Zero-Hopf bifurcation in the general Van der Pol-Duffing equation

TL;DR

This paper analytically investigates the coexistence of multiple periodic solutions and invariant tori in the general Van der Pol-Duffing oscillator, employing averaging methods and supporting findings with numerical examples.

Contribution

It introduces an analytical approach using averaging methods to study complex dynamics in the Van der Pol-Duffing oscillator, including coexistence phenomena.

Findings

Multiple periodic solutions coexist with invariant tori.

Analytical results are validated through numerical examples.

The study advances understanding of nonlinear oscillatory behavior.

Abstract

We study analytically the coexistence of multiple periodic solutions and invariant tori in the general Van der Pol-Duffing oscillator equations. We use several results related to the averaging method in order to analytically obtain our results. We also provide numerical examples for all the analytical results that we provide.

Figures

Equations

$$\dot{x}=$$

$$\dot{y}=$$

$$\dot{z}=$$

$$h(\epsilon )$$

$$\alpha (\epsilon )$$

$$\mu (\epsilon )$$

$${\delta }_a=k_0\left(2k_0{\mu }_1{\nu }_0^2-{\alpha }_1{\omega }^2\right),$$

$${\delta }_c=k_0\left({\alpha }_1{\omega }^2+k_0{\mu }_1{\nu }_0^2\right),$$

$${\lambda }_1$$

$${\lambda }_2$$

$$\begin{split}A&=\frac{\left(k_{0}\nu_{0}+2\omega^{2}\right)\left(4\alpha_{1}\omega^{2}+k\nu_{0}(5\alpha_{1}+2\mu_{1}\nu_{0})\right)}{8k_{0}\nu_{0}\omega^{3}},\\ B&=\dfrac{1}{\alpha_{1}^{2}k_{0}^{2}\nu_{0}^{2}}\Big{(}64\alpha_{1}^{2}\omega^{8}+k_{0}^{4}\nu_{0}^{4}\left(145\alpha_{1}^{2}+276\alpha_{1}\mu_{1}\nu_{0}+36\mu_{1}^{2}\nu_{0}^{2}\right)\\ &+4k^{3}\nu_{0}^{3}\omega^{2}\left(153\alpha_{1}^{2}+136\alpha_{1}\mu_{1}\nu_{0}+12\mu_{1}^{2}\nu_{0}^{2}\right)\\ &+4k^{2}\nu_{0}^{2}\omega^{4}\left(221\alpha_{1}^{2}+84\alpha_{1}\mu_{1}\nu_{0}+4\mu_{1}^{2}\nu_{0}^{2}\right)+32\alpha_{1}k_{0}\nu_{0}\omega^{6}(15\alpha_{1}+2\mu_{1}\nu_{0})\Big{)}.\end{split}$$

$$\begin{array}{rl}{\ell }_{1,1}& =-\frac{h_0^2{\nu }_0\left(5k_0^2{\nu }_0^2+16k_0{\nu }_0{\omega }^2+8{\omega }^4\right)}{\left(k_0{\nu }_0+{\omega }^2\right)},\\ {\hat{\mu }}_1& =-\frac{{\alpha }_1\left(5k_0\nu +4{\omega }^2\right)}{2k_0{\nu }_0^2}.\end{array}$$

$${\alpha }_0=0,{\beta }_0=\frac{k_0{\nu }_0+{\omega }^2}{h_0}\text{mbox}and{\mu }_0=0.$$

$${\lambda }_{\pm }=A\pm \frac{{\alpha }_1}{8{\omega }^3}\sqrt{B}.$$

$$h_0=0,k_0=-\frac{{\mu }_0^2{\nu }_0^2+{\omega }^2}{{\nu }_0},{\alpha }_0={\mu }_0{\nu }_0,$$

$${\beta }_0=0,k_0=-\frac{{\mu }_0^2{\nu }_0^2+{\omega }^2}{{\nu }_0},{\alpha }_0={\mu }_0{\nu }_0.$$

$${\alpha }_0=0,{\beta }_0=\frac{{\omega }^2}{h_0},{\nu }_0=0.$$

$${\alpha }_0=0,k_0=-\frac{{\omega }^2}{{\nu }_0},h_0=0,{\mu }_0=0,$$

$$\dot{\mathbf{x}}=\sum _{i=1}^k{\epsilon }^i{\mathbf{F}}_i(t,\mathbf{x})+{\epsilon }^{k+1}\tilde{\mathbf{F}}(t,\mathbf{x},\epsilon ),(t,\mathbf{x},\epsilon )\in \mathbb{R}\times \Omega \times (-{\epsilon }_0,{\epsilon }_0),$$

$$\mathbf{x}(t,\mathbf{z},\epsilon )=\mathbf{z}+\sum _{i=1}^k{\epsilon }^i\frac{{\mathbf{y}}_i(t,\mathbf{z})}{i!}+\mathcal{O}({\epsilon }^{k+1}),$$

$$\Pi (\mathbf{z},\epsilon )=\mathbf{z}+\sum _{i=1}^k{\epsilon }^i{\mathbf{g}}_i(\mathbf{z})+\mathcal{O}({\epsilon }^{k+1}).$$

$${\mathbf{g}}_i(\mathbf{z})=\frac{{\mathbf{y}}_i(T,\mathbf{z})}{i!}$$

$${\mathbf{g}}_i(\mathbf{z})=\frac{{\mathbf{y}}_i(T,\mathbf{z})}{i!},$$

$${\mathbf{y}}_1(t,\mathbf{z})=$$

$${\mathbf{y}}_2(t,\mathbf{z})=$$

$${\mathbf{y}}_3(t,\mathbf{z})=$$

$$+3\frac{{\partial }^2{\mathbf{F}}_1}{\partial {\mathbf{x}}^2}(\tau ,\mathbf{z}){\mathbf{y}}_1(\tau ,\mathbf{z}{)}^2+3\frac{\partial {\mathbf{F}}_1}{\partial \mathbf{x}}(\tau ,\mathbf{z}){\mathbf{y}}_2(\tau ,\mathbf{z})\mathrm{d}\tau ,$$

$${\mathbf{y}}_4(t,\mathbf{z})=$$

$$12\frac{\partial {\mathbf{F}}_2}{\partial \mathbf{x}}(\tau ,\mathbf{z}){\mathbf{y}}_2(\tau ,\mathbf{z})+12\frac{{\partial }^2{\mathbf{F}}_1}{\partial {\mathbf{x}}^2}(\tau ,\mathbf{z}){\mathbf{y}}_1(\tau ,\mathbf{z})\odot {\mathbf{y}}_2(\tau ,\mathbf{z})$$

