\lambda}^{12}=\Psi_{\lambda}|_{M_{\lambda}^{12}}$ ( resp. $\Psi_{\lambda}^{2}=\Psi_{\lambda}|_{M_{\lambda}^{2}}$ ). Furthermore,

[TABLE]

Remark 4.4

Assertions (1) and (2) are only slight modifications of some classical results in the geometric theory of evolution equations (see e.g. [34, Chap. II, Theorem 2.1]). A sketch of the proof can be found in [17]. (3) is a parameterized version of a corresponding result in [34, Chap. II, Theorem 3.1].

Remark 4.5

Using $\psi_{\lambda}^{\prime}(0)=0$ and $\rho_{\lambda}^{\prime}(0)=0$ it is easy to deduce that for any $\varepsilon>0$, there exists $r>0$ such that

[TABLE]

for all $x\in$B${}^{1}_{r}\oplus\mbox{B}^{2}_{r}$, $x_{2}\in\mbox{B}^{2}_{r}$ and $\lambda\in J_{0}$. It follows that

[TABLE]

uniformly with respect to $\lambda\in J_{0}$.

Remark 4.6

It is known (see e.g. [34, Section 2.2]) that $\psi_{\lambda}$ satisfies equation

[TABLE]

for $x\in V_{1}\oplus V_{2}$, and $\rho_{\lambda}$ satisfies

[TABLE]

for $x_{2}\in V_{2}$. Here (and below) $P^{i}=P_{\lambda_{0}}^{i},$ $P^{ij}=P_{\lambda_{0}}^{ij},$ and

[TABLE]

Let

[TABLE]

Define $\phi_{\lambda}:U_{\lambda}^{12}\rightarrow E^{3}_{\lambda}$ and $\xi_{\lambda}:U_{\lambda}^{2}\rightarrow E^{13}_{\lambda}$ as below:

[TABLE]

[TABLE]

As a direct consequence of Theorem 4.3, we have the following.

Theorem 4.7

Assume the hypotheses (H1)-(H3). Then

1. $(1)$

the following two sets are local invariant manifolds of system (4.1):

[TABLE]

[TABLE] 2. $(2)$

$S\subset U_{\lambda}=T_{\lambda}^{-1}V$* ( where $V=V_{1}\oplus V_{2}\oplus V_{3}$ ) is an isolated invariant set of the semiflow $\Phi_{\lambda}$ of (4.1) iff * $S\subset{\mathcal{M}}^{12}_{\lambda}$ ( resp. $S\subset{\mathcal{M}}^{2}_{\lambda}$  ) and is an isolated invariant set of $\Phi_{\lambda}^{12}=\Phi_{\lambda}|_{{\mathcal{M}}_{\lambda}^{12}}$ ( resp. $\Phi_{\lambda}^{2}=\Phi_{\lambda}|_{{\mathcal{M}}_{\lambda}^{2}}$ ). Furthermore,

[TABLE]

Let $u(t)=x(t)+x_{3}(t)$ be a solution of (4.2) (or (4.1)) lying in ${\mathcal{M}}_{\lambda}^{12}$, where

[TABLE]

Then $x_{3}=\phi_{\lambda}(x)$, and therefore $x$ satisfies

[TABLE]

Similarly one can also obtain the equation corresponding to ${\mathcal{M}}_{\lambda}^{2}$:

[TABLE]

(4.19) and (4.20) will be referred to as the reduction equations of (4.2) (or (4.1)) on ${\mathcal{M}}_{\lambda}^{12}$ and ${\mathcal{M}}_{\lambda}^{2}$, respectively.

Denote ${\mathcal{R}}_{\lambda}^{12}$ and ${\mathcal{R}}_{\lambda}^{2}$ the local semiflow on $U_{\lambda}^{12}$ and $U_{\lambda}^{2}$ generated by (4.19) and (4.20), respectively. Then ${\mathcal{R}}_{\lambda}^{12}$ and ${\mathcal{R}}_{\lambda}^{2}$ conjugate with $\Phi_{\lambda}^{12}$ and $\Phi_{\lambda}^{2}$, respectively. More precisely, we have

[TABLE]

[TABLE]

Remark 4.8

By (4.22), (4.22) and Theorem 4.7 it is easy to see that $S\subset U_{\lambda}$ is an (isolated) invariant set of $\Phi_{\lambda}$ iff $S_{12}=P_{\lambda}^{12}S$ (resp. $S_{2}=P_{\lambda}^{2}S$) is an (isolated) invariant set of ${\mathcal{R}}_{\lambda}^{12}$ (resp. ${\mathcal{R}}_{\lambda}^{2}$).

5 Reduction Theorem of Equilibrium Index near Equilibrium Points

Our main purpose in this section is to establish a reduction theorem for equilibrium indices of isolated invariant sets of system (4.1) near equilibrium points.

We follow the same notations as in Section 4 and assume the function $f_{\lambda}(x)$ in (4.1) satisfies all the regularity hypotheses in Section 4. Suppose also that $f_{\lambda}(0)\equiv 0$ for $\lambda\in\mathbb{R}$. The main result is the following theorem.

Theorem 5.1

Assume the hypotheses (H1)-(H3) in Section 4 are fulfilled. Then there exists an open neighborhood $U$ of [math] in $E$ (independent of $\lambda\in J_{0}$) such that for every isolated invariant set $S$ of $\Phi_{\lambda}$ $(\lambda\in J_{0})$ in $U$, we have

[TABLE]

where $\mathfrak{m}_{i}=\dim\,(X^{i})$ $(i=1,2)$.

Proof. Some basic ideas and techniques used here are borrowed from [34] (see [34, Chap. II, Theorem 3.1] and its proof).

For notational simplicity, in the following we will drop the subscript “$\lambda$” and simply rewrite $B_{\lambda}$, $h_{\lambda}$, $\psi_{\lambda}$, $\rho_{\lambda}$, $\zeta_{\lambda}$ and $\Psi_{\lambda}$ as $B$, $h$, $\cdots$, respectively, unless we need to emphasize the dependence on $\lambda$. We split the argument into several steps. *Step *1. We begin with system (4.8). Let $V_{i}$ ($i=1,2,3$) be the neighborhood of [math] in $E^{i}$ given in Theorem 4.3, and let

[TABLE]

For each fixed $\lambda\in J_{0}$, define a family of mappings $Q_{\theta}:V\rightarrow E$ ($\theta\in[0,1]$) as

[TABLE]

Note that if $u=x+x_{3}\in M^{12}=M^{12}_{\lambda}$ then

[TABLE]

Using the same argument as in Step 1 in the proof of [34, Chap. II, Theorem 3.1], one can easily verify that $Q_{\theta}(V):=\widetilde{V}_{\theta}$ is open; furthermore, $Q_{\theta}$ has a continuous inverse $Q_{\theta}^{-1}:\widetilde{V}_{\theta}\rightarrow V$ given by

[TABLE]

where $x=P^{12}v\in\Omega$, and $y_{3}=P^{3}v\in E^{3}$. (Here and below $P^{i}$ and $P^{ij}$ denote $P^{i}_{\lambda_{0}}$ and $P^{ij}_{\lambda_{0}}$, respectively, except otherwise statement.)

Note that $Q_{\theta}$ depends upon $\theta$ and $\lambda$ continuously. Consequently $\widetilde{V}_{\theta}=Q_{\theta}(V)$ is continuous in ($\theta$, $\lambda$) in the sense of Hausdorff distance. Using this simple fact one easily checks that there is a neighborhood $\widetilde{\mathcal{O}}$ of [math] in $E$ such that

[TABLE]

By Lemma 4.2 we can pick a number $r>0$ such that

[TABLE]

($\mbox{B}^{i}_{r}$ denotes the ball in $E^{i}$). In view of (4.13), it can be assumed that $r$ is chosen sufficiently small so that for all $\lambda\in J_{0}$, we have

[TABLE]

Define a local semiflow $\widetilde{\Psi}_{\theta}$ on $\widetilde{V}_{\theta}$ to be the “ image” of the local semiflow $\Psi=\Psi_{\lambda}$ of (4.8) under $Q_{\theta}$, namely,

[TABLE]

Then by (5.4) $\widetilde{\Psi}_{\theta}$ is well defined on the domain $\Xi(r)$ for all $\theta\in[0,1]$ and $\lambda\in J_{0}$. Making us of the relation in (4.14), it can be shown by some simple computations that $\widetilde{\Psi}_{\theta}$ is precisely the local semiflow generated by system

[TABLE]

where $x=P^{12}v$, $y_{3}=P^{3}v$, $N_{\theta}=P^{12}hQ_{\theta}^{-1}$, and

[TABLE]

Note that (5.5) reduces to system (4.8) when $\theta=0$. Hence $\widetilde{\Psi}_{0}$ coincides with $\Psi$.

