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# Homoclinic solutions for a forced generalized Liénard system

- Yongxin Zhang
^{1}Email author

**2012**:94

https://doi.org/10.1186/1687-1847-2012-94

© Zhang; licensee Springer 2012

**Received:**9 February 2012**Accepted:**1 June 2012**Published:**29 June 2012

## Abstract

In this article, we find a special class of homoclinic solutions which tend to 0 as $t\to \pm \mathrm{\infty}$, for a forced generalized Liénard system $\ddot{x}+{f}_{1}(x)\dot{x}+{f}_{2}(x){\dot{x}}^{2}+g(x)=p(t)$. Since it is not a small perturbation of a Hamiltonian system, we cannot employ the well-known Melnikov method to determine the existence of homoclinic solutions. We use a sequence of periodically forced systems to approximate the considered system, and find their periodic solutions. We prove that the sequence of those periodic solutions has an accumulation which gives a homoclinic solution of the forced Liénard type system.

**MSC:**34A34, 34C99.

## Keywords

- homoclinic
- bounded solution
- non-Hamiltonian
- accumulation

## 1 Introduction

As a special bounded solution, homoclinic solution is one of important subject in the study of qualitative theory of differential equations. In recent decades, many works (see *e.g.*, [1–5]) contributed to homoclinic solutions and heteroclinic solutions for small perturbation of integrable systems, where either the Melnikov method or the Liapunov-Schmidt reduction was used.

*heteroclinic orbits*if it connects two different equilibria. It is called a

*homoclinic orbit*if the two equilibria coincide. For autonomous Hamiltonian systems homoclinic (heteroclinic) orbits can be found from the invariant surfaces (curves) of identical energy containing saddles. In 1990, Rabinowitz [8] considered a nonautonomous Hamiltonian system

*e.g.*, [9, 10]) to obtain periodic solutions for those auxiliary equations, and prove that the desired homoclinic solution is just an accumulation of those periodic solutions. Later, several different types of Hamiltonian system were also studied for homoclinic orbits (see

*e.g.*, [11, 12]). Based on these works, some efforts were made to find homoclinic orbits for nonlinear systems with a time-dependent force. Izydorek and Janczewska [13] considered (1) with a bounded time-dependent force $f(t)$

*i.e.*

In addition, they also found in [14] such a kind of special solutions for a similar equation to (2). As pointed in [14], the limit $(0,0)$ is not a solution of the system, they called the solution in (3) is *homoclinic* to zero. Later, some authors studied the existence of this special solution of some Hamiltonian systems (see *e.g.*, [14–17]).

by studying the convergence of a series of periodic solutions to the auxiliary periodic systems, he got a homoclinic solution which is an accumulation of the series of periodic solutions.

where ${f}_{1}$${f}_{2}$ and *g* are continuous functions on **R** and *p* is a bounded continuous function on **R**. This generalized equation is frequently encountered as a mathematical model of most dynamics processes in electromechanical systems of physics and engineering [23]. When ${f}_{2}(x)\equiv 0$, the equation becomes the equation in [22].

*et al.*[32] got the uniqueness of periodic solution to the system

To study the existence of homoclinic solutions to Equation (4), we still use a sequence of periodic forced systems to approximate Equation (4), and find their periodic solutions. Because our system and those approximating systems are not Hamiltonian, we use the fixed point theory and Massera’s theorem (see [33]) instead of the variational method in finding those periodic solutions. We prove that the sequence of those periodic solutions has an accumulation which gives an homoclinic solution of the forced Liénard type system. We need the following hypotheses: (H_{1}) = *p* is nonzero and continuous bounded function on **R** and $\parallel p\parallel :={sup}_{t\in \mathbf{R}}|p(t)|=b>0$, where $b\le a,1\le a:={sup}_{t\in \mathbf{R}}a(x)<+\mathrm{\infty}$, such that $k\in \mathbf{Z}$, $p(-k)=p(k)$.; (H_{2}) = For each $x,y\in \mathbf{R}$, there exists a constant $r>0$ such that $|{f}_{i}(x)-{f}_{i}(y)|\le r|x-y|$ ($i=1,2$) and ${f}_{1}(x)-|{f}_{2}(x)|(M+\frac{aM}{2})\ge a$, where $M=max\{{M}_{0},2,b\}$, ${M}_{0}={g}^{-1}(b)$.; (H_{3}) = $g\in {C}^{1}(\mathbf{R},\mathbf{R})$, $g(0)=0$, $g(\pm \mathrm{\infty})=\pm \mathrm{\infty}$ and $0<{g}^{\prime}(x)\le \frac{a}{M}$ for each $x\in \mathbf{R}$..

Our main result will be given as follows.