$$+4\frac{{\partial }^3{\mathbf{F}}_1}{\partial {\mathbf{x}}^3}(\tau ,\mathbf{z}){\mathbf{y}}_1(\tau ,\mathbf{z}{)}^3+4\frac{\partial {\mathbf{F}}_1}{\partial \mathbf{x}}(\tau ,\mathbf{z}){\mathbf{y}}_3(\tau ,\mathbf{z})\mathrm{d}\tau .$$

$$\mathcal{Z}=\{{\mathbf{z}}_u=(u,\mathcal{B}(u)):u\in \overline{V}\}\subset U,$$

$$\left\{\begin{array}[]{l}0=u-\pi\Pi(u,v,\varepsilon),\vspace{0.2cm}\\ 0=v-\pi^{\perp}\Pi(u,v,\varepsilon).\\ \end{array}\right.$$

$$\mathcal{F}(u,\epsilon )=\sum _{i=1}^{k-1}{\epsilon }^i{f}_i(u)+\mathcal{O}({\epsilon }^{k+1}),$$

$$D{\mathbf{g}}_{\ell }({\mathbf{z}}_u)=\left(\begin{array}{cc}{\Lambda }_u& {\Gamma }_u\\ B_u& {\Delta }_u\end{array}\right)$$

$$\begin{array}[]{rl}\gamma_{1}(u)=&-\Delta_{u}^{-1}\pi^{\perp}{\bf g}_{\ell+1}({\bf z}_{u}),\vspace{0.3cm}\\ f_{1}(u)=&\Gamma_{u}\gamma_{1}(u)+\pi{\bf g}_{\ell+1}({\bf z}_{u}),\\ \end{array}$$

$$\begin{array}[]{rl}\gamma_{2}(u)=&-\Delta_{u}^{-1}\Bigg{(}\dfrac{\partial^{2}\pi^{\perp}{\bf g}_{\ell}}{\partial b^{2}}({\bf z}_{u})\gamma_{1}(u)^{2}+2\dfrac{\partial\pi^{\perp}{\bf g}_{\ell+1}}{\partial b}({\bf z}_{u})\gamma_{1}(u)+2\pi^{\perp}{\bf g}_{\ell+2}(u)\Bigg{)},\vspace{0.3cm}\\ f_{2}(u)=&\dfrac{1}{2}\Gamma_{u}\gamma_{2}(u)+\dfrac{1}{2}\dfrac{\partial^{2}\pi{\bf g}_{\ell}}{\partial b^{2}}({\bf z}_{u})\gamma_{1}(u)^{2}+\dfrac{\partial\pi{\bf g}_{\ell+1}}{\partial b}(z_{u})\gamma_{1}(u)+\pi{\bf g}_{\ell+2}({\bf z}_{u}).\vspace{0.3cm}\\ \end{array}$$

Full text

1. Introduction and statements of the main results

In three-dimensional autonomous differential systems a zero-Hopf equilibrium is a equilibrium point which has a zero eigenvalue and a pair of purely imaginary eigenvalues. Generically, a zero-Hopf bifurcation is a two-parameter unfolding of a three-dimensional autonomous differential equation with a zero-Hopf equilibrium. The scientific literature about zero-Hopf bifurcation is rich and varied, containing works of several authors for instance Guckenheimer and Holmes [4, 5], Scheurle and Marsden [9], Han [7] and many others.

However, there is no general theory that is able to describe completely the variety of invariant sets that may bifurcate from zero-Hopf equilibria. Consequently, most of the systems exhibiting a zero-Hopf bifurcation require to be studied directly. The complexity of the invariant sets that can emerge from zero-Hopf equilibria make this kind of bifurcation a classical indication of local birth of chaos in differential systems. Although, the proof of generic unfolding of Hopf-zero singularities displaying strange attractors is fairly recent (see [1]).

In this paper we investigate the existence of periodic solutions and invariant tori emerging from the origin of the three-dimensional differential system

[TABLE]

where $\alpha$, $h$, $\beta$, $k$, $\nu$ and $\mu$ are real parameters. This system is a generalization of the Van der Pol-Duffing oscillator proposed by Zhao et. al. in [10].

The dynamical behaviour of system (1) was investigated by Matouk and Agiza in [8] and Zhao et. al. in [10]. They verified the existence of periodic solutions, gave the stability conditions for the equilibrium points and investigated the presence of chaos and hidden attractors. These papers used a combination of analytical and numerical methods for obtaining their results.

Our approach is focused in the analytic detection of multiple invariant sets bifurcating from the origin of system (1). In our main result, three periodic orbits and two invariant tori are detected bifurcating simultaneously from the origin of system (1). As far as we know, this is the first time that this kind of phenomena has been reported for system (1). The technique used in this paper is a natural combination of the averaging theory and Lyapunov coefficient method that was recently developed in [8].

Take $\omega>0$, it can be easily verified that there are five parameter families such that the origin of system (1) is a zero-Hopf equilibrium point. Namely, the parameter families are

• $i$)

$h=0$, $k=-\dfrac{\mu^{2}\nu^{2}+\omega^{2}}{\nu}$ and $\alpha=\mu\nu$;

• $ii)$

$\beta=0$, $k=-\dfrac{\mu^{2}\nu^{2}+\omega^{2}}{\nu}$, and $\alpha=\mu\nu$;

• $iii)$

$\alpha=0$, $\beta=\dfrac{k\nu+\omega^{2}}{h}$ and $\mu=0$;

• $iv)$

$\alpha=0$, $\beta=\frac{\omega^{2}}{h}$ and $\nu=0$;

• $v)$

$\alpha=0$, $k=-\dfrac{\omega^{2}}{\nu}$, $h=0$ and $\mu=0$.

In order to study system (1) with coefficients that are $\varepsilon$-close to the families presented above we are going to assume hereafter that the coefficients of system (1) depend smoothly on $\varepsilon$, accordingly

[TABLE]

Our first result will be on the coexistence of multiple tori and periodic solutions in system (1). In this case, three periodic solutions will coexists if the quantities

[TABLE]

are simultaneously positive. In this case, two periodic solutions will be symmetric with respect to the central one. The stability of the central periodic solution will be determined by the sign of the following values

[TABLE]

The other two symmetric periodic solutions will have their stability determined by the following coefficients

[TABLE]

When $A$ passes through zero these periodic solutions will change their stability and a Neimark-Sacker bifurcation may occur around each symmetric periodic solution. In this case, two invariant tori will emerge. In this situation the bifurcation parameters are

[TABLE]

Our first main result can be stated as follows.