Take a number $\varepsilon>0$ (independent of $\lambda$ and $\theta$) such that $\Xi(\varepsilon)\subset V$ and

[TABLE]

Let $S\subset{\Xi(\varepsilon/2)}$ be an isolated invariant set of $\Psi$, and write

[TABLE]

Then (5.6) implies that $S_{\theta}$ is an isolated invariant set of $\widetilde{\Psi}_{\theta}$ in $\Xi(r)$ for all $\theta\in[0,1]$ (and $\lambda\in J_{0}$). It is trivial to verify that $(\widetilde{\Psi}_{\theta},S_{\theta})$ is a static coninuation. Thus by Theorem 3.6 we deduce that

[TABLE]

Step 2. For $\theta=1$, system (5.5) reads

[TABLE]

where

[TABLE]

Since $Q_{1}^{-1}(x+y_{3})=x+(y_{3}+\psi(x))$, it is clear that

[TABLE]

Consider on the domain $\Xi(r)$ the homotopy of system (5.8):

[TABLE]

Lemma 5.2

There exist a small open neighborhood $\widetilde{U}=\widetilde{U}_{1}\oplus\widetilde{U}_{2}\oplus\widetilde{U}_{3}$ of [math] in $E$ with $\widetilde{U}\subset\Xi(r)$ and constants $C,\mu>0$ such that for all $\theta\in[0,1]$ and $\lambda\in J_{0}$, if $u(t)=x(t)+y_{3}(t)$ is a solution of (5.10) on $[t_{0},t_{1}]$ lying in $\widetilde{U}$ then

[TABLE]

Proof. The proof is a slight modification of that of [34, Chap. II, Lemma 3.3]. We omit the details. $\Box$ Denote $G_{\theta}$ the local semiflow on $\widetilde{U}$ generated by (5.10). Then (5.9) implies that $W:=\widetilde{U}_{1}\oplus\widetilde{U}_{2}$ is a local invariant manifold of $G_{\theta}$ for all $\theta\in[0,1]$. Further by (5.9) and (5.10) we deduce that $G_{\theta}|_{W}$ is actually independent of $\theta$.

On the other hand, if $K\subset\widetilde{U}$ is an invariant set of $G_{\theta}$, then by Lemma 5.2 we find that $K\subset W$. Using this fact and the independence of $G_{\theta}|_{W}$ upon $\theta$ it is easy to see that all the local semiflows $G_{\theta}$ ($\theta\in[0,1]$) share the same invariant sets in $\widetilde{U}$. Consequently $K\subset\widetilde{U}$ is an isolated invariant set of $G_{\theta_{0}}$ for some $\theta_{0}\in[0,1]$ iff it is an isolated invariant set of $G_{\theta}$ for all $\theta\in[0,1]$.

Now we pick a positive number $\eta<\varepsilon/2$ such that

[TABLE]

Let $S\subset\Xi(\eta)$ be an isolated invariant set of $\Psi$. Then

[TABLE]

is an isolated invariant set of $\widetilde{\Psi}_{1}$ in $\widetilde{U}$. (The second equality in (5.12) follows from (5.3) and the fact that $S\subset M^{12}_{\lambda}$.) Since $\widetilde{\Psi}_{1}$ coincides with $G_{1}$, by what we have proved above we see that $S_{1}$ is an isolated invariant set of $G_{\theta}$ for all $\theta\in[0,1]$. Further one can easily verify that $(G_{\theta},S_{\theta})$ is a static coninuation. Thus by Theorem 3.6 and (5.7) we obtain that

[TABLE]

where $G_{0}$ is the local semiflow on $\widetilde{U}$ generated by the system

[TABLE]

Step 3. We now calculate the index $\mbox{Ind}(G_{0},S_{1})$. For this purpose, consider the following homotopy of system (5.14):

[TABLE]

where $B^{3}_{\tau}=\tau B^{3}+(1-\tau)I$. Since $\mbox{Re}\,\sigma(B^{3})\geq 2\delta>0$ (see (4.10)), we have

[TABLE]

Denote $\Pi_{\tau}$ the local semiflow generated by (5.15). Then $\Pi_{1}=G_{0}$. Hence $S_{1}$ is an isolated invariant set for $\Pi_{1}$. Further repeating the same argument as above with minor modifications it is easy to deduce that $S_{1}$ is an isolated invariant set of $\Pi_{\tau}$ for all $\tau\in[0,1]$. Now as in the preceding steps, by Theorem 3.6 we get

[TABLE]

Step 4. For $\tau=0$, (5.15) reads

[TABLE]

An equivalent form of the system reads

[TABLE]

where $w=x+y_{3}$. Take a number $a>0$ sufficiently large so that $aI+B:=B_{a}$ has bounded inverse $B_{a}^{-1}$. Rewrite (5.18) as

[TABLE]

Since $L_{\lambda}$ commutes with $P_{\lambda}^{ij}$, by (4.6) it is trivial to verify that $B=B_{\lambda}:=T_{\lambda}L_{\lambda}T_{\lambda}^{-1}$ commutes with $P^{ij}=P_{\lambda_{0}}^{ij}$ for all $\lambda\in J_{0}$. Hence $P^{12}B_{a}=B_{a}P^{12}$. Using this fact one can easily show that $\widetilde{B}_{a}:=P^{12}B_{a}+P^{3}$ is invertible with

[TABLE]

Define an operator ${\mathcal{K}}$ on $\widetilde{U}=(\widetilde{U}_{1}\oplus\widetilde{U}_{2})\oplus\widetilde{U}_{3}:=\widetilde{\Omega}\oplus\widetilde{U}_{3}$ as

[TABLE]

Then

[TABLE]

Hence ${\mathcal{K}}:\widetilde{U}\rightarrow E$ is a finite dimensional operator. Therefore by the definition of the equilibrium index and the reduction property of the Leray-Schauder degree, we deduce that

[TABLE]

where $N\subset\widetilde{U}$ is a closed isolating neighborhood of $S_{1}$, and $\widetilde{N}=N\cap\widetilde{\Omega}$. Thus by (5.13) and (5.16) we have

[TABLE]

Denote $\Psi^{12}$ the local semiflow generated by the first equation in (5.17):

[TABLE]

($\Psi^{12}$ is well-defined on $\Omega=V_{1}\oplus V_{2}$.) As in Remark 4.8 we know that $S_{1}=P^{12}S$ is an isolated invariant set of $\Psi^{12}$ with $\widetilde{N}$ being an isolating neighborhood. Hence we have

[TABLE]

where

[TABLE]

On the other hand, it is trivial to verify that $(aI+B^{12})^{-1}=P^{12}B_{a}^{-1}|_{E^{12}}.$ Therefore

[TABLE]

Combing this with (5.22) and (5.20) one concludes that

[TABLE]

Step 5. Now we pay some attention to the calculation of $\mbox{Ind}(\Psi^{12},S_{1})$. At this point, we are in a quite similar situation as we were at the beginning of calculating the index $\mbox{Ind}(\Psi,S)$ with system (4.8) replaced by (5.21).

Write $x=x_{1}+x_{2}$, where $x_{i}=P^{i}x$. Then (5.21) can be reformulated as

[TABLE]

Repeating a similar argument as in Steps 1 and 2 leading to (5.13) with some corresponding modifications, it can be shown that there exists $\beta>0$ (independent of $\lambda\in J_{0}$) such that for any isolated invariant set $K$ of (5.21) with $K\subset\mbox{B}^{1}_{\beta}\oplus\mbox{B}^{2}_{\beta}$,

[TABLE]

where $\pi$ is the local semiflow generated by

[TABLE]

where $\zeta(x_{2})=\zeta_{\lambda}(x_{2})$ is the *invariant manifold * mapping given in (4.11).

Let

[TABLE]

Pick a closed isolating neighborhood $N$ of $K$ with $N\subset\mbox{B}^{1}_{\beta}\oplus\mbox{B}^{2}_{\beta}$. Recalling that $\mbox{Re}\,\sigma(B^{1})<0$, a simple calculation yields

[TABLE]

where $\Psi^{2}$ is the local semiflow on $V_{2}$ generated by the system

[TABLE]

Step 6. We first take a positive number $\beta^{\prime}<\beta$ such that $\Xi(\beta^{\prime})\subset\widetilde{U}$, where $\widetilde{U}$ is the neighborhood of [math] given in Lemma 5.2. Then choose a positive number $\eta^{\prime}<\eta$ ($\eta$ is the number in (5.11)) small enough so that

[TABLE]

where $Q_{\theta}$ is the mapping defined in (5.2).

