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# Homoclinic solutions for a forced generalized Liénard system

*Advances in Difference Equations*
**volume 2012**, Article number: 94 (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.

## 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.

As indicated in [6, 7], an orbit is referred to as a *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

where $t\in \mathbf{R}$$q:\mathbf{R}\to {\mathbf{R}}^{n}$ and $V:\mathbf{R}\times {\mathbf{R}}^{n}\to \mathbf{R}$ is a differentiable function such that $V(t,0)\equiv 0$, and gave the existence of its homoclinic solutions. His strategy is to construct a sequence of periodic auxiliary systems to approximate the Hamiltonian system (1), and apply the variational method (see *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.*

where $t\in \mathbf{R}$$q:\mathbf{R}\to {\mathbf{R}}^{n}$ and $V:\mathbf{R}\times {\mathbf{R}}^{n}\to \mathbf{R}$ and $f:\mathbf{R}\to {\mathbf{R}}^{n}$, and found a solution ${q}_{0}(t)$ which satisfies

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]).

To deal with some non-Hamiltonian systems with a time-dependent force, which is independent of the state variable but cannot be regarded as a small perturbation, the topological degree theory [18, 19] and the fixed point theory [20, 21] are also applied to give the existence of periodic solutions and almost periodic solutions. Applying the Rabinowitz’s strategy and the fixed point theory, Zhang [22] considered the existence of homoclinic solutions to the equation

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.

In this article, we consider the existence of homoclinic solutions of the forced generalized Liénard type system

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].

Equation (4) is equivalent to the system

or

where

There are some results of boundedness and the oscillation of the solutions to Equation (4) [24–30]. In 2007, Hesaaraki and Moradifam [31] studied the global asymptotic and oscillation and existence of periodic solution to a type of generalized Liénard system

In 2009, by using the theory of topological degree, Zhou *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*$t\to \pm \mathrm{\infty}$.

When ${f}_{2}(x)\equiv 0$, the conditions of Theorem 1.1 are not equivalent to the conditions of the results in [22]. Obviously, let

and

Then we get

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

as shown in Figure 1, where

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 $\mathrm{\Gamma}\times \mathbf{R}$.

If $({x}_{0},{y}_{0})\in \overline{AB}$, by (H_{2}), we have

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

If $({x}_{0},{y}_{0})\in \overline{BC}$, since $x\le M$, by (H_{2}), we have $|g(x)|\ge b$. Then

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 $t>{t}_{0}$.

If $({x}_{0},{y}_{0})\in \overline{D{A}^{\prime}}$, by (H_{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.

Now we consider the following two periodic equations

and

where *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.

For each $k\in \mathbf{N}$, let ${p}_{k}:\mathbf{R}\to \mathbf{R}$ be a 2*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

or periodic systems

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

and

For $|{\dot{y}}_{k}(t)-{\dot{y}}_{k}({t}_{0})|$, since ${x}_{k}(t)$ is bounded, we can define

By the conditions (H_{2}), we can get the estimation as follows.

Let

we have

Moreover, by (H_{3}), we have

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

Hence, we obtain

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 $[a,b]$.

To be convenient, we still represent ${\{{q}_{i}\}}_{i\in \mathbf{N}}$ by $\{{q}_{k}\}$. For each $k\in \mathbf{N}$, let ${E}_{2k}\subset {C}_{b}(\mathbf{R},\mathbf{R})$, denote the Banach space of continuous 2*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.

Step 1. We show that ${q}_{0}$ is a solution of (9). For every $k\in \mathbf{N}$ and $t\in \mathbf{R}$ we have

For each fixed $a,b\in \mathbf{R}$, $a<b$, since ${q}_{k}\to {q}_{0}$ and ${p}_{k}\to p$ on $[a,b]$ uniformly, we have

on $[a,b]$ uniformly. Hence there exists a constant ${k}_{0}\in \mathbf{N}$ such that (10) can be transformed into

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 ${C}_{b}(\mathbf{R},{\mathbf{R}}^{2})$.

Step 2. We prove that ${x}_{0}(t)\to 0$, as $t\to \pm \mathrm{\infty}$. We note that

by Lemmas 2.1 and 2.2, we have

from (H_{2}), we obtain

Integrating (12) from −*k* to *k*, we have

So

Hence, by (H_{1}), there is a constant ${d}_{1}>0$ independent of *k* such that

On the other hand, we have

From Lemmas 2.1 and 2.2, there exists a constant ${m}_{1}>0$ independent of *k* such that

Let ${m}_{2}={min}_{(x,\dot{x})\in D}{g}^{\prime}(x)$, since ${g}^{\prime}(x)>0$, we have ${m}_{2}>0$ and $|g({x}_{k})|\ge {m}_{2}|{x}_{k}|$. Integrating (14) from −*k* to *k*, we have

Then there exists a constant ${d}_{2}>0$ independent of *k* such that

We note that

Obviously,

holds for every $i\in \mathbf{N}$ and $k\in \mathbf{N}$, $k\ge i$. Since ${x}_{k}\to {x}_{0}$ on $[-i,i]$ uniformly. Let $k\to +\mathrm{\infty}$, we obtain

and let $i\to +\mathrm{\infty}$, we have

Hence,

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.

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## Acknowledgements

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

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Zhang, Y. Homoclinic solutions for a forced generalized Liénard system.
*Adv Differ Equ* **2012, **94 (2012). https://doi.org/10.1186/1687-1847-2012-94

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### Keywords

- homoclinic
- bounded solution
- non-Hamiltonian
- accumulation