Theorem 1**.**

Consider system (1) with the parameters (2) satisfying the following relation

[TABLE]

Then, the following statements hold:

• a)

If $\delta_{a}>0,$ $\delta_{b}>0,$ $\delta_{c}>0$ and $\delta_{d}>0,$ for $\varepsilon>0$ sufficiently small, system (1) will have three periodic solutions $\varphi_{0}(t,\varepsilon),$ $\varphi_{+}(t,\varepsilon)$ and $\varphi_{-}(t,\varepsilon)$ bifurcating from the origin.

• b)

If $\lambda_{1}<0$ and $\lambda_{2}<0$, then the periodic solution $\varphi_{0}(t,\varepsilon)$ will be asymptotic stable. On the other hand, if $\lambda_{1}>0$, or $\lambda_{2}>0$ the periodic solution $\varphi_{0}(t,\varepsilon)$ will be unstable. Analogously, we define

[TABLE]

If $\Re(\lambda_{+})<0$ and $\Re(\lambda_{-})<0$, then both periodic solutions $\varphi_{\pm}(t,\varepsilon)$ will be stable. In case that $\Re(\lambda_{+})>0$, or $\Re(\lambda_{-})>0$ both periodic solutions will be unstable.

• c)

If $\ell_{1,1}\neq 0$, then there exists a smooth curve $\widehat{\mu}(\varepsilon)$, satisfying $\widehat{\mu}(\varepsilon)=\varepsilon\widehat{\mu}_{1}+\mathcal{O}(\varepsilon^{2})$, such that each of the periodic orbits $\varphi_{\pm}(t,\varepsilon)$ bifurcates into an unique invariant torus whenever $\ell_{1,1}(\mu_{1}-\widehat{\mu}(\varepsilon))<0$. Moreover, if $\ell_{1,1}>0$ (resp. $\ell_{1,1}<0$) these tori are unstable (resp. asymptotically stable), whereas the periodic orbits $\varphi_{\pm}(t,\varepsilon)$ are asymptotically stable (resp. unstable).

The proof of Theorem 1 will be provided in Section 3 (see Proposition 7 and 12). The next result provides information about the bifurcation of periodic solutions associated with the parametric families $(i),$ $(ii),$ $(iv)$ and $(v)$. We introduce the following assumptions.

• H1.

If $\delta_{1}=\omega^{2}(\mu_{0}\nu_{1}-\alpha_{1}+\mu_{1}\nu_{0})-\beta_{0}h_{1}\mu_{0}\nu_{0}$, then the coefficients (2) of system (1) satisfy

[TABLE]

where $\delta_{1}\nu_{0}>0$ and $\beta_{1}h_{1}\mu_{0}\neq 0$.

• H2.

If $\delta_{2}=\omega^{2}(\mu_{0}\nu_{1}-\alpha_{1}+\mu_{1}\nu_{0})-\beta_{1}h_{0}\mu_{0}\nu_{0}$, then the coefficients (2) of system (1) satisfy

[TABLE]

Moreover, $\alpha_{1}=0$, $\alpha_{2}=0$, $\beta_{0}=\dfrac{\omega^{2}}{h_{0}}$ and $\mu_{0}=0$ where $\delta_{2}\nu_{0}>0$ and $\beta_{1}h_{0}\mu_{0}\neq 0$.

• H3.

If $\delta_{3}=\sqrt{\mu_{1}}(h_{1}k_{0}-h_{0}k_{1})h_{0}\omega$, then the coefficients (2) of system (1) satisfy

[TABLE]

Moreover, $\alpha_{1}=0,$ $\mu_{0}=0$, $\alpha_{2}=0$, where $\delta_{3}>0$ and $\left(2k_{0}\mu_{1}\nu_{1}^{2}-\alpha_{3}\omega^{2}\right)\neq 0$.

• H4.

If $\delta_{4}=\nu_{0}(\mu_{1}\nu_{0}-\alpha_{1})$, then the coefficients (2) of system (1) satisfy

[TABLE]

where $\delta_{4}>0$ and $\beta_{0}h_{1}\left(2\alpha_{1}^{2}-3\alpha_{1}\mu_{1}\nu_{0}+\mu_{1}^{2}\nu_{0}^{2}\right)\neq 0$.

Our second main result is stated as follows.

Theorem 2**.**

Assume that system (1) with coefficients (2) satisfies one of the hypothesis H1-H4. Then for $\varepsilon>0$ sufficiently small, system (1) will have a periodic solution emerging from the origin of coordinates.

The next section will be dedicated to the averaging method that will be used along the paper for obtaining our results. The proofs of Theorems 1 and 2 will be presented in Section 3. Finally, in Section 4, for each of the results stated in Theorems 1 and 2 we present a numerical example.

2. Averaging Theory: Periodic Solutions and Invariant Tori

In this work we use two aspects of the averaging theory. The first aspect of the theory is the detection of periodic solutions. In this context we will use classical averaging theory for detecting periodic solutions. Moreover, a new version of averaging theory combined with Lyapunov-Schmidt reduction will be used for finding periodic solutions that are undetectable by the classical method [2] (see Proposition 11). The second aspect of the averaging theory is the detection of invariant tori. The result we will used for this task is also a new feature of averaging theory combined with the analysis of Lyapunov constants, see [2].

This section is dedicated to present all results that will be used for proving Theorems (1) and (2).

2.1. Existence of Periodic Solutions

Consider the non-autonomous differential systems written in the following standard form:

[TABLE]

where $\Omega$ is an open bounded subset of $\mathbb{R}^{n}$ and $\varepsilon_{0}$ is a small positive real number. It is assumed that ${\bf F}_{i},$ $i=1,\ldots,k,$ and $\widetilde{\bf F}$ are sufficiently smooth functions and $T$-periodic in the variable $t.$ The periodicity of system (4) allow us to see it as defined in the cylinder $(t,{\bf x})\in\mathbb{S}^{1}\times\Omega,$ where $\mathbb{S}^{1}\equiv\mathbb{R}/T\mathbb{Z}.$

From the qualitative theory of ordinary differential equations, the solution ${\bf x}(t,{\bf z},\varepsilon)$ of (4), satisfying ${\bf x}(0,{\bf z},\varepsilon)={\bf z}$, can be written as

[TABLE]

where the expressions for ${\bf y}_{i}$ are obtained by solving recursively the system of equations obtained from (4) (see [2, Lemma 5]). Hence, the correspondent Poincaré map $\Pi({\bf z},\varepsilon)={\bf x}(T,{\bf z},\varepsilon)$ of system (4) can be written as

[TABLE]

It is well know, again from the qualitative theory of ordinary differential equations, that a fixed point of the Poincaré map (5), that is, a point satisfying $\Pi({\bf z}(\varepsilon),\varepsilon)={\bf z}(\varepsilon)$, corresponds to a branch of isolated $T$-periodic solutions of system (4). The averaging method is based on using the coefficient functions of the Poincaré map (5) in order to obtain fixed points for it.