Let $S\subset\Xi(\eta^{\prime})$ be an isolated invariant set of $\Psi$ in $V$. Clearly

[TABLE]

Since $S_{1}=P^{12}S\subset E^{12}$ (see (5.12)), the inclusion $S_{1}\subset\Xi(\beta^{\prime})$ implies that

[TABLE]

Thus by (5.23) and (5.27) we obtain that

[TABLE]

Step 7. Let $U_{\lambda}=T_{\lambda}^{-1}V$. By continuity of $T_{\lambda}^{-1}$ in $\lambda$ one can easily verify that there exists a neighborhood ${\mathcal{O}}$ of [math] in $E$ such that ${\mathcal{O}}\subset U_{\lambda}$ for all $\lambda\in J_{0}$. Take a number $\gamma>0$ sufficiently small such that $U:=\Xi(\gamma)\subset{\mathcal{O}}$; furthermore,

[TABLE]

The reduction equation (4.20) on ${\mathcal{M}}_{\lambda}^{2}$ reads

[TABLE]

where $L^{2}=L_{\lambda}^{2}$, $g=g_{\lambda}$, and $\xi=\xi_{\lambda}=T^{-1}_{\lambda}\zeta_{\lambda}T_{\lambda}$. Write

[TABLE]

Noticing that $T_{\lambda}P^{2}_{\lambda}=P^{2}T_{\lambda}$ (recall that $P^{2}=P^{2}_{\lambda_{0}}$), we find that

[TABLE]

where $\widetilde{h}(y_{2})=h\left(y_{2}+\zeta(y_{2})\right)$, and $h=h_{\lambda}=T_{\lambda}gT_{\lambda}^{-1}$ is the function in equation (4.8). Since $B^{2}=B^{2}_{\lambda}=T_{\lambda}L_{\lambda}^{2}T_{\lambda}^{-1}$, by Proposition 3.4 we deduce that for any isolated invariant set $K$ of the local semiflow $\Psi^{2}=\Psi_{\lambda}^{2}$ of (5.28), one has

[TABLE]

where ${\mathcal{R}}_{\lambda}^{2}$ is the local semiflow of (4.20) on $U_{\lambda}^{2}$.

Now let $S\subset U$ be an isolated invariant set of $\Phi_{\lambda}$. Set $K=T_{\lambda}S$. Then by Proposition 3.4 and (5.30) we conclude that

[TABLE]

which completes the proof of the validity of the second equality in (5.1). (The last equality in (5.32) is due to the fact that $T_{\lambda}^{-1}P^{2}=P^{2}_{\lambda}T_{\lambda}^{-1}$.)

The first equality in (5.1) follows from Proposition 3.4 and the first equality in (5.30). The proof of the theorem is complete. $\Box$

6 An Index Formula for Bifurcating Invariant Sets near Equilibrium Points

As in Section 5, we follow the same notations as in Section 4 and assume the function $f_{\lambda}(x)$ in (4.1) satisfies all the regularity hypotheses in Section 4. Suppose also that $f_{\lambda}(0)\equiv 0$ for $\lambda\in\mathbb{R}$. The main result is the following theorem.

Theorem 6.1

In addition to (H1)-(H3) (see Section 4), assume that

1. (H4)

for $\lambda\in J_{0}$, we have

[TABLE]

Suppose also that $S_{0}=\{0\}$ is an isolated invariant set of $\Phi_{\lambda_{0}}$.

Then there exist closed isolating neighborhood $N$ of $S_{0}$ with respect to $\Phi_{\lambda_{0}}$ and $\varepsilon>0$ such that $\Phi_{\lambda}$ has a maximal compact invariant set $K_{\lambda}$ ($K_{\lambda}$ may be void) in $N\setminus S_{0}$ for each $\lambda\in[\lambda_{0}-\varepsilon,\,\lambda_{0}+\varepsilon]$, $\lambda\neq\lambda_{0}$. Furthermore,

[TABLE]

where $\Phi_{\lambda}^{2}=\Phi_{\lambda}|_{{\mathcal{M}}_{\lambda}^{2}}$, and $\mathfrak{m}_{i}=\mbox{\em dim}\,(X^{i})$.

Remark 6.2

The interested reader is referred to [17, Theorem 4.3] on when the bifurcating invariant set $K_{\lambda}$ is nonvoid. We also infer from the proof of [17, Theorem 4.3] that $K_{\lambda}$ is upper semicontinuous in $\lambda$.

Remark 6.3

For convenience, we will call the number $\mathfrak{m}_{2}$ in Theorem 6.1 the crossing number at $\lambda=\lambda_{0}$ in case (H4) is fulfilled.

Proof of Theorem 6.1. (1) Pick a closed isolating neighborhood $N$ of $S_{0}$ with $N\subset U$, where $U$ is the neighborhood of [math] given in Theorem 5.1. Then by a very standard argument (see e.g. [34, Chap. I]) it can be shown that there exists $\varepsilon>0$ such that $N$ is an isolating neighborhood of $\Phi_{\lambda}$ for all $\lambda\in[\lambda_{0}-\varepsilon,\,\lambda_{0}+\varepsilon]$. Denote $S_{\lambda}$ the maximal compact invariant set of $\Phi_{\lambda}$ in $N$. Then we infer from Theorem 4.7 that $S_{\lambda}\subset{\mathcal{M}}^{2}_{\lambda}$ and is an isolated invariant set of $\Phi_{\lambda}^{2}$ on ${\mathcal{M}}_{\lambda}^{2}$.

For convenience in statement, let us temporarily forget the conjugacy between $\Phi_{\lambda}^{2}$ and the local semiflow ${\mathcal{R}}_{\lambda}^{2}$ of the reduction equation (4.20) and identify the two local semiflows.

Let $\lambda\in[\lambda_{0}-\varepsilon,\lambda_{0})$. Then by (H4) we deduce that $\mbox{Re}\,\sigma(L_{\lambda}^{2})<0$, where $L_{\lambda}^{2}$ is the linear operator in the reduction equation (4.20). Hence $S_{0}=\{0\}$ is a repeller of $\Phi_{\lambda}^{2}$. It follows by the standard Morse decomposition theory of invariant sets that $S_{\lambda}$ has a Morse decomposition ${\mathscr{M}}=\{M_{0},M_{1}\}$ with $M_{1}=S_{0}$. Note that $M_{0}:=K_{\lambda}$ is the maximal compact invariant set of $\Phi_{\lambda}$ in $S_{\lambda}$. By maximality of $S_{\lambda}$ it can be easily seen that $K_{\lambda}$ is also the maximal compact invariant set of $\Phi_{\lambda}$ in $N\setminus S_{0}$. (It may occur that $S_{\lambda}=S_{0}$. In such a case we have $K_{\lambda}=M_{0}=\emptyset$.)

If $\lambda\in(\lambda_{0},\lambda_{0}+\varepsilon]$ then $S_{0}$ is an attractor of $\Phi_{\lambda}^{2}$, and a parallel argument applies to show that $\Phi_{\lambda}$ has a maximal compact invariant set $K_{\lambda}$ in $N\setminus S_{0}$.

(2) Let $K_{\lambda}$ be the maximal compact invariant set of $\Phi_{\lambda}$ in $N\setminus S_{0}$. Since $0\not\in K_{\lambda}$, we can pick a $\beta=\beta(\lambda)>0$ with $\mbox{B}_{2\beta}\subset N$ such that $\mbox{B}_{2\beta}\cap K_{\lambda}=\emptyset$, where $\mbox{B}_{r}$ denotes the ball in $E$ centered at [math] with radius $r$. Then $N_{\beta}:=N\setminus\mbox{B}_{\beta}$ is a closed neighborhood of $K_{\lambda}$. Further by maximality of $K_{\lambda}$ in $N\setminus S_{0}$ we deduce that $N_{\beta}$ is an isolating neighborhood of $K_{\lambda}$.