**Theorem 1.1** *Suppose that conditions* (*H*_{1})-(*H*_{3}) *hold*. *Then system* (4) *has a nontrivial homoclinic solution*$x(t)$, *which satisfies that*$(x(t),\dot{x}(t))\to (0,0)$*as*$t\to \pm \mathrm{\infty}$.

and the conditions (H_{1})-(H_{3}) hold.

## 2 Proof of theorem

**Lemma 2.1** *Suppose that the conditions* (*H*_{1})-(*H*_{3}) *hold*, *there is a region* *D* *surrounded by a Jordan curve*, *such that every solution of* (4) *which starts from the point of* *D* *is bounded uniformly*.

*Proof*Let

*D*be the closure of the region surrounded by the closed curve

Let $\mathrm{\Xi}:=\{(x,y,t)|(x,y)\in D,t\in \mathbf{R}\}$. For every ${t}_{0}\in \mathbf{R}$, $({x}_{0},{y}_{0})\in \mathrm{\Gamma}$, we claim that ${q}_{k}(t,{t}_{0},{x}_{0},{y}_{0})\in \mathrm{\Xi}$ for all $t>{t}_{0}$. We only need to prove this claim for every solution starting with the cylindrical surface $\mathrm{\Gamma}\times \mathbf{R}$.

Then the curve cannot leave Ξ from $\overline{AB}\times \mathbf{R}$ for $t>{t}_{0}$.

Then the curve cannot leave Ξ from $\overline{BC}\times \mathbf{R}$ for $t>{t}_{0}$.

If $({x}_{0},{y}_{0})\in \overline{CD}$, from $\dot{x}=y>0$, so the curve cannot leave Ξ from $\overline{CD}\times \mathbf{R}$ for $t>{t}_{0}$.

_{2}),(H

_{3}), we have

Then the curve cannot leave Ξ from $\overline{D{A}^{\prime}}\times \mathbf{R}$ for $t>{t}_{0}$.

Since the region *D* is symmetrical, we can prove that the curve cannot leave Ξ from $\overline{{A}^{\prime}{B}^{\prime}}\cup \overline{{B}^{\prime}{C}^{\prime}}\cup \overline{{C}^{\prime}{D}^{\prime}}\cup \overline{{D}^{\prime}A}\times \mathbf{R}$ for $t>{t}_{0}$. Then we complete the proof of the claim and this lemma. □

From the combination of Lemma 2.2 and the Massera’s theorem (see [33]), we have the following lemma.

**Lemma 2.2** *Suppose that* (*H*_{1})-(*H*_{3}) *hold*, *and* *p* *is a* *T*-*periodic function*. *Then* (4) *has a* *T*-*periodic solution*$x(t)$*satisfying*$|x(t)|+|\dot{x}(t)|\le B$, *where* *B* *depends on* *D*.

**Remark 2.3** From the construction of *D*, we can take $(2+\frac{1}{a}+\frac{a}{2})M$ as a value of *B* in Lemma 2.2.

*p*is ${\omega}_{1}$-periodic function and

*ϕ*is ${\omega}_{2}$-periodic function. Suppose that all conditions in Theorem 2.1 hold and

Then, we obtain two periodic solutions $({x}_{1}(t),{\dot{x}}_{1}(t))$ of (7) and $({x}_{2}(t),{\dot{x}}_{2}(t))$ of (8), from the construction of *D* in Theorem 2.1, we can obtain two Jordan domains ${D}_{p}$ of (7) and ${D}_{\varphi}$ of (8) such that ${D}_{p}\subset {D}_{\varphi}$. The trajectory of $({x}_{1}(t),{\dot{x}}_{1}(t))$ is contained in ${D}_{p}$ and the trajectory of $({x}_{2}(t),{\dot{x}}_{2}(t))$ is contained in ${D}_{\varphi}$. Hence, we can find a region ${D}_{0}$ which is independent of ${D}_{p}$ and contains both the two trajectories.

*k*-periodic function such that ${p}_{k}(t)=p(t)$ for all $t\in [-k,k]$. Since $p(-k)=p(k)$ as indicated in (H

_{1}), we see that ${p}_{k}$ is continuous on the whole

**R**. Now, we consider a series of periodic equations

Noting that ${sup}_{t\in \mathbf{R}}|{p}_{k}(t)|\le {sup}_{t\in \mathbf{R}}|p(t)|<+\mathrm{\infty}$ for every $k\in \mathbf{N}$, and from the discussion above we can get results as follows.

**Lemma 2.4** *Suppose that conditions* (*H*_{1})-(*H*_{3}) *hold*. *Then system* (9) *possesses a* 2*k*-*periodic solution*${q}_{k}$*for every*$k\in \mathbf{N}$. *Moreover*, *there exists a constant* *B* *independent of* *k* *such that*$|{q}_{k}|:={max}_{t\in [-k,k]}\{|{x}_{k}(t)|,|{y}_{k}(t)|\}\le B$.