We define by

[TABLE]

the averaged functions of order $i$ of system (4). If for some $m\in\{1,2,\ldots,k\}$ we have that ${\bf g}_{0}=\cdots={\bf g}_{m-1}=0$ and ${\bf g}_{m}\neq 0$, then a simple zero of ${\bf g}_{m}({\bf z})$ provides a branch of fixed points ${\bf z}(\varepsilon)$ for the map (5).

In fact, with this terminology the classical averaging method for finding periodic solutions can be summarized by the following theorem.

Theorem 3** ([6]).**

Assume that, for some $m\in\{1,\ldots,k\}$, ${\bf g}_{0}=\cdots{\bf g}_{m-1}=0$ and ${\bf g}_{m}\neq 0$. If there exists ${\bf z}^{*}\in\Omega$ such that ${\bf g}_{m}({\bf z}^{*})=0$ and $|D{\bf g}_{m}({\bf z}^{*})|\neq 0$, then for $|\varepsilon|\neq 0$ sufficiently small there exists an isolated $T$-periodic solution $\varphi(t,\varepsilon)$ of system (4) such that $\varphi(0,0)={\bf z}^{*}$.

Now we present the expressions of the functions ${\bf y}_{i},$ for $i=1,\dots,4$ that will be needed in this work. For the general expressions, the reader is addressed to [2]. Consider the vector ${\bf y}=(y^{1},\dots,y^{n})\in\mathbb{R}^{n}$. We denote {\bf y}^{m}=\big{(}{\bf y},\cdots,{\bf y}\big{)}\in\mathbb{R}^{mn}. Then

[TABLE]

where

[TABLE]

The previous result cannot be used when the zero of the first non vanishing averaged function ${\bf g}_{m}$ is not isolated. In this context we can use the method presented in [2] that we reproduce here.

Let $\pi:\mathbb{R}^{m}\times\mathbb{R}^{n-m}\rightarrow\mathbb{R}^{m}$ and $\pi^{\perp}:\mathbb{R}^{m}\times\mathbb{R}^{n-m}\rightarrow\mathbb{R}^{n-m}$ be the projections onto the first $m$ coordinates and onto the last $n-m$ coordinates, respectively. Denote ${\bf z}\in\Omega$ as ${\bf z}=(u,v)\in\mathbb{R}^{m}\times\mathbb{R}^{n-m}$. Let $\ell$ be the first subindex such that ${\bf g}_{\ell}\not\equiv 0$. Moreover, assume that the averaged function ${\bf g}_{\ell}$ vanishes on the set

[TABLE]

where $m<n$ are positive integers, $V$ is an open bounded subset of $\mathbb{R}^{m},$ and $\mathcal{B}\colon\overline{V}\rightarrow\mathbb{R}^{n-m}$ is a $\mathcal{C}^{k}$ function. Thus, $\mathcal{Z}$ is a set of non-isolated zeros of ${\bf g}_{\ell}$ and, consequently, Theorem 3 cannot be applied. Nevertheless, in this case we follow [2], and the Lyapunov-Schmidt reduction can be used to obtain sufficient conditions for the existence of isolated $T$-periodic solutions bifurcating from $\mathcal{Z}$ as follows. First, notice that the equation ${\bf z}=\Pi({\bf z},\varepsilon)$ is equivalent to the following system of equations

[TABLE]

Under convenient assumptions, the implicit function theorem can be used to find a function $\overline{\mathcal{B}}(u,\varepsilon),$ satisfying $\overline{\mathcal{B}}(u,0)=\mathcal{B}(u),$ which solves the second line of system (8), that is, $\overline{\mathcal{B}}(u,\varepsilon)=\pi^{\perp}\Pi(u,\overline{\mathcal{B}}(u,\varepsilon),\varepsilon)$. Then, substituting $v=\overline{\mathcal{B}}(u,\varepsilon)$ into the first line of system (8), we obtain a single equation to be solved, namely $\mathcal{F}(u,\varepsilon)=u-\pi\Pi(u,\overline{\mathcal{B}}(u,\varepsilon),\varepsilon)=0.$ Expanding $\mathcal{F}(u,\varepsilon)$ around $\varepsilon=0,$ we get the bifurcation functions $f_{i}$, for $i=1,\ldots,k-1,$

[TABLE]

which will be given in terms of the derivatives $\gamma_{j}(u)=(\partial^{j}\overline{\mathcal{B}}/\partial\varepsilon^{j})(u,0),$ $j=1,\ldots,i,$ and averaged functions ${\bf g}_{j},$ $j=1,\ldots,i+1.$ Denote $f_{0}=0.$ Notice that, if for some $r\in\{1,\ldots,k\}$ we have $f_{0}=\cdots=f_{r-1}=0$ and $f_{r}\neq 0$, then a simple zero of $f_{r}(u)$ provides a branch of zeros $u(\varepsilon)$ of $\mathcal{F}$, that is, $\mathcal{F}(u(\varepsilon),\varepsilon)=0.$ Consequently, {\bf z}(\varepsilon)=\big{(}u(\varepsilon),\overline{\mathcal{B}}(u(\varepsilon),\varepsilon)\big{)} is a branch of solutions of system (8), that is, fixed points for the map (5). Again, ${\bf x}(t,{\bf z}(\varepsilon),\varepsilon)$ corresponds to a branch of isolated $T$-periodic solutions of system (4). In this paper, we present the second order version of Theorem [2, Theorem $A$], this will be the content of Theorem 4.

The next result uses some bifurcation functions $\gamma_{i}$ and $f_{i}$ for $i=1,2$ that will be defined as follows. Denote

[TABLE]

where $\Lambda_{u}=\partial_{a}\pi{\bf g}_{\ell}(z_{u})$, $\Gamma_{u}=\partial_{b}\pi{\bf g}_{\ell}(z_{u})$, $B_{u}=\partial_{a}\pi^{\perp}{\bf g}_{\ell}(z_{u})$ and $\Delta_{u}=\partial_{b}\pi^{\perp}{\bf g}_{\ell}(z_{u})$. The bifurcation function of order $i$ $f_{i},$ for $i=1,\ldots 4,$ are defined as

[TABLE]

[TABLE]

Then we can define the function

[TABLE]

and state the following theorem.