We infer from the Morse decomposition theory that $S_{\lambda}$ is the union of $S_{0}$, $K_{\lambda}$ and the connecting orbits between $S_{0}$ and $K_{\lambda}$. Thus one concludes that ${\mathscr{E}}(S_{\lambda})\subset K_{\lambda}\cup S_{0}.$ (Recall that ${\mathscr{E}}(S)$ denotes the set of equilibrium points in $S$.) Therefore

[TABLE]

On the other hand, by homotopy property of the equilibrium index we have

[TABLE]

Hence by (5.1) and Proposition 3.4 we deduce that

[TABLE]

It then follows by (6.2) that

[TABLE]

Simple calculations show that

[TABLE]

Therefore

[TABLE]

We also infer from (6.3) that

[TABLE]

Combining this with (6.2) and (6.5) one immediately concludes the validity of the second equality in (6.1). $\Box$

7 A Global Static Bifurcation Theorem

We follow the same notations in the preceding sections. Furthermore, we assume the function $f_{\lambda}(x)$ in (4.1) satisfies all the regularity hypotheses in Section 4.

Let ${\mathscr{X}}=E\times\mathbb{R}$. ${\mathscr{X}}$ is equipped with metric $d$ defined as

[TABLE]

Given ${\mathcal{Z}}\subset{\mathscr{X}}$ and $\lambda\in\mathbb{R}$, denote ${\mathcal{Z}}[\lambda]$ the $\lambda$-section of ${\mathcal{Z}}$:

[TABLE]

Let $\Phi_{\lambda}$ be the local semiflow of (4.1), and denote $\Pi$ the skew-product flow of the family $\Phi_{\lambda}$ ($\lambda\in\mathbb{R}$) on ${\mathscr{X}}$,

[TABLE]

As in the proof of Theorem 3.5 it can be shown that the set $\{f_{\lambda}(u):\,\,(u,\lambda)\in{\mathcal{B}}\}$ is bounded in $X$ for any bounded subset ${\mathcal{B}}$ of ${\mathscr{X}}$. Using this simple fact and applying some fundamental theory on abstract evolution equations (see e.g.  [10, Chap. 3] and [34, Chap. I, Theorem 4.4]), it can be shown by very standard argument that $\Pi$ is asymptotically compact. Consequently each bounded closed invariant set ${\mathcal{K}}$ of $\Pi$ is necessarily compact.

Assume $f_{\lambda}(0)\equiv 0$ for all $\lambda\in\mathbb{R}$, hence $u=0$ is always a trivial equilibrium of $\Phi_{\lambda}$. Given ${\mathcal{U}}\subset{\mathscr{X}}$, denote

[TABLE]

Definition 7.1

Let ${\mathcal{U}}\subset{\mathscr{X}}$ be a closed neighborhood of $(0,\mu)$. The (static) bifurcation branch $\Gamma_{\mathcal{U}}(0,\mu)$ of $\Phi_{\lambda}$ from $(0,\mu)$ in ${\mathcal{U}}$ is defined to be the component of ${{\mathscr{C}}({\mathcal{U}})}$ which contains $(0,\mu)$.

7.1 Existence of local bifurcation branch

Let us first give an existence result of a nontrivial local bifurcation branch.

Let ${\mathcal{M}}_{\lambda}^{2}$ and ${\mathcal{M}}_{\lambda}^{12}$ be the local invariant manifolds of $\Phi_{\lambda}$ given in Theorem 4.7, and denote $\Phi_{\lambda}^{2}=\Phi_{\lambda}|_{{\mathcal{M}}_{\lambda}^{2}}$, $\Phi_{\lambda}^{12}=\Phi_{\lambda}|_{{\mathcal{M}}_{\lambda}^{12}}$.

Theorem 7.2

Assume (H1)-(H4) (see Section 4 and Theorem 6.1) are fulfilled. Suppose $S_{0}=\{0\}$ is an isolated invariant set of $\Phi_{\lambda_{0}}$, where $\lambda_{0}$ is the number appearing in (H4). Let $N$ be the isolating neighborhood of $S_{0}$ given in Theorem 6.1. Then there exists $\varepsilon>0$ such that the following assertions hold.

1. $(1)$

If $\chi\left(h(\Phi_{\lambda_{0}}^{2},S_{0})\right)\neq(-1)^{\mathfrak{m}_{2}}$ then $\Gamma[\lambda_{0}+\varepsilon]\neq\emptyset$. 2. $(2)$

If $\chi\left(h(\Phi_{\lambda_{0}}^{2},S_{0})\right)\neq 1$ then $\Gamma[\lambda_{0}-\varepsilon]\neq\emptyset.$

Here $\Gamma=\Gamma_{\mathcal{N}}(0,\lambda_{0})$, and ${\mathcal{N}}=N\times[\lambda_{0}-\varepsilon,\,\lambda_{0}+\varepsilon]$.

Proof. For simplicity, we set $\lambda_{0}=0$. Choose an $\varepsilon>0$ such that the assertions in Theorem 6.1 hold. Let $S_{\lambda}$ be the maximal compact invariant set of $\Phi_{\lambda}$ in $N$, and $K_{\lambda}$ the maximal compact invariant set of $\Phi_{\lambda}$ in $N\setminus S_{0}$. We may restrict $\varepsilon$ small enough so that $N$ is an isolating neighborhood of $S_{\lambda}$ for all $\lambda\in[-\varepsilon,\varepsilon]$.

For definiteness we assume $\chi\left(h(\Phi_{0}^{2},S_{0})\right)\neq(-1)^{\mathfrak{m}_{2}}$ and prove that

[TABLE]

Let us first show that for any $0<\mu<\varepsilon$, there is a component ${\mathcal{Z}}$ of ${\mathscr{C}}({\mathcal{N}}_{\mu})$, where ${\mathcal{N}}_{\mu}=N\times[\mu,\varepsilon]$, such that

[TABLE]

Set $\Lambda=[\mu,\varepsilon]$, and let ${\mathcal{K}}=\bigcup_{\lambda\in\Lambda}K_{\lambda}\times\{\lambda\}$. Then ${\mathcal{K}}$ is a bounded invariant set of the skew-product flow $\Pi$. By upper semicontinuity of $K_{\lambda}$ in $\lambda$ (see Remark 6.2) one can easily verify that ${\mathcal{K}}$ is closed. Remark 2.6 then asserts that ${\mathcal{K}}$ is compact in ${\mathscr{X}}$. As $(0,\lambda)\not\in{\mathcal{K}}$ for $\lambda\in\Lambda$, we have

[TABLE]

It is also trivial to deduce that $\min_{\lambda\in\Lambda}d(K_{\lambda},\partial N):=r_{N}>0$. Pick a number $0<r<r_{0}$. Then $\Omega:=N\setminus\mbox{B}_{r}$ is a closed isolating neighborhood of $K_{\lambda}$ for all $\lambda\in\Lambda$.

Take a number $a>0$ sufficiently large so that $\mbox{Re}\,\sigma(aI+A)>0$. Set

[TABLE]

As in the proof of Theorem 3.5 we know that $\bigcup_{\lambda\in\Lambda}F_{\lambda}(B)$ is precompact in $E$ for any bounded subset $B$ of $E$.

We infer from Theorem 6.1 that

[TABLE]

This allows us to apply the classical Leray-Schauder continuation theorem (see e.g. Mawhin [23, Section 2]) to deduce that there is a component ${\mathcal{Z}}\in{\mathscr{C}}(\Omega\times\Lambda)$ such that ${\mathcal{Z}}[\mu]\neq\emptyset\neq{\mathcal{Z}}[\varepsilon]$, from which (7.2) immediately follows.

We are now ready to complete the proof of the theorem. Take a sequence of positive numbers $\mu_{k}\rightarrow 0$. For each $\mu_{k}$, pick a connected component ${\mathcal{Z}}_{k}$ of ${\mathscr{C}}({\mathcal{O}}_{\mu_{k}})$ such that \begin{array}[]{ll}{\mathcal{Z}}_{k}[\mu_{k}]\neq\emptyset\neq{\mathcal{Z}}_{k}[\varepsilon].\end{array} By Lemma 2.3 we may assume that ${\mathcal{Z}}_{k}$ converges in the sense of Hausdorff distance $\delta_{\mbox{\tiny H}}(\cdot,\cdot)$ to a compact set ${\mathcal{Z}}_{0}$. Then ${\mathcal{Z}}_{0}$ is a continuum in ${\mathscr{C}}({\mathcal{N}})$ with ${\mathcal{Z}}_{0}[0]\neq\emptyset\neq{\mathcal{Z}}_{0}[\varepsilon]$. On the other hand, by the choice of $N$ we have ${\mathscr{C}}({\mathcal{N}})[0]=\{0\}$. Therefore one concludes that $0\in{\mathcal{Z}}_{0}$.  $\Box$

7.2 Global static bifurcation theorem

For simplicity, in this subsection we rewrite $\mbox{B}_{E}(0,r)=\mbox{B}_{r}$ . Given ${\mathcal{C}}\subset{\mathscr{X}}$ and $J\subset\mathbb{R}$, we also denote

[TABLE]

Let $M_{0}=(0,\lambda_{0})$ be a bifurcation point.

