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Homoclinic solutions for a forced generalized Liénard system
Advances in Difference Equations volume 2012, Article number: 94 (2012)
In this article, we find a special class of homoclinic solