Boundedness of solutions for Duffing equation with low regularity in time

TL;DR

This paper proves that solutions to a Duffing equation with low regularity time-dependent coefficients are bounded, extending understanding of solution behavior under less smooth conditions.

Contribution

It establishes boundedness of solutions for a Duffing equation with coefficients of low regularity in time, a novel result in the study of nonlinear oscillators.

Findings

All solutions are bounded under specified regularity conditions.

Boundedness holds even with coefficients of limited smoothness.

The result broadens the class of Duffing equations with predictable solution behavior.

Abstract

It is shown that all solutions are bounded for Duffing equation $\ddot{x}+ x^{2n+1}+\sum_{j=0}^{2n}P_{j}(t)x^{j}=0,$ provided that for each $n+1\le j\le 2n$, $P_j(t)\in C^{\gamma}(\mathbb T)$ with $\gamma>1-\frac1n$ and for each $0\le j\le n$, $P_j\in L(\mathbb T^1)$.

Equations

$$\ddot{x}+\beta x^3+\alpha x=P(t),P\in C({\mathbb{T}}^1),{\mathbb{T}}^1:=\mathbb{R}/\mathbb{Z},$$

$$\ddot{x}+2x^3=P(t).$$

$$\ddot{x}+x^{2n+1}+\sum _{j=0}^{2n}{P}_j(t)x^j=0,n\ge 1,$$

$$P_j\in C^{\nu },\nu \ge 1+\frac{4}{n}+[{\log }_2^n]\to \infty ,\nLeftarrow \text{mbox}asn\to \infty .$$

$$\ddot{x}+x^{2n+1}+a(t)x+P(t)=0,a(t)\in C^0({\mathbb{T}}^1),P(t)\in C^0({\mathbb{T}}^1).$$

$$P_j\in C^{5+\epsilon }({\mathbb{T}}^1),\epsilon >0.$$

$$\ddot{x}+x^{2n+1}+\sum _{j=0}^{2n}{P}_j(t)x^j=0,n\ge 1,$$

$$\underset{t\in \mathbb{R}}{\sup }(|x(t)|+|\dot{x}(t)|)<C<\infty ,$$

$$A\ddot{x}+A^{2n+1}{x}^{2n+1}+\sum _{j=0}^{2n}{P}_j(t)x^j{A}^j=0,$$

$$\ddot{x}+A^{2n}{x}^{2n+1}+\sum _{j=0}^{2n}{P}_j(t)x^j{A}^{j-1}=0.$$

$$y=A^{-n}\dot{x},\text{mbox}or\dot{x}=A^{n}y.$$

$$\dot{y}$$

$$\dot{x}=\frac{\partial H}{\partial y},\dot{y}=-\frac{\partial H}{\partial x},$$

$$H=A^n\left(\frac{1}{2}{y}^2+\frac{1}{2(n+1)}{x}^{2(n+1)}\right)+\sum _{j=0}^{2n}\frac{P_j(t)}{j+1}{x}^{j+1}{A}^{j-n-1}.$$

$${\mathbb{T}}_s^1=\{t\in \mathbb{C}/\mathbb{Z}:|\mathrm{Im}t|<s\}\text{mbox}foranys>0.$$

$$\dot{x}=\frac{\partial H_0}{\partial y},\dot{y}=-\frac{\partial H_0}{\partial x},H_0=\frac{1}{2}{y}^2+\frac{1}{2(n+1)}{x}^{2(n+1)}.$$

$$(n+1)y_0^2(t)+x_0^{2n+2}(t)\equiv 1,t\in \mathbb{R},$$

$$\Psi_{0}:\;\;\left\{\begin{array}[]{ll}x=c^{\alpha}I^{\alpha}x_{0}(\theta T_{0}),\\ \\ y=c^{\beta}I^{\beta}y_{0}(\theta T_{0}),\end{array}\right.$$

$$\dot{\theta }=\frac{\partial H}{\partial I},\dot{I}=-\frac{\partial H}{\partial \theta },$$

$$H_0(I)=d\cdot A^n\cdot I^{2\beta }=d\cdot A^n\cdot I^{\frac{2(n+1)}{n+2}},d=\frac{c^{2\beta }}{2(n+1)},$$

$$R(I,\theta ,t)=\sum _{j=0}^{2n}\frac{P_j(t)}{j+1}(c^{\frac{1}{n+1}}{x}_0(\theta T_0){)}^{j+1}{A}^{j-n-1}{I}^{\frac{j+1}{n+2}}.$$

$$\Vert f{\Vert }_{C^{\mu },X}=\underset{0<|x-y|<1}{\sup }\frac{\Vert f(x)-f(y){\Vert }_X}{|x-y{|}^{\mu }}+\underset{x\in {\mathbb{T}}^n}{\sup }\Vert f(x){\Vert }_X.$$

$$\Vert T{\Vert }_{X_{\alpha }}=\sup \{||T(u_1,u_2,\cdots ,u_{|\alpha |})|{|}_X:\Vert u_i\Vert =1,1\le i\le |\alpha |\}.$$

$$||f|{|}_{C^{\ell }}=\underset{|\alpha |\le \ell }{\sup }||{\partial }^{\alpha }f|{|}_{C^{\mu },X_{\alpha }}$$

$$f_{\sigma }(x):=K_{\sigma }\ast f=\frac{1}{{\sigma }^n}{\int }_{{\mathbb{T}}^n}K(\frac{x-y}{\sigma })f(y)dy.$$

$$\Delta_{\sigma}^{n}:=\{x\in\mathbb{C}^{n}\big{|}|\mathrm{Im}x_{j}|\leq\sigma,\;1\leq j\leq n\},$$

$$\underset{x\in {\Delta }_{\sigma }^n}{\sup }||{\partial }^{\alpha }{f}_{\sigma }(x)-\sum _{|\beta |\le \ell -|\alpha |}\frac{{\partial }^{\beta +\alpha }f(\mathrm{Re}x)}{\beta !}(\sqrt{-1}\mathrm{Im}x{)}^{\beta }|{|}_{X_{\alpha }}\le C||f|{|}_{C^{\ell }}{\sigma }^{\ell -|\alpha |},$$

$$\underset{x\in {\Delta }_s^n}{\sup }\Vert {\partial }^{\alpha }{f}_{\sigma }(x)-{\partial }^{\alpha }{f}_s(x){\Vert }_{X_{\alpha }}\le C||f|{|}_{C^{\ell }}{\sigma }^{\ell -|\alpha |}.$$

$$\underset{t\in {\mathbb{T}}^1}{\sup }|P_{j,\epsilon }(t)-P_j(t)|\le C{\epsilon }^{\gamma }||P_j|{|}_{C^{\gamma }},$$

$$\underset{t\in {\mathbb{T}}_{\epsilon }^1}{\sup }|P_{j,\epsilon }(t)|\le C||P_j|{|}_{C^{\gamma }}.$$

Full text

1 Introduction

In 1962, Moser [6] proposed to study the boundedness of all solutions (Lagrange stability) for Duffing equation

[TABLE]

where $\beta>0,$ $\alpha\in\mathbb{R}$ are constants.

In 1976, Morris [5] proved the boundedness of all solutions for

[TABLE]

Subsequently, Morris’ boundedness results was, by Dieckerhoff-Zehnder [1] in 1987, extended to a wider class of systems

[TABLE]

where

[TABLE]

Then they remarked that

”It is not clear whether the boundedness phenomenon is related to the smoothness in the $t$-variable or whether this requirement is a shortcoming of our proof.”

In 1989 and 1992, Liu [4, 3] proved the boundedness for

[TABLE]

In 1991, Laederich-Levi [2] relaxed the smoothness requirement of $P_{j}(t)\;(j=0,1,\cdots,2n)$ for (1.1) to

[TABLE]

In his PhD thesis (1995), the present author further relaxed the requirement to $C^{2}$. See [12],[13] and [14].

In the present paper, we will relax the smoothness requirement to $C^{1}.$ More exactly, we have the following theorem

Theorem 1.1**.**

For Arbitrary given constant $\gamma>1-\frac{1}{n}$, assume $P_{j}(t)\in C^{\gamma}(\mathbb{T}^{1})$ for $n+1\leq j\leq 2n$ and $P_{j}(t)\in L(\mathbb{T}^{1})$ for $0\leq j\leq n$. Then every solution $x(t)$ of the equation (1.1),

[TABLE]

is bounded, i.e. it exists for all $t\in\mathbb{R}$ and

[TABLE]

where $C=C(x(0),\dot{x}(0))$ depends the initial data $(x(0),\dot{x}(0)).$

Remark 1**.**

In [11], it is proved that there is a continuous periodic function $p(t)$ such that the Duffing equation $\frac{d^{2}x}{dt^{2}}+x^{2n+1}+p(t)x^{l}=0$ with $p(t)\in C^{0}(\mathbb{T}^{1})$, $n\geq 2$, $2n+1>l\geq n+2$ possesses an unbounded solution, which shows that the smoothness of the coefficients $P_{j}(t)$’s does influence the boundedness of solutions. Therefor, the result of theorem 1.1 is sharp without considering the derivative of non-integral order.

2 Action-Angle Variable

Replacing $x$ by $Ax$ in (1.1), we get

[TABLE]

where $A$ is a constant large enough. That is,

[TABLE]

Let

[TABLE]

Then

[TABLE]

Thus,

[TABLE]

where

[TABLE]

Let

[TABLE]

Consider an auxiliary Hamiltonian system

[TABLE]

Let $(x_{0}(t),y_{0}(t))$ be the solution to (2.5) with initial $(x_{0}(0),y_{0}(0))=(1,0).$ Then this solution is clearly periodic. Let $T_{0}$ be its minimal positive period. By Energy conservation, we has

[TABLE]

by which, we construct the following symplectic transformation

[TABLE]

where $\alpha=\frac{1}{n+2},\;\beta=1-\alpha=\frac{n+1}{n+2},\;c=\frac{1}{\alpha T_{0}}$ and where $(I,\theta)\in\mathbb{R}^{+}\times\mathbb{T}^{1}$ is action-angle variables. By calculation, $\text{det}\,\frac{\partial\,(x,y)}{\partial(\theta,I)}=1$. Thus the transformation is indeed symplectic. Clearly $\Psi_{0}(I,\theta)$ is analytic in $(I,\theta)\in\mathbb{R}^{+}\times\mathbb{T}^{1}_{s_{0}}$ with some constant $s_{0}>0.$

Under $\Psi_{0},$ (2.3) is changed

[TABLE]

where $H=H_{0}(I)+R(I,\theta,t)$ with

[TABLE]

and

[TABLE]

Clearly, $R(I,\theta,t)=O(A^{n-1})$ for $A\to\infty$ and fixed $I$ belongs to some compact intervals.

3 Approximation Lemma

First, we cite an approximation lemma. See [9] and [10], for the detail. We start by recalling some definitions and setting some new notations. Assume $X$ is a Banach space with the norm $||\cdot||_{X}$. First recall that $C^{\mu}(\mathbb{R}^{n};X)$ for $0<\mu<1$ denotes the space of bounded Hölder continuous functions $f:\mathbb{R}^{n}\mapsto X$ with the form

[TABLE]

If $\mu=0$ then $\|f\|_{C^{\mu},X}$ denotes the sup-norm. For $\ell=k+\mu$ with $k\in\mathbb{N}$ and $0\leq\mu<1,$ we denote by $C^{\ell}(\mathbb{R}^{n};X)$ the space of functions $f:\mathbb{R}^{n}\mapsto X$ with Hölder continuous partial derivatives, i.e., $\partial^{\alpha}f\in C^{\mu}(\mathbb{R}^{n};X_{\alpha})$ for all muti-indices $\alpha=(\alpha_{1},\cdots,\alpha_{n})\in\mathbb{N}^{n}$ with the assumption that $|\alpha|:=|\alpha_{1}|+\cdots+|\alpha_{n}|\leq k$ and $X_{\alpha}$ is the Banach space of bounded operators $T:\prod^{|\alpha|}(\mathbb{R}^{n})\mapsto X$ with the norm

[TABLE]

We define the norm

[TABLE]

Theorem 3.1**.**

(Jackson) Let $f\in C^{\ell}(\mathbb{R}^{n};X)$ for some $\ell>0$ with finite $C^{\ell}$ norm over $\mathbb{R}^{n}.$ Let $\phi$ be a radical-symmetric, $C^{\infty}$ function, having as support the closure of the unit ball centered at the origin, where $\phi$ is completely flat and takes value 1, let $K=\widehat{\phi}$ be its Fourier transform. For all $\sigma>0$ define

[TABLE]

Then there exists a constant $C\geq 1$ depending only on $\ell$ and $n$ such that the following holds: For any $\sigma>0,$ the function $f_{\sigma}(x)$ is a real-analytic function from $\mathbb{C}^{n}$ to $X$ such that if $\Delta_{\sigma}^{n}$ denotes the $n$-dimensional complex strip of width $\sigma,$

[TABLE]

then for $\forall\alpha\in\mathbb{N}^{n}$ with $|\alpha|\leq\ell$ one has

[TABLE]

and for all $0\leq s\leq\sigma,$

[TABLE]

The function $f_{\sigma}$ preserves periodicity (i.e., if $f$ is T-periodic in any of its variable $x_{j}$, so is $f_{\sigma}$).