Theorem 4**.**

Let $\Delta_{\alpha}$ denote the lower right corner $(n-m)\times(n-m)$ matrix of the Jacobian matrix $D\,g_{i}(z_{\alpha})$. In additional to hypothesis (Ha) we assume that

• $(i)$

for each $\alpha\in\mbox{{Cl}}(V)$, $\det(\Delta_{\alpha})\neq 0$;

• $(ii)$

$f_{1}$* is not identically zero;*

• $(iii)$

there exists a small parameter $\varepsilon_{0}>0$ such that for each $\varepsilon\in[-\varepsilon_{0},\varepsilon_{0}]$ there exists $a_{\varepsilon}\in V$ satisfying $\mathcal{F}^{2}(a_{\varepsilon},\varepsilon)=0$;

• $(iv)$

there exist a constant $P_{0}>0$ and a positive integer $l\leq 2$ such that

[TABLE]

Then, for $|\varepsilon|\neq 0$ sufficiently small, there exists $z(\varepsilon)$ such that $g(z(\varepsilon),\varepsilon)=0$ with $|\pi^{\perp}z(\varepsilon)-\pi^{\perp}z_{a_{\varepsilon}}|=\mathcal{O}(\varepsilon)$ and $|\pi\,z(\varepsilon)-\pi\,z_{a_{\varepsilon}}|=\mathcal{O}(\varepsilon^{k+1-l}).$

Remark 5**.**

Let ${\bf z}(\varepsilon)$ be a fixed point of the Poincaré map, using the higher order averaged functions in order to obtain an expression

[TABLE]

for the fixed point. Consider also the Jacobian matrix

[TABLE]

If all eigenvalues of $M(\varepsilon)$ are simple, we can use the Taylor expansion of the eigenvalues of $M(\varepsilon)$ in order to determine the stability of the periodic solution ${\bf x}(T,{\bf z}(\varepsilon),\varepsilon)$. This method was used for finding the stable periodic solutions presented in Section 4. Note that this can be done only if all eigenvalues are simple.

2.2. Bifurcation of an Invariant Torus

In this section we consider a parameter dependence on (4) and take $n=2$ obtaining the following two-parameter family of non-autonomous differential systems:

[TABLE]

where $\Omega$ is an open bounded subset of $\mathbb{R}^{2},$ $J$ is a open interval, and $\varepsilon_{0}$ is a small positive real number. Notice that system (11) is written in the standard form (4) of the averaging theory with an additional distinguished parameter $\mu$. It has been provided generic conditions on the averaged functions (7) guaranteeing the existence of a codimension-one bifurcation curve $\mu(\varepsilon)$ in the parameter space $(\mu,\varepsilon)$ characterized by the birth of an invariant torus from a periodic solution of (11) in $\mathbb{S}^{1}\times\Omega$. In this section, we first introduce the main result obtained in [3], and then we apply it to conclude the proof of Theorem 1.

The strategy followed by [3] consisted in looking for conditions that ensure a Neimark-Sacker Bifurcation in the Poincaré map $\Pi({\bf z},\varepsilon,\mu)={\bf x}(T,{\bf z},\varepsilon,\mu),$ ${\bf z}\in\Sigma=\{t=0\},$ of system (11). Here, ${\bf x}(t,{\bf z},\varepsilon,\mu)$ denotes the solution of (11) satisfying ${\bf x}(0,{\bf z},\varepsilon,\mu)={\bf z}.$ In discrete dynamical system theory, this bifurcation is characterized by the birth of an invariant closed curve from a fixed point, as the fixed point changes stability. As it well known, an invariant torus corresponds to an invariant closed curve $\Gamma\subset\Sigma$ of $\Pi({\bf z},\varepsilon,\mu)$, that is, $\Pi(\Gamma,\varepsilon,\mu)=\Gamma.$

Recall that the first-order averaged function can be written as

[TABLE]

Consider the following hypotheses:

• A1.

There exists a continuous curve $\mu\in J\mapsto{\bf x}_{\mu}\in\Omega,$ defined in an interval $J\ni\mu_{0},$ such that ${\bf g}_{1}({\bf x}_{\mu},\mu)=0$ for every $\mu\in J\subset\mathbb{R},$ the pair of complex conjugated eigenvalues $\alpha(\mu)\pm i\beta(\mu)$ of $D_{\bf x}{\bf g}_{1}({\bf x}_{\mu},\mu)$ satisfies $\alpha(\mu_{0})=0$ and $\beta(\mu_{0})=\omega_{0},$ and $D_{\bf x}{\bf g}_{1}({\bf x}_{\mu_{0}},0)$ is in its real Jordan normal form;

• A2.

Let $\alpha(\mu)\pm i\beta(\mu)$ be the pair of complex conjugated eigenvalues of $D_{\bf x}{\bf g}_{1}({\bf x}_{\mu},\mu)$ such that $\alpha(\mu_{0})=0,$ $\beta(\mu_{0})=\omega_{0}>0.$ Assume that $\alpha^{\prime}(\mu_{0})\neq 0$.

Finally, define the number

[TABLE]

We observe that hypothesis A1 implies the existence of a neighborhood $J_{0}\subset J$ of $\mu_{0}$ and $\varepsilon_{1},$ $0<\varepsilon_{1}<\varepsilon_{0}$ such that, for every $(\mu,\varepsilon)\in J_{0}\times(-\varepsilon_{1},\varepsilon_{1}),$ the differential equation (11) admits a unique $T$-periodic solution $\varphi(t,\mu,\varepsilon)$ satisfying $\varphi(0,\mu,\varepsilon)\to{\bf x}_{\mu}$ as $\varepsilon\to 0$ (see [3, Lemma 1]). When the differential equation (11) is defined in the extended phase space $\mathbb{S}^{1}\times\Omega,$ such a periodic solution is given by $\Phi(t,\mu,\varepsilon)=(t,\varphi(t,\mu,\varepsilon)).$

Theorem 6** ([3]).**

In addition to hypotheses A1 and A2, assume that $\ell_{1}\neq 0.$ Then, for each $\varepsilon>0$ sufficiently small, there exists a $C^{1}$ curve $\mu(\varepsilon)\in J_{0},$ with $\mu(0)=\mu_{0},$ neighborhoods $\mathcal{U}_{\varepsilon}\subset\mathbb{S}^{1}\times\Omega$ of the periodic solution $\Phi(t,\mu(\varepsilon),\varepsilon),$ and intervals $J_{\varepsilon}\subset J_{0}$ of $\mu(\varepsilon)$ for which the following statements hold.