Definition 7.3

The global (static) bifurcation branch $\Gamma$ of $M_{0}$ is defined to be the bifurcation branch of $M_{0}$ in ${\mathscr{X}}$.

The right-hand side global bifurcation branch $\Gamma^{+}$ of $M_{0}$ is defined as

[TABLE]

where $\Gamma^{+}_{\varepsilon}$ denotes the bifurcation branch of $M_{0}$ in ${\mathscr{X}}\setminus\left(\mbox{\em B}_{\varepsilon}\times(-\varepsilon,0)\right)$.

Similarly one can define the left-hand side global bifurcation branch $\Gamma^{-}$.

Remark 7.4

One easily verifies that both $\Gamma^{-}$ and $\Gamma^{+}$ are connected; moreover, we have $\Gamma=\Gamma^{+}\cup\Gamma^{-}$.

Our main result in this section is the following global bifurcation theorem.

Theorem 7.5

Assume (H1)-(H4) are fulfilled. Suppose $S_{0}=\{0\}$ is an isolated invariant set of $\Phi_{\lambda_{0}}$; furthermore,

[TABLE]

Then one of the following cases occurs (see Figures 6.3-6.5).

1. $(1)$

$\Gamma$* is unbounded. * 2. $(2)$

There exists $\lambda_{1}\neq\lambda_{0}$ such that $(0,\lambda_{1})\in\Gamma$ . 3. $(3)$

$\Gamma^{+}\cap\Gamma^{-}\neq\{M_{0}\}$, in which case both $\Gamma^{-}$ and $\Gamma^{+}$ return back to $M_{0}$.

Figure 6.3:  $\Gamma$ is unbounded.           Figure 6.4:   $\Gamma$ connects to $(0,\lambda_{1})$.

Figure 6.5:  $\Gamma^{\pm}$ return back to $M_{0}$.

Proof. We may assume $\lambda_{0}=0$. We argue by contradiction and suppose that none of the cases (1)-(3) occurs. Then both $\Gamma^{\pm}$ are bounded closed subsets of ${\mathscr{X}}=E\times\mathbb{R}$. Moreover,

[TABLE]

[TABLE]

see Fig. LABEL:fig7-4. By asymptotic compactness of $\Pi$ we know that $\Gamma^{\pm}$ are compact.

Step 1. By Theorem 7.2 we have either $\Gamma^{+}\setminus\{M_{0}\}\neq\emptyset$, or $\Gamma^{-}\setminus\{M_{0}\}\neq\emptyset$. Let

[TABLE]

We may assume ${\mathcal{R}}\neq\emptyset\neq{\mathcal{L}}.$ In the case when ${\mathcal{R}}=\emptyset$ or ${\mathcal{L}}=\emptyset$, the argument is simpler and can be obtained by slightly modifying the one below. In the following we first show that ${\mathcal{R}}$ and ${\mathcal{L}}$ are closed and hence are compact. For this purpose it suffices to check that $M_{0}$ is an isolated point in both the sets $\Gamma^{+}|_{\mathbb{R}^{-}}$ and $\Gamma^{-}|_{\mathbb{R}^{+}}$.

We argue by contradiction and suppose that $M_{0}$ is not isolated in, say, $\Gamma^{+}|_{\mathbb{R}^{-}}$. Then there exists a sequence $(u_{n},\lambda_{n})\in{\mathcal{R}}$ converges to $M_{0}$. Since $u=0$ is an isolated equilibrium of $\Phi_{0}$, it is clear that $\lambda_{n}\neq 0$ (hence $\lambda_{n}<0$) for all $n$ sufficiently large.

Pick a closed statically isolating neighborhood $V$ of [math] with respect to $\Phi_{0}$. There is $\varepsilon>0$ such that $V$ is statically isolating with respect to $\Phi_{\lambda}$ for $\lambda\in[-\varepsilon,\varepsilon]$. It can be assumed that $\lambda_{n}<0$ and $(u_{n},\lambda_{n})\in{\mathcal{V}}:=V\times[-\varepsilon,\varepsilon]$ for all $n$.

Denote ${\mathcal{C}}_{n}$ the continuum of ${\mathcal{V}}\cap\Gamma^{+}|_{\mathbb{R}^{-}}$ containing $(u_{n},\lambda_{n})$. If $M_{0}\in{\mathcal{C}}_{n}$ then by definition we find that ${\mathcal{C}}_{n}\subset\Gamma^{-}$. Hence $(u_{n},\lambda_{n})\in\Gamma^{+}\cap\Gamma^{-}$, which contradicts (7.5). Thus we deduce that $M_{0}\not\in{\mathcal{C}}_{n}$. As $u=0$ is the only equilibrium of $\Phi_{0}$ in $V$, it follows that ${\mathcal{C}}_{n}\cap(V\times\{0\})=\emptyset.$ Further, because $\Gamma^{+}\cap(\partial V\times[-\varepsilon,\varepsilon])=\emptyset$, by connectedness of $\Gamma^{+}$ one concludes that ${\mathcal{C}}_{n}[-\varepsilon]\neq\emptyset$.

By Lemma 2.3 we may assume that ${\mathcal{C}}_{n}$ converges in the sense of Hausdorff distance to a compact subset ${\mathcal{C}}_{0}$. Then ${\mathcal{C}}_{0}$ is a continuum in $\Gamma^{+}$. Clearly $M_{0}\in{\mathcal{C}}_{0}$ and ${\mathcal{C}}_{0}[-\varepsilon]\neq\emptyset$. Hence ${\mathcal{C}}_{0}[\lambda]\neq\emptyset$ for all $\lambda\in[-\varepsilon,0]$. We infer from (7.4) that

[TABLE]

Thereby ${\mathcal{C}}_{0}\subset\Gamma^{-}$. But this and (7.6) contradict (7.5).

Step 2. By (7.4) and the definition of ${\mathcal{R}}$ and ${\mathcal{L}}$ it is clear that $(0,\lambda)\not\in{\mathcal{R}}\cup{\mathcal{L}}$ for all $\lambda\in\mathbb{R}$. Thus by compactness of ${\mathcal{R}}$ and ${\mathcal{L}}$, there exists $\delta_{0}>0$ such that

[TABLE]

Let $N\subset E$ and $\varepsilon>0$ be given as in Theorem 6.1. We may assume $N$ is chosen sufficiently small so that $N\subset\mbox{B}_{\delta_{0}}$. Clearly

[TABLE]

see Fig. LABEL:. Pick a number $0<\delta_{1}<\delta_{0}$ with $\mbox{B}_{4\delta_{1}}\subset N$. Denote $S_{\lambda}$ the maximal invariant set of $\Phi_{\lambda}$ in $N$. Since

[TABLE]

there exists $0<a\leq\varepsilon$ such that

[TABLE]

By compactness of $\Gamma^{\pm}$ it is easy to verify that $\Gamma^{\pm}[\lambda]$ are upper semicontinuous in $\lambda$. Thus we can restrict $a$ sufficiently small so that

[TABLE]

[TABLE]

(Note that $\Gamma^{+}[0]=S_{0}\cup{\mathcal{R}}[0]$, and $\Gamma^{-}[0]=S_{0}\cup{\mathcal{L}}[0]$.) Let

[TABLE]

By (7.7), (7.11), (7.12) and the choice of $\delta_{1}$ we clearly have

[TABLE]

Thus by (7.5) one finds that ${\mathcal{R}}_{a}\cap\Gamma^{-}=\emptyset$ and ${\mathcal{L}}_{a}\cap\Gamma^{+}=\emptyset.$ The choice of $\delta_{1}$ and (7.7) also imply

[TABLE]

where ${\mathcal{N}}:=N\times J_{a}$. Therefore we can pick a $0<{\delta_{2}}<\delta_{1}/2$ such that

[TABLE]

and

[TABLE]

It is easy to check that

[TABLE]

Let ${\mathcal{Z}}:=\Gamma\cap{\mathcal{N}}$. Then since $\Gamma=\Gamma^{+}\cup\Gamma^{-}$, it follows from (7.16) that

[TABLE]

Step 3. By (7.9) there is $c>0$ with $c<\frac{1}{2}\min\{\delta_{2},a\}$ such that

[TABLE]

where $S_{\lambda}$ is the maximal invariant set of $\Phi_{\lambda}$ in $N$.