By this theorem, for each $P_{j}(t)\in C^{\gamma}(\mathbb{T}^{1}),\;j=n+1,1,\cdots,2n,$ and any $\varepsilon>0,$ there is a real analytic function 111A complex value function $f(t)$ of complex variable $t$ in some domain in $\mathbb{C}$ is called real analytic if it is analytic in the domain and is real for real argument $t$   $P_{j,\varepsilon}(t)$ from $\mathbb{T}^{1}_{\varepsilon}$ to $\mathbb{C}$ such that

[TABLE]

and

[TABLE]

Write

[TABLE]

where

[TABLE]

[TABLE]

Now let us restrict $I$ belongs to some compact intervals, $[1,4],$ say. Let

[TABLE]

For a sufficiently small $\varepsilon_{0}>0$, letting

[TABLE]

by Theorem 3.1, we have the following facts:

(i)

$R^{\varepsilon}(I,\theta,t)$ is real analytic in $(I,\theta)\in[1,4]\times\mathbb{T}^{1}_{s_{0}}$ for fixed $t\in\mathbb{T}^{1}$ and $R^{\varepsilon}(I,\theta,\cdot)\in L^{1}(\mathbb{T}^{1})$ for fixed $(I,\theta)\in[1,4]\times\mathbb{T}^{1}_{s_{0}},$ and

[TABLE]

where $C$ is a constant 222Denote by $C$ a universal constant which may be different in different place. depending on only $||P_{j}||_{C^{\gamma}}$.

(ii)

$R_{\varepsilon}(I,\theta,t)$ is real analytic in $(I,\theta,t)\in[1,4]\times\mathbb{T}^{1}_{s_{0}}\times\mathbb{T}^{1}_{\varepsilon}$ and

[TABLE]

where $C$ is a constant depends on only $||P_{j}||_{C^{\gamma}}.$ Therefore, we have

[TABLE]

4 Symplectic transformations

We will look for a series of symplectic transformations $\Psi_{1},\cdots,\Psi_{N}$ such that

[TABLE]

where $H_{0}^{N}(\mu)\approx A^{n}\mu^{\frac{2(n+1)}{n+2}}$ and that Moser’s twist works for $H^{(N)}.$

To this end, let $\Psi_{1}:(\mu,\phi)\mapsto(I,\theta)$ is implicitly defined by

[TABLE]

with $S_{1}=S_{1}(\mu,\theta,t)$ to be specified latter. If $\Psi_{1}$ is well-defined, then it is symplectic, since

[TABLE]

The transformed Hamiltonian function $H^{(1)}(\mu,\phi,t)=H\circ\Psi_{1}(\mu,\phi,t).$ We express temporarily in the variable $(\mu,\theta)$ instead of $(\mu,\phi)$:

[TABLE]

By Taylor’s formula and (3.11)

[TABLE]

where

[TABLE]

Let

[TABLE]

Then

[TABLE]

where

[TABLE]

We are now in position to solve (4.4).

[TABLE]

By (3.8) and (3.10), $S_{1}$ is well-defined in $(\mu,\theta,t)\in[1,4]\times\mathbb{T}_{s_{0}}\times\mathbb{T}_{\varepsilon},$ and analytic in the domain, and

[TABLE]

Thus, by the implicit function theorem, $\Psi_{1}(\mu,\phi,t):[1+O(A^{-1}),4-O(A^{-1})]\times\mathbb{T}^{1}_{s_{0}/2}\times\mathbb{T}^{1}_{\varepsilon}\to[1,4]\times\mathbb{T}^{1}_{s_{0}}\times\mathbb{T}^{1}_{\varepsilon}.$

Estimate of $H^{1}_{0}(\mu,t).$

By (3.10), we have that $H^{1}_{0}(\mu,t)$ is analytic in $[1,4]\times\mathbb{T}_{\varepsilon},$ and

[TABLE]

and by Cauchy’s estimate

[TABLE]

• 2.