Now, for the sequence of periodic solutions ${\{{q}_{k}\}}_{k\in \mathbf{N}}$ determined in Lemma 2.4, we obtain the following lemma by using the Ascoli-Arzela theorem.

**Lemma 2.5** *There exist a subsequence*${\{{q}_{i}\}}_{i\in \mathbf{N}}$*of*${\{{q}_{k}\}}_{k\in \mathbf{N}}$*and continuous functions*${q}_{0}:\mathbf{R}\to {\mathbf{R}}^{2}$*and*${\tilde{q}}_{0}:\mathbf{R}\to {\mathbf{R}}^{2}$*such that*${q}_{i}\to {q}_{0}$, ${\dot{q}}_{i}\to {\tilde{q}}_{0}$, *as*$i\to +\mathrm{\infty}$, *in*${C}_{b}(\mathbf{R},{\mathbf{R}}^{2})$.

*Proof*For each $t,{t}_{0}\in \mathbf{R}$, $t\ge {t}_{0}$, by Lemma 2.4, there exists a constant ${B}_{1}>0$ such that $|-{f}_{1}({x}_{k}(s)){y}_{k}(s)-{f}_{2}(x(s)){y}_{k}^{2}(s)-g({x}_{k}(s))|\le {B}_{1}$. We note that

_{3}), we have

*p*is continuous and bounded, there is a constant ${r}_{3}>0$ such that

So ${\{{q}_{k}\}}_{k\in \mathbf{N}}$ and ${\{{\dot{q}}_{k}\}}_{k\in \mathbf{N}}$ are both equicontinuous. On the other hand, from Lemma 2.4, ${\{{q}_{k}\}}_{k\in \mathbf{N}}$ is bounded uniformly, and so does ${\{{\dot{q}}_{k}\}}_{k\in \mathbf{N}}$. Hence we obtain the existence of a subsequence ${\{{q}_{i}\}}_{i\in \mathbf{N}}$ convergent to a certain ${q}_{0}:=({x}_{0}(t),{y}_{0}(t))$ in ${C}_{b}(\mathbf{R},{\mathbf{R}}^{2})$ by Ascoli-Arzela theorem. □

**Remark 2.6** We cannot conclude that the convergence of the subsequence ${\{{q}_{i}\}}_{i\in \mathbf{N}}$ is uniform for $t\in \mathbf{R}$. However, for every $a,b\in \mathbf{R}$, $a<b$, the uniform convergence holds on $[a,b]$.

*k*-periodic functions on

**R**with values in

**R**under the norm

Now, we prove the main result that ${q}_{0}$ is the desired homoclinic solution of (5).

**Lemma 2.7** *The function*${q}_{0}$*determined by Lemma* 2.5 *is a nontrivial homoclinic solution of* (5).

*Proof* The proof will be divided into three steps.

for each $k\ge {k}_{0}$ and $t\in [a,b]$. So ${\dot{q}}_{k}$ is continuous on $[a,b]$ for $k\ge {k}_{0}$. Note the fact that ${\dot{q}}_{k}$ is a derivative of ${q}_{k}$ in $[a,b]$ for every $k\ge {k}_{0}$ and ${\dot{q}}_{k}$ converges to ${\tilde{q}}_{0}$ uniformly in $[a,b]$ by Lemma 2.5. Since (11) holds and ${q}_{k}\to {q}_{0}$ uniformly on $[a,b]$, we have $w={\dot{q}}_{0}$ in $(a,b)$. Because *a* and *b* are arbitrary, we conclude that $w={\dot{q}}_{0}$ in **R** and ${q}_{0}$ satisfies (5). Moreover, we have actually proved that $\{{q}_{k}\}$ converges to ${q}_{0}$ in ${C}_{b}(\mathbf{R},{\mathbf{R}}^{2})$.

_{2}), we obtain

*k*to

*k*, we have

_{1}), there is a constant ${d}_{1}>0$ independent of

*k*such that

*k*such that

*k*to

*k*, we have

*k*such that

This implies that ${x}_{0}(t)\to 0$ as $t\to \pm \mathrm{\infty}$. Similarly, ${y}_{0}(t)\to 0$ as $t\to \pm \mathrm{\infty}$.

Step 3. $p(t)\not\equiv 0$ implies that ${q}_{0}$ is nontrivial.

We complete this lemma. □

Finally, Theorem 1.1 is proved by summarizing the results in Lemmas 2.1 and 2.7.

## Declarations

### Acknowledgements

This work was supported by the Sichuan Provincial Department of Education Fund (12ZA068), and the projects of Leshan Normal University (Z1164, Z1006).

## Authors’ Affiliations

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