Let $A_{c}:=\bigcup_{|\lambda|\geq c/2}\Gamma[\lambda]$. Then by compactness of $\Gamma$ we deduce that $A_{c}$ is a compact subset of $E$. As $0\not\in A_{c}$, one has

[TABLE]

Take a number $0<r<\frac{1}{2}\min\{c,\kappa\}.$ By some standard argument (see e.g. the proof of [17, Theorem 6.2], one can find a bounded closed neighborhood ${\mathcal{O}}$ of $\Gamma$ with ${\mathcal{O}}\subset\mbox{B}_{\mathscr{X}}(\Gamma,r)$ such that

[TABLE]

Let

[TABLE]

Define

[TABLE]

We infer from (7.14) and (7.15) that ${\mathcal{F}}$, ${\mathcal{G}}$ and ${\mathcal{H}}$ are disjointed. Recalling that ${\mathcal{O}}\subset\mbox{B}_{{\mathscr{X}}}(\Gamma,r)$ and $r<c/2<\delta_{2}/4$, one trivially verifies that

[TABLE]

Furthermore, the following basic facts hold true.

Lemma 7.6

Let $F_{\lambda}={\mathcal{F}}[\lambda],$ $G_{\lambda}={\mathcal{G}}[\lambda],$ and $H_{\lambda}={\mathcal{H}}[\lambda]$. Then

[TABLE]

[TABLE]

[TABLE]

where $\mbox{\em\mbox{B}}_{r}$ denotes the ball in $E$ of center [math] and radius $r$.

Proof. (1)  As $\delta_{2}<\delta_{1}/2$, the inequality (7.25) directly follows from (7.13).

(2)  Note that ${\mathcal{Z}}[\lambda]\subset S_{\lambda}$ for all $\lambda\in J_{a}$. Thus by (7.10) we have ${\mathcal{Z}}\subset\mbox{B}_{\delta_{1}}\times J_{a}$. Observing that if $||u||_{\alpha}>2\delta_{1}$ then

[TABLE]

one concludes that $\overline{}\mbox{B}_{{\mathscr{X}}_{0}}({\mathcal{Z}},\delta_{2})\subset\overline{}\mbox{B}_{2\delta_{1}}\times J_{a},$ from which (7.23) immediately follows.

(3) It is obvious that ${\mathcal{O}}[\lambda]\supset F_{\lambda}\cup G_{\lambda}\cup H_{\lambda}$. So we only need to verify the converse inclusion for each $\lambda\in J_{c}$. For this purpose, it suffices to check that if $(u,\lambda)\in{\mathcal{O}}$ and $|\lambda|\leq c$ then $(u,\lambda)\in{\mathcal{F}}\cup{\mathcal{G}}\cup{\mathcal{H}}$.

Indeed, let $M=(u,\lambda)\in{\mathcal{O}}$, $|\lambda|\leq c$. Then for any $M^{\prime}=(u^{\prime},\lambda^{\prime})\in{\mathscr{X}}$ with $|\lambda^{\prime}|>a$, we have

[TABLE]

Therefore since $d(M,\Gamma)<r$, we deduce that $d(M,\Gamma)=d\left(M,\Gamma|_{J_{a}}\right).$ Thus by (7.17) one has

[TABLE]

which implies

[TABLE]

Hence by (7.21) we see that $M\in{\mathcal{F}}\cup{\mathcal{G}}\cup{\mathcal{H}}$. This finishes the proof of (7.24). $\Box$

Step 4. For the sake of definiteness, we assume

[TABLE]

and focus our attention on the interval $(0,\infty)$. Because ${\mathcal{F}}$ is a neighborhood of $M_{0}:=(0,\lambda_{0})=(0,0)$ in ${\mathscr{X}}$, there exist $\sigma,\eta>0$ such that $\mbox{B}_{\sigma}\times J_{\eta}\subset{\mathcal{F}},$ i.e.,

[TABLE]

By (7.9) we can find a number $0<\mu\leq\frac{1}{2}\min\{\eta,c\}$ such that

[TABLE]

where $K_{\lambda}$ is the bifurcating invariant set given in Theorem 6.1.

Let $K=\bigcup_{\mu\leq\lambda\leq c}K_{\lambda}$. Then as in (7.3) we deduce that

[TABLE]

Take a number $\Lambda>0$ such that

[TABLE]

Let $0<\rho<\min(\kappa^{\prime},\kappa)$, where $\kappa$ is the number given in (7.19). Let ${\mathscr{Y}}=E\times[\mu,\Lambda]$, and define

[TABLE]

Clearly ${\mathcal{V}}$ is closed in ${\mathscr{Y}}$. Since $\mbox{B}_{\rho}\times[\mu,\Lambda]$ is open in ${\mathscr{Y}}$, we see that ${\mathcal{W}}$ is closed in ${\mathscr{Y}}$ as well. We claim that

[TABLE]

To see this, by definition it suffices to show that if $\lambda\in[\mu,\Lambda]$, then any equilibrium $e\neq 0$ of $\Phi_{\lambda}$ in ${\mathcal{V}}[\lambda]$ is contained in ${\mathcal{W}}[\lambda]$.

Figure 6.7                                             Figure 6.8

We first consider $\lambda\in[\mu,c]$. Because $K_{\lambda}$ contains all nonzero equilibria of $\Phi_{\lambda}$ in $N$, in this case one finds (by the choice of $\rho$) that there are no equilibrium points of $\Phi_{\lambda}$ in $\mbox{B}_{\rho}$ other than [math]. Thus by the definition of ${\mathcal{W}}$ we see that $e\in{\mathcal{W}}[\lambda]$.

Now assume $\lambda>c$. We show that ${\mathcal{O}}[\lambda]\cap\mbox{B}_{\rho}=\emptyset$, hence ${\mathcal{V}}[\lambda]={\mathcal{W}}[\lambda]={\mathcal{O}}[\lambda]$, and the conclusion immediately follows. Let $u\in{\mathcal{O}}[\lambda]$. As ${\mathcal{O}}\subset\mbox{B}_{\mathscr{X}}(\Gamma,r)$, there is a $(v,\lambda^{\prime})\in\Gamma$ such that

[TABLE]

which implies $\lambda^{\prime}\geq c/2$ (otherwise $|\lambda-\lambda^{\prime}|>c/2>r$). Hence by (7.19) we have $||v||_{a}\geq 4\kappa$. Therefore

[TABLE]

We thus infer from the choice of $\rho$ that $u\not\in\mbox{B}_{\rho}$. This verifies that ${\mathcal{O}}[\lambda]\cap\mbox{B}_{\rho}=\emptyset$ and completes the proof of (7.30).

It is trivial to check that ${\mathcal{V}}$ is a neighborhood of ${\mathscr{C}}={\mathscr{C}}({\mathcal{V}})={\mathscr{C}}({\mathcal{W}})$ in ${\mathscr{Y}}=E\times[\mu,\Lambda]$. Pick a number $0<\rho<\min(\kappa^{\prime},\kappa)$ sufficiently small so that ${\mathcal{W}}$ is a neighborhood of ${\mathscr{C}}$ in ${\mathscr{Y}}$. Theorem 3.5 then asserts that

[TABLE]

On the other hand, if $\mu\leq\lambda\leq 2\mu<c$ then

[TABLE]

Setting

[TABLE]

one obtains that

[TABLE]

for $\lambda\in[\mu,2\mu]$. It follows by (7.31) that

[TABLE]

We infer from (7.28) and the choice of $\rho$ that $K_{\lambda}\subset F_{\lambda}\setminus\mbox{B}_{\rho}=F_{\lambda}^{\prime}$ for $0<\lambda\leq 2\mu$. Recalling that $K_{\lambda}$ contains all nontrivial equilibria in $N$ (and hence in $F_{\lambda}^{\prime}$) and noticing that $\ell_{0}[\lambda]={\mathscr{E}}_{\Phi_{\lambda}}(F_{\lambda}^{\prime})$, we deduce that

[TABLE]

Thus by (7.32) we have

[TABLE]

Step 5. Finally we show that the left-hand side of (7.33) equals [math], thus obtain a contradiction. For this purpose, consider the domain ${\mathcal{G}}$ in ${\mathscr{X}}_{1}=E\times(-\infty,a]$ defined in (7.21). Let $\widetilde{\mathcal{G}}={\mathcal{G}}\cap{\mathscr{W}}$, where ${\mathscr{W}}=E\times[-\Lambda,c]$, and $\Lambda$ is the number in (7.29). We claim that the boundary $\partial_{{\mathscr{W}}}\widetilde{\mathcal{G}}$ of $\widetilde{\mathcal{G}}$ in ${\mathscr{W}}$ contains no equilibrium points. To see this, by (7.20) it suffices to check that $\partial_{{\mathscr{W}}}\widetilde{\mathcal{G}}\subset\partial{\mathcal{O}}$.