• $(i)$

For $\mu\in J_{\varepsilon}$ such that $\ell_{1}(\mu-\mu(\varepsilon))\geq 0,$ the periodic solution $\Phi(t,\mu(\varepsilon),\varepsilon)$ is unstable (resp. asymptotically stable), provided that $\ell_{1}>0$ (resp. $\ell_{1}<0$), and the differential equation (11) does not admit any invariant tori in $\mathcal{U}_{\varepsilon}.$

• $(ii)$

For $\mu\in J_{\varepsilon}$ such that $\ell_{1}(\mu-\mu(\varepsilon))<0,$ the differential equation (11) admits a unique invariant torus $T_{\mu,\varepsilon}$ in $\mathcal{U}_{\varepsilon}$ surrounding the periodic solution $\Phi(t,\mu,\varepsilon).$ Moreover, $T_{\mu,\varepsilon}$ is unstable (resp. asymptotically stable), whereas the periodic solution $\Phi(t,\mu,\varepsilon)$ is asymptotically stable (resp. unstable), provided that $\ell_{1}>0$ (resp. $\ell_{1}<0$).

• $(iii)$

$T_{\mu,\varepsilon}$* is the unique invariant torus of the differential equation (11) bifurcating from the periodic solution $\Phi(t,\mu(\varepsilon),\varepsilon)$ in $\mathcal{U}_{\varepsilon}$ as $\mu$ passes through $\mu(\varepsilon).$*

This result will be used for proving Theorem $1(c)$ (see Proposition (12)).

3. Proofs of Theorems 1 and 2

We are going to use the classical averaging method, Theorem 3 for detecting periodic solutions stated in Theorem 1$(a)$.

Proposition 7**.**

Consider system (1) with coefficients (2) satisfying the hypotheses of Theorem 1$(a)-(b)$. Then for $|\varepsilon|\neq 0$ sufficiently small, system (1) has three periodic solutions $\varphi_{0}(t,\varepsilon)$ and $\varphi_{\pm}(t,\varepsilon)$ satisfying $\varphi_{0}(t,\varepsilon)\to(0,0,0)$ and $\varphi_{\pm}(t,\varepsilon)\to(0,0,0)$ when $\varepsilon\to 0$.

Proof.

Assume that the hypothesis in Theorem 1 hold. We will use the regular algorithm for applying averaging method. First we will use the following linear change of variables in order to have the linear part of system (1) in its real Jordan normal form

[TABLE]

In addition, rescaling the system by taking $(\overline{X},\overline{Y},\overline{Z})=\varepsilon(X,Y,Z)$, we get

[TABLE]

Now, consider the cylindrical variables

[TABLE]

Notice that $\dot{\theta}=\omega+\mathcal{O}(\varepsilon),$ which is positive for $\varepsilon>0$ sufficiently small. Therefore, taking $\theta$ as the new time of the system, system (1) becomes the following non-autonomous differential system

[TABLE]

Observe that the non-autonomous differential system (14) is written in the standard form (4) for applying the averaging theorem. Thus, identifying

[TABLE]

we compute the first-order averaged function (7), {\bf g}_{1}(r,z)=\big{(}{\bf g}_{1}^{1}(r,z),{\bf g}_{1}^{2}(r,z)\big{)}, as

[TABLE]

We are interested in the solutions of the non-linear system ${\bf g}_{1}(r,z)=(0,0)$ with $r>0$. In this case, since $\delta_{a}>0,$ $\delta_{b}>0,$ and $\delta_{c}>0$, there are three solutions $(r_{0},0)$, $(r_{1},z_{1})$ and $(r_{1},z_{2})$, namely

[TABLE]

Moreover, the Jacobian determinant of ${\bf g}_{1}$ at $(r_{0},0)$ and $(r_{1},z_{i})$ for $i=1,2$ is given respectively by

[TABLE]

and from our assumptions we have that $\det\left(r_{0},0)\right)\neq 0$ and $\det\left(r_{1},z_{i})\right)\neq 0$. Thus, the result follows by applying Theorem 3 and going back through the changes of variables (13). ∎

The bifurcation of invariant tori stated in Theorem 1$(c)$ will be proved using Theorem 6.

Proposition 8**.**

Consider $\ell_{1}$ as defined in (12) and assume that system (1) satisfies the hypotheses of Proposition 7. If $\ell_{1}\neq 0$, then there exists a smooth curve $\mu(\varepsilon),$ defined for $\varepsilon>0$ sufficiently small and satisfying

[TABLE]

and intervals $J_{\varepsilon}$ containing $\mu(\varepsilon)$ such that a unique invariant torus bifurcates from the periodic solution $\varphi(t,\mu(\varepsilon),\varepsilon)$ as $\mu_{1}$ passes through $\mu(\varepsilon).$ Such a torus exists whenever $\mu_{1}\in J_{\varepsilon}$ and $\ell_{1}(\mu_{1}-\mu(\varepsilon))<0,$ and surrounds the periodic solution $\varphi(t,\mu_{1},\varepsilon).$ In addition, if $\ell_{1}>0$ (resp. $\ell_{1}<0$) such a torus is unstable (resp. asymptotically stable), whereas the periodic solution $\varphi(t,\mu_{1},\varepsilon)$ is asymptotically stable (resp. unstable).

Proof.

Consider the first-order averaging function ${\bf g}_{1}$, defined in (15), of the non-autonomous differential system (14). The proof will be basically, putting the truncated averaging system

[TABLE]

into the correct coordinates in order to check hypotheses A1 and A2 and applying Theorem 6. We point out that we will proof our result for ${\bf y}_{\mu_{1}}=(r_{1},z_{1})$. We observe that the case ${\bf y}_{\mu_{1}}=(r_{1},z_{2})$ can be done analogously.

First, we have to write $D{\bf g}_{1}({\bf y}_{\widehat{\mu}_{1}},\widehat{\mu}_{1})$ in its normal Jordan form. Accordingly, consider the linear change of variables

[TABLE]

Denoting ${\bf x}_{\mu_{1}}=T^{-1}\,{\bf y}_{\mu_{1}}$ we have

[TABLE]

and we see that the Jacobian matrix $D\tilde{\bf g}_{1}({\bf x}_{\widehat{\mu}_{1}},\widehat{\mu}_{1})$ is in its normal Jordan form. Moreover, taking ${\bf x}=(u,v)$, we get

[TABLE]

Now, in order to check hypothesis A1, let

[TABLE]

We compute the characteristic polynomial of the Jacobian matrix $D\tilde{\bf g}_{1}({\bf x}_{\widehat{\mu}_{1}},\widehat{\mu}_{1}),$ obtaining

[TABLE]

where $\delta_{a}$, $\delta_{b}$ and $\delta_{d}$ are defined in (3). Denoting the roots of $p(\lambda)$ by $\lambda(\mu_{1})=a(\mu_{1})\pm i\,b(\mu_{1})$, it is straightforward to see that $a(\widehat{\mu}_{1})=0$ and $b(\widehat{\mu}_{1})=\omega_{0}$. So, hypothesis A1 holds.