Estimate of $R^{1}_{\varepsilon}(\mu,\theta,t).$

By (4.8) and the Cauchy estimate,

[TABLE]

where

[TABLE]

By assuming $1\geq\gamma>1-\frac{1}{n}$,

[TABLE]

By (3.10) and noting $H_{0}(\mu)=dA^{n-1}\mu^{\frac{2n+2}{n+2}},$ we have

[TABLE]

where

[TABLE]

By (4.8) and the implicit function theorem, there exist $U_{1}(\mu,\phi,t),\;V_{1}(\mu,\phi,t)$ analytic in $\mathcal{D}_{1}$ such that

[TABLE]

and

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

Similarly, let

[TABLE]

with $S_{2}=S_{2}(\lambda,\phi,t)$ is defined by

[TABLE]

By (4.9) and (4.13),

[TABLE]

It follows from the implicit function theorem that $\Psi_{2}:(\lambda,\widetilde{\phi})\in\mathcal{D}_{2}\to\mathcal{D}_{1}$ is well-defined, where $\mathcal{D}_{2}=[1+O(A^{-1}),4-O(A^{-1})]\times\mathbb{T}_{s_{0}/4}\times\mathbb{T}_{\varepsilon/4}.$ By Cauchy estimate,

[TABLE]

Let

[TABLE]

where

[TABLE]

By (4.20),

[TABLE]

Let

[TABLE]

It follows that

[TABLE]

• 3.

Estimate of $H^{2}_{0}(\lambda,t).$

By (4.9), (4.10) and (4.13),

[TABLE]

[TABLE]

• 4.

Estimate of $R^{2}_{\varepsilon}(\lambda,\phi,t).$

By (4.13), (4.21), (4.22) and (4.24), (4.27),

[TABLE]

Take $N\in\mathbb{N}$ with $n-\varpi\,{N}\leq-1.$ Repeating the above procedure $N$ times, we get a series of symplectic transformations $\Psi_{1},\cdots,\Psi_{N}$ such that

[TABLE]

where $(\rho,\xi,t)\in[1+O(A^{-1}),4-O(A^{-1})]\times\mathbb{T}_{s_{0}/2^{N}}^{1}\times\mathbb{T}_{\varepsilon/2^{N}}^{1},$ and

[TABLE]

and

[TABLE]

and $H_{0}^{N}(\rho,t)$ satisfies

[TABLE]

[TABLE]

and $R_{\varepsilon}^{N}(\rho,\xi,t)$ satisfies that for $0\leq p+q\leq 6.$

[TABLE]

where $C$ depends on $N$ and we have assumed that $A$ is large enough such that

[TABLE]

Let

[TABLE]

Then by (3.9), (4.37), (4.33) and (4.34)

[TABLE]

Now,

[TABLE]

5 Proof of theorem

For $H^{N}$ the Hamiltonian equation is

[TABLE]

Note

[TABLE]

By suing Picard iteration and Gronwall’s inequality and noting (4.39), we get that the time-1 map of (5.1) is of the form

[TABLE]

with

[TABLE]

and

[TABLE]

Since (5.1) is Hamiltonian, the map $P$ is symplectic. By Moser’s twist theorem at pp.50-54 of [7], $\mathcal{P}$ has an invariant curve $\Gamma$ in the annulus $[2,3]\times\mathbb{T}^{1}.$ Since $A$ can be arbitrarily large, it follows that the time-1 map of the original system has an invariant curve $\Gamma_{A}$ in the annulus $[2\,A+C,3\,A-C]\times\mathbb{T}^{1}$ with $C$ is a constant independent of $A$. Choosing a sequence $A=A_{k}\to\infty$ as $k\to\infty$, we have that there are countable many invariant curves $\Gamma_{A_{k}}$, clustering at $\infty.$ Therefore any solution of the original system is bounded. This completes the proof of Theorem.

Remark 2**.**

Any solutions starting from the invariant curves $\Gamma_{A_{k}}$ ($k=1,2,...$) are quasi-periodic with frequencies $(1,\omega_{k})$ in time $t$, where $(1,\omega_{k})$ satisfies Diophantine conditions and $\omega>C\,A_{k}^{n}$.