Indeed, since ${\mathscr{W}}\subset{\mathscr{X}}_{1}$, we infer from Lemma 2.1 that

[TABLE]

On the other hand, by (7.22) we find that ${\mathcal{G}}\subset\mbox{B}_{{\mathscr{X}}_{1}}({\mathcal{R}}_{a},\delta_{2})=\mbox{int}_{{\mathscr{X}}_{1}}Q,$ where $Q=\overline{}\mbox{B}_{{\mathscr{X}}_{1}}({\mathcal{R}}_{a},\delta_{2})$. Therefore by Lemma 2.1 we deduce that

[TABLE]

Hence $\partial_{{\mathscr{W}}}\widetilde{\mathcal{G}}\subset\partial{\mathcal{O}}$.

Let $\wp={\mathscr{C}}(\widetilde{\mathcal{G}})$. Then $\wp$ is a compact invariant set of the skew-product flow $\Pi$ of the family $\Phi_{\lambda}$ ($\lambda\in[-\Lambda,c]$) on ${\mathscr{W}}$. We infer from the claim proved above that $\wp\cap\partial_{\mathscr{W}}\widetilde{\mathcal{G}}=\emptyset$. Hence $\wp$ is a static isolating set of $\Pi$. Thanks to Theorem 3.5,

[TABLE]

Therefore

[TABLE]

A parallel argument as above applies to show that $\mbox{Ind}(\Phi_{\lambda},\ell_{2}[\lambda])=0$ for $\lambda\in[\mu,2\mu].$ Finally combining the above results with (7.33) we conclude $\mbox{Ind}(\Phi_{\lambda},K_{\lambda})=0$. This and (7.26) contradict Theorem 6.1.

The proof of the theorem is finished.  $\Box$

Remark 7.7

If the third case (3) in Theorem 7.5 occurs then both $\Gamma^{\pm}$ are nontrivial; furthermore, we have $d(0,\Gamma[0]\setminus\{0\})>0$. Therefore as depicted in Fig. LABEL:, it is easy to see that there is a two-sided neighborhood $I_{0}$ of $\lambda_{0}$ such that for each each $\lambda\in I_{0}$ with $\lambda\neq\lambda_{0}$, the system $\Phi_{\lambda}$ has at least two distinct nontrivial equilibria. Consequently we have a weaker version of Theorem 7.5:

Theorem 7.8

Assume the hypotheses in Theorem 7.5. Then either there is a two-sided neighborhood $I_{0}$ of $\lambda_{0}$ such that for each $\lambda\in I_{0}\setminus\{\lambda_{0}\}$, $\Phi_{\lambda}$ has at least two distinct nontrivial equilibria, or one of the following two assertions holds:

1. $(1)$

$\Gamma$* is unbounded. * 2. $(2)$

There exists $\lambda_{1}\neq\lambda_{0}$ such that $(0,\lambda_{1})\in\Gamma$ .

8 The case $\mathfrak{m}_{2}=2$

We now pay some attention to a particular case, namely, the case $\mathfrak{m}_{2}=2$. An easy example will also be included to illustrate our theoretical results.

8.1 A local and global bifurcation theorem

In what follows, by a $k$-dimensional topological sphere it means the boundary $\partial D$ of any contractible open subset $D$ of a $(k+1)$-dimensional manifold ${\mathcal{M}}$ without boundary. Denote $\mathbb{S}^{k}$ any $k$-dimensional topological sphere.

The main results in this section are summarized in the following theorem.

Theorem 8.1

Assume (H1)-(H4) are fulfilled with $\mathfrak{m}_{2}=2$. Suppose $S_{0}=\{0\}$ is an isolated invariant set of $\Phi_{\lambda_{0}}$ and $\chi\left(h(\Phi_{\lambda_{0}}^{2},S_{0})\right)\neq 1.$

Then one of the following two assertions holds.

1. (1)

There is a one-sided neighborhood $I_{1}$ of $\lambda_{0}$ such that for each $\lambda\in I_{1}\setminus\{0\}$, the system $\Phi_{\lambda}$ has a compact invariant set $S_{\lambda}=\mathbb{S}^{1}$ with $0\not\in S_{\lambda}$, and $S_{\lambda}$ consists of either a closed orbit, or some nontrivial equilibrium points of $\Phi_{\lambda}$ and connecting orbits between them. 2. (2)

$\Phi_{\lambda}$* undergoes a static bifurcation as stated in Theorem 7.8. *

To prove the theorem, we need a basic result on the planar system

[TABLE]

Denote $\pi$ the local semiflow of (8.1).

Assume $f(0)=0$, and suppose $S_{0}=\{0\}$ is an isolated invariant set of $\pi$.

Lemma 8.2

Suppose $S_{0}$ is neither an attractor nor a repeller. Then

[TABLE]

Proof. It is known that $H_{q}(h(\pi,S_{0}))=0$ for $q>2$. So one only needs to verify the validity of (8.2) for $q=0,2$.

We infer from [7, Theorem 1.5] that $S_{0}$ has an isolating block $B$ with smooth boundary $\partial B$. Note that $\partial B$ consists of Jordan curves. Therefore there is at least one Jordan curve $C\subset\partial B$ such that $0\in\mbox{int}\,N$, where $N$ denotes the bounded closed domain with $\partial N=C$. It is easy to understand that $N$ is an isolating block of $S_{0}$; moreover, $N$ is contractible.

Because $S_{0}$ is neither an attractor nor a repeller, one has $N^{e}\neq\emptyset\neq N^{i}.$ (See Subsection 2.3 for the definition of $N^{e}$, $N^{i}$ and $N^{\pm}$.) Thus we see that $N^{-}$ is the union of at most countably infinitely many disjoint curve segments $C_{i}$ ($1\leq i\leq n_{0}\leq\infty$). For each $i$, we fix a point $p_{i}\in C_{i}$. Then $p_{i}$ is a strong deformation retract of $C_{i}$. It follows that $P=\{p_{i}:\,\,1\leq i\leq n_{0}\}$ is a strong deformation retract of $N^{-}$. One can easily check that

[TABLE]

Recall that we have the exact sequence

[TABLE]

By (8.3) one concludes that $j_{*}$ is an isomorphism. Therefore

[TABLE]

Since $N$ is contractible, $N/N^{-}$ is a path-connected space. Hence we have

[TABLE]

which completes the proof of the lemma. $\Box$

Now we turn to the proof of Theorem 8.1. Proof of Theorem 8.1. If $S_{0}$ is an attractor/repeller of $\Phi_{\lambda_{0}}^{2}$, then the system undergoes an attractor/repeller bifurcation, and the conclusions in assertion (1) follow from the attractor bifurcation theory in Ma and Wang [22] (see also [17, Theorem 4.2]). So we assume $S_{0}$ is neither an attractor nor a repeller of $\Phi_{\lambda_{0}}^{2}$.

Let $\beta_{q}=\mbox{rank}\,\left(H_{q}(h(\Phi_{\lambda_{0}}^{2},S_{0}))\right)$ be the $q$-th Betti number of $h(\Phi_{\lambda_{0}}^{2},S_{0})$. By Lemma 8.2 we have $\beta_{q}=0$ for all $q\neq 1$. Hence

[TABLE]

By Theorem 7.8 one concludes that assertion (2) holds. $\Box$

8.2 An example

Finally let us give a simple example to illustrate our theoretical results.

Consider the periodic problem on $J=[-\pi,\pi]$:

[TABLE]

where $a\in C^{2}(J)$, and $h\in C^{2}(J\times\mathbb{R})$. Moreover,

[TABLE]

uniformly for $x\in J$.

Let $X=L^{2}(J)$. Define an operator $A$ on $X$ to be the differential operator $-\frac{d^{2}}{dx^{2}}$ associated with the periodic boundary condition in (8.4). Then

[TABLE]

The first eigenvalue $\lambda_{0}$ is simple with an eigenfunction $e^{O}(x)=1/\sqrt{2\pi}$, and all the others are of multiplicity $2$. For $n\geq 1$, $A$ has a pair of eigenfunctions

[TABLE]

pertaining to $\lambda_{n}$. The system $\{e^{0},e^{1}_{1},e^{1}_{2},\cdots,e^{n}_{1},e^{n}_{2},\cdots\}$ forms a normal orthogonal basis of $X$.