In order to verify hypothesis A2, we compute the derivative of the real part of the eigenvalues of (3) at $\mu_{1}=\widehat{\mu}_{1}$, and we obtain the relation

[TABLE]

which verifies hypothesis A2. Finally, taking (12) into account, we compute

[TABLE]

Hence, we conclude the proof of Theorem 1$(c)$ by applying Theorem 6. ∎

The classical averaging method, Theorem 3, will be used for proving the next two propositions.

Proposition 9**.**

Assume that system (1) with coefficients in (2) satisfies hypothesis H1. Then, for $|\varepsilon|\neq 0$ sufficiently small, system (1) has a periodic solution $\varphi(t,\varepsilon)$ satisfying $\varphi(t,\varepsilon)\to(0,0,0)$ when $\varepsilon\to 0$.

Proof.

Following the same method used in the proof of (7), the first step is to write system (1) in the standard form (4) in order to use the averaging theory for detecting its periodic solutions. So we consider the linear change of variables

[TABLE]

In addition, taking $(\overline{X},\overline{Y},\overline{Z})=\sqrt{\varepsilon}(X,Y,Z)$, we see that the unperturbed system (that is, system with $\varepsilon=0$) in these new variables can be written as \big{(}\dot{X},\dot{Y},\dot{Z}\big{)}=\big{(}-\omega Y,\omega X,0\big{)}. Thus, using cylindrical coordinates $(X,Y,Z)=(r\cos\theta,r\sin\theta,z)$, we see that $\dot{\theta}=\omega+\mathcal{O}(\varepsilon),$ which is positive for $|\varepsilon|$ sufficiently small. Therefore, we take $\theta$ as the new time of the system so that system (1) becomes a $T$-periodic non-autonomous differential system.

Observe that the non-autonomous differential system (16) is written in the standard form (4) for applying the averaging theorem. Thus, identifying

[TABLE]

we compute the first order averaged function obtaining the first averaged function

[TABLE]

This function has the following simple zero

[TABLE]

In fact, we have the following Jacobian determinant

[TABLE]

which is non-zero by hypothesis. The proof of the proposition follows from Theorem 3 and going back through the changes of variables. ∎

Proposition 10**.**

Assume that system (1) with coefficients in (2) satisfies the hypothesis H2. Then, for $|\varepsilon|\neq 0$ sufficiently small, system (1) has a periodic solution $\varphi(t,\varepsilon)$ satisfying $\varphi(t,\varepsilon)\to(0,0,0)$ when $\varepsilon\to 0$.

Proof.

Again we perform the linear change of variables

[TABLE]

in order to put the linear part of system (1) in its Jordan normal form. In addition, we take $(\overline{X},\overline{Y},\overline{Z})=\sqrt{\varepsilon}(X,Y,Z)$, and the unperturbed system becomes \big{(}\dot{X},\dot{Y},\dot{Z}\big{)}=\big{(}-\omega Y,\omega X,0\big{)}. Thus, taking cylindrical coordinates $(X,Y,Z)=(r\cos\theta,r\sin\theta,z)$, we have $\dot{\theta}=\omega+\mathcal{O}(\varepsilon),$. Therefore, we can take $\theta$ as the new time of the system so that system (1) becomes the following non-autonomous differential system

[TABLE]

where $(\theta,r,z)\in\mathbb{R}\times\mathbb{R}_{+}\times\mathbb{R}$. Identifying

[TABLE]

we have that this non-autonomous differential system is in the standard form (4) for applying the averaging theorem. Thus, we compute the first-order averaged function (7) obtaining {\bf g}_{1}(r,z)=\big{(}g^{1}_{1}(r,z),g^{2}_{1}(r,z)\big{)} where

[TABLE]

This function has the following simple zero

[TABLE]

In fact, we have the following Jacobian determinant

[TABLE]

which is non-zero by hypothesis. The proof of the proposition follows from Theorem 3 and going back through the changes of variables. ∎

In order to be able to detect a periodic solution of system (1) under the hypothesis H3 we need to apply Theorem 4, that combines the Brouwer degree with the Lyapunov-Schimidt reduction.

Proposition 11**.**

Assume that system (1) with coefficients in (2) satisfies the hypothesis H3. Then, for $|\varepsilon|\neq 0$ sufficiently small, system (1) has a periodic solution $\varphi(t,\varepsilon)$ satisfying $\varphi(t,\varepsilon)\to(0,0,0)$ when $\varepsilon\to 0$.

Proof.

We start by writing the linear part of system (1) in its Jordan normal form, so consider the linear change of variables

[TABLE]

In addition, taking $(\overline{X},\overline{Y},\overline{Z})=\varepsilon(X,Y,Z)$, we see that the unperturbed system (that is, the system with $\varepsilon=0$) can be written as \big{(}\dot{X},\dot{Y},\dot{Z}\big{)}=\big{(}-\omega Y,\omega X,0\big{)}. Thus, taking cylindrical coordinates $(X,Y,Z)=(r\cos\theta,r\sin\theta,z)$, we see that $\dot{\theta}=\omega+\mathcal{O}(\varepsilon),$ which is positive for $|\varepsilon|$ sufficiently small. Therefore, by doing a time-rescaling, $\theta$ can be taken as the new time of the system so that system (1) becomes the following non-autonomous differential system

[TABLE]

where $(\theta,r,z)\in\mathbb{R}\times\mathbb{R}_{+}\times\mathbb{R}$. Due to the extension of the expressions of $F_{i}^{j}(\theta,r,z)$, $i=1,\dots,4$ and $j=1,2$, we shall omit them here. However, they are trivially computed in terms of the parameters $\omega$, $h_{i},\alpha_{i},\mu_{i},k_{i},\beta_{i}$, and $\nu_{i}.$

Observe that the non-autonomous differential system (16) is written in the standard form (4) for applying the averaging theorem. Thus, identifying

[TABLE]

we compute the first-order averaged function (7) obtaining ${\bf g}_{1}(r,z)=\left(0,0\right).$ The second averaged function is

[TABLE]

for $(r,z)\in\mathbb{R}_{+}\times\mathbb{R}.$ This function has three continuum of zeros, namely

[TABLE]

Moreover, the Jacobian matrix of ${\bf g}_{2}$ can be written as

[TABLE]

In order to apply Theorem 4, we compute the third and fourth order averaged functions using the formulas (7), obtaining {\bf g}_{3}(r,z)=\big{(}g^{1}_{3}(r,z),g^{2}_{3}(r,z)\big{)} and {\bf g}_{4}(r,z)=\big{(}g^{1}_{4}(r,z),g^{2}_{4}(r,z)\big{)}, as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We are going to find a periodic solution bifurcating from