Fix a number $\alpha\in[0,1)$, and let $E=X^{\alpha}$. Denote $||\cdot||_{\alpha}$ and $||\cdot||$ the norms on $E$ and $X$, respectively. Define $g:E\rightarrow X$ as

[TABLE]

Then (8.4) can be written in an abstract form

[TABLE]

Now we turn to the bifurcation of (8.6) at each eigenvalue $\lambda_{k}$, $k\geq 1$.

Consider the corresponding evolution equation

[TABLE]

where $L_{\lambda}=A-\lambda I$. We fix $k\geq 1$ and, for simplicity, rewrite

[TABLE]

Denote $X_{1}=\mbox{span}\{e_{1},e_{2}\}$, and let $X_{2}=X_{1}^{\perp}$. Then $X=X_{1}\oplus X_{2}$. Set

[TABLE]

Then $E_{1}=X_{1}$, and the norms $||\cdot||_{\alpha}$ and $||\cdot||$ are equivalent on $E_{1}$. Theorem 4.3 asserts that there is a convex neighborhood $V_{1}$ of [math] in $E_{1}$ as well as a continuously differentiable mapping $\zeta:V_{1}\rightarrow E_{2}$ such that ${\mathcal{M}}^{c}=\{u_{1}+\zeta(u_{1}):\,\,u_{1}\in V_{1}\}$ is a local center manifold of (8.7) at $\lambda=\lambda_{k}$, and the reduction equation on ${\mathcal{M}}^{c}$ reads

[TABLE]

The mapping $\zeta$ can be approximated by some simpler ones. Indeed, we infer from [34, Chap. II, Theorem 2.3] that if $\phi:V_{1}\rightarrow E_{2}$ is a $C^{1}$-mapping with Lipschitz derivative $\phi^{\prime}$ such that $\phi(V_{1})\subset D(A)$ and

[TABLE]

for some constants $M>0$ and $\beta>1$, where

[TABLE]

with $L=L_{\lambda_{k}}$, then

[TABLE]

We observe that if $\phi^{\prime}(u_{1})=O(||u_{1}||)$ as $||u_{1}||\rightarrow 0$, then since $LE_{1}=\{0\}$,

[TABLE]

For every $u_{1}=c_{1}e_{1}+c_{2}e_{2}\in V_{1}$, we also have

[TABLE]

Note that $w_{i}\in X_{2}$ ($i=0,1,2$). As

[TABLE]

is a one-one mapping, there exist $v_{i}\in D(L_{2})\subset E_{2}$ such that

[TABLE]

Thus if we define $\phi(u_{1})$ as

[TABLE]

then

[TABLE]

Combining this with (8.10), one finds that (8.9) is fulfilled with $\beta=3$. Hence

[TABLE]

Let $u_{1}(t)=c_{1}(t)e_{1}+c_{2}(t)e_{2}$. Then the reduction equation (8.8) reads

[TABLE]

or equivalently

[TABLE]

where

[TABLE]

Now we can state the following bifurcation result.

Theorem 8.3

Suppose that the bilinear form $B_{i}$ is positive definite for $i=1$ or $2$, i.e., there exists $\gamma>0$ such that

[TABLE]

Then one of the following alternatives occurs.

1. (1)

There is a two-sided neighborhood $I_{2}$ such that for each $\lambda\in I_{2}\setminus\{0\}$, the problem (8.4) has at least two distinct nontrivial solutions. 2. (2)

The global static bifurcation branch $\Gamma$ of $(0,\lambda_{k})$ is either unbounded, or meets another bifurcation point $(0,\lambda_{m})\neq(0,\lambda_{k}).$

Proof. By the assumption of the theorem it is trivial to check that $u_{1}=0$ is an isolated equilibrium of (8.11). Consequently $u=0$ is an isolated equilibrium of $\Phi_{\lambda_{k}}$, where $\Phi_{\lambda}$ denotes the local semifow of (8.7). Since the system (8.7) has a Lyapunov function $V(u)$ which is precisely the variational functional of the problem (8.4), we easily deduce that $S_{0}=\{0\}$ is an isolating invariant set of $\Phi_{\lambda_{k}}$.

By positivity of $B_{i}$ we see that

[TABLE]

unless $c_{1}(t)=0=c_{2}(t)$, which implies that $S_{0}$ is neither an attractor nor a repeller of (8.11). Now the conclusion of the theorem immediately follows from Theorem 8.1. $\Box$

Remark 8.4

A simple example in which the bilinear form $B_{1}$ is positive definite is the one where the function $a(x)=\sin kx$ or $\cos kx$ for some $k\geq 0$. Indeed, if, say, $a(x)=\sin kx$, then

[TABLE]

and

[TABLE]

from which it is obvious that $B_{1}$ is positive definite.

Appendix A: Isomorphisms Induced by Projections

Let $X_{\lambda}^{i},$ $X_{\lambda}^{ij}$, $P_{\lambda}^{i}$ and $P_{\lambda}^{ij}$ be the same as in Section 4.1. Since $P^{3}_{\lambda}=I-(P^{1}_{\lambda}+P^{2}_{\lambda})$, the continuity of $P^{1}_{\lambda}$ and $P^{2}_{\lambda}$ implies that $P_{\lambda}^{3}$ is continuous in $\lambda$ as well.

By (H3) we can assume $J_{0}$ is chosen sufficiently small so that

[TABLE]

Then

[TABLE]

As before, we drop the subscript “$\lambda_{0}$” and rewrite

[TABLE]

Proposition A1.  *For each $i=1,2,3$, the restriction $P^{i}|_{X^{i}_{\lambda}}$ of $P^{i}$ on $X_{\lambda}^{i}$ is an isomorphism between $X^{i}_{\lambda}$ and $X^{i}$. * Proof. To prove Pro. A1, let us first verify that $P^{i}|_{X^{i}_{\lambda}}$ are one-to-one mappings.

As $P_{\lambda}^{3}=I-P_{\lambda}^{12}$, we deduce that

[TABLE]

In what follows we argue by contradiction and suppose $P^{i}|_{X^{i}_{\lambda}}$ fails to be a one-to-one mapping for some $i$. Then there would exist $x_{i}\in X^{i}_{\lambda}$ with $x_{i}\neq 0$ such that $P^{i}x_{i}=0$. Further by (A1) and (A2) we see that

[TABLE]

a contradiction !

Now we show that $P^{i}|_{X^{i}_{\lambda}}$ are isomorphisms. Since $P^{i}|_{X^{i}_{\lambda}}$ are one-to-one mappings, by (4.4) one immediately concludes that $P^{i}|_{X^{i}_{\lambda}}$ are isomorphisms for $i=1,2$. So we only need to consider the case where $i=3$.

Let $Q=P^{3}+P_{\lambda}^{12}$. Then

[TABLE]

Because

[TABLE]

and $||P^{12}-P_{\lambda}^{12}||<1$, by the basic knowledge in linear functional analysis, we know that $Q:X\rightarrow X$ is an isomorphism. To show that $P^{3}|_{X^{3}_{\lambda}}$ is an isomorphism, there remains to check that $Q{X^{3}_{\lambda}}=X^{3}$. For this purpose, it suffices to show that $Q^{-1}X^{3}\subset X^{3}_{\lambda}$.

We argue by contradiction and suppose the contrary. There would exist $u\not\in X^{3}_{\lambda}$ such that $Qu\in X^{3}$. Let $u=x_{\lambda}+x^{3}_{\lambda}$, where $x_{\lambda}\in X_{\lambda}^{12}$, and $x_{\lambda}^{3}\in X_{\lambda}^{3}$. Then $x_{\lambda}\neq 0$. We observe that

[TABLE]

Hence $x_{\lambda}\in X^{3}$. Thereby we have $x_{\lambda}\in X_{\lambda}^{12}\cap X^{3}$. It follows that

[TABLE]

Thus

[TABLE]

This leads to a contradiction and completes the proof of the proposition. $\Box$

Now we define for each $\lambda\in J_{0}$ a linear operators $T_{\lambda}$ on $X$ as follows:

[TABLE]

It is trivial to check that $T_{\lambda}$ is an isomorphism with $T_{\lambda_{0}}=I$. Clearly $T_{\lambda}$ is continuous in $\lambda$, and

[TABLE]

Thus we have Proposition A2. Under the assumptions (H1)-(H3), there exists a family of isomorphisms $T_{\lambda}$ ($\lambda\in J_{0}$) on $X$ depending continuously on $\lambda$ with $T_{\lambda_{0}}=I$, such that

[TABLE]