[TABLE]

Thus we use ${\bf z}_{r_{+}}$ to compute the coefficients of the bifurcation function (9) obtaining

[TABLE]

having $\Delta_{r}=\dfrac{4\pi\mu_{1}\nu_{1}}{\omega}.$ Now we compute the zeroes of the function

[TABLE]

and we obtain

[TABLE]

where

[TABLE]

Then, under these conditionsthe hypothesis $(i)$, $(ii)$ and $(iii)$ of Theorem 4 hold. The hypothesis $(iv)$ also holds since we have

[TABLE]

where $\kappa(h_{0},h_{1},k_{0},k_{1},\mu_{1},\mu_{2},\nu_{1},\nu_{2},\alpha_{4},\omega)=$

[TABLE]

Thus for $\varepsilon>0$ sufficiently small we can find $P_{0}>0$ satisfying

[TABLE]

The proof of the proposition follows from applying Theorem 4 and going back through the changes of coordinates. The same result will be obtained in $\mathcal{Z}_{-}$, although both periodic orbits cannot coexist. For $\mathcal{Z}_{0}$ the result is not conclusive. ∎

The next result will be obtained by direct analysis of the Poincaré map of the system (1) under the hypothesis H4.

Proposition 12**.**

Assume that system (1) in coefficients (2) satisfies the hypothesis H4. Then, for $|\varepsilon|\neq 0$ sufficiently small, system (1) has a periodic solution $\varphi(t,\varepsilon)$ satisfying $\varphi(t,\varepsilon)\to(0,0,0)$ when $\varepsilon\to 0$.

Proof.

Using the linear change of variables

[TABLE]

and taking $(\overline{X},\overline{Y},\overline{Z})=\sqrt{\varepsilon}(X,Y,Z)$, the unperturbed part of system (1) becomes \big{(}\dot{X},\dot{Y},\dot{Z}\big{)}=\big{(}-\omega Y,\omega X,0\big{)}. Thus, we use cylindrical coordinates $(X,Y,Z)=(r\cos\theta,r\sin\theta,z)$, and $\dot{\theta}=\omega+\mathcal{O}(\varepsilon),$ for taking $\theta$ as the new time of the system so that system (1) becomes the following non-autonomous differential system

[TABLE]

where $(\theta,r,z)\in\mathbb{R}\times\mathbb{R}_{+}\times\mathbb{R}$. Again we omit the expressions of $F_{i}^{j}(\theta,r,z)$, $i=1,2$ because they are very large.

The non-autonomous differential system is in the standard form (4). Thus, identifying

[TABLE]

we compute the first-order averaged function (7) obtaining

[TABLE]

for $(r,z)\in\mathbb{R}_{+}\times\mathbb{R}.$ This function has a continuum of zeroes with positive $r$, namely

[TABLE]

Moreover, the Jacobian matrix of ${\bf g}_{1}$ can be written as

[TABLE]

We are going to find a fixed point for the Poincaré map

[TABLE]

As the first averaged function has no simple zero, we compute the second order averaged function {\bf g}_{2}(r,z)=\big{(}g^{1}_{2}(r,z),g^{2}_{2}(r,z)\big{)} where

[TABLE]

Thus, we use the Poincaré map to define the displacement map

[TABLE]

We consider now the function $\xi\colon\mathbb{R}^{2}\times(-\varepsilon_{0},\varepsilon_{0})\mapsto\mathbb{R}^{2}$,

[TABLE]

and composing it with the displacement map we obtain

[TABLE]

with the first coefficient function $H_{1}(r,z)=\left(H^{1}_{1}(r,z),H^{2}_{1}(r,z)\right)$ satisfying

[TABLE]

Obviously, a zero of H\big{(}(r,z),\varepsilon\big{)} is a fixed point of the Poincaré map $\Pi((r,z),\varepsilon)$. Then since the coefficient function $H^{1}_{1}(r,z)$ has the simple zero

[TABLE]

where

[TABLE]

it follows from the Implicit Function Theorem that H\big{(}(r,z),\varepsilon\big{)} has a branch of zeroes $(\tilde{r}(\varepsilon),\tilde{z}(\varepsilon))$ satisfying $(\tilde{r}(0),\tilde{z}(0))=(\tilde{r},\tilde{z}).$ Consequently, the Poincaré map has a fixed point such that

[TABLE]

The proof of the proposition follows from going back through the changes of variables. ∎

4. Numerical Examples

In this section we present a numerical example for each result stated in Section 1.

4.1. Example 1

Consider system (1) with the coefficients (2) satisfying the following relations

[TABLE]

Take $\varepsilon=\frac{1}{70},$ and we observe that $\ell_{1}=-\frac{6403\pi}{1040}.$ As stated in Theorem 1, Figure 1 shows the existence of three periodic solutions and two invariant tori coexisting near the origin of coordinates of system (1).

4.2. Example 2

Consider system (1) with the coefficients (2) satisfying the following relations

[TABLE]

Take $\varepsilon=\dfrac{1}{25}.$ In this case system (1) satisfies the hypothesis H1. Thus, as stated in Theorem 2, Figure 2 shows the existence of a periodic solutions near the origin of coordinates of system (1).

4.3. Example 3

Consider system (1) with the coefficients (2) satisfying the following relations

[TABLE]

Take $\varepsilon=\dfrac{1}{25}.$ In this case system (1) satisfies the hypothesis H2. Thus, as stated in Theorem 2, Figure 3 shows the existence of a periodic solutions near the origin of coordinates of system (1).

4.4. Example 4

Consider system (1) with the coefficients (2) satisfying the following relations

[TABLE]

Take $\varepsilon=\dfrac{1}{150}.$ In this case system (1) satisfies the hypothesis H3. Thus, as stated in Theorem 2, Figure 4 shows the existence of a periodic solution near the origin of coordinates of system (1).

4.5. Example 5

Consider system (1) with the coefficients (2) satisfying the following relations

[TABLE]

Here, we take $\varepsilon=\dfrac{1}{150}.$ In this case system (1) satisfies the hypothesis H4. Thus, as stated in Theorem 2, Figure 5 shows the existence of a periodic solution near the origin of coordinates of system (1).

Acknowledgements

This work started when MRC was doing a research visit to the Instituto Superior Técnico - Universidade de Lisboa, supported by the grant Becas Iberoamérica, Santander Investigación. MRC is gratefully indebted to Prof. C. Valls for her supervision during this period the for the fruitful collaboration. MRC is partially supported by FAPESP grants 2018/07344-0 and 2019/05657-4. CV is partially supported by FCT/Portugal through UID/MAT/ 04459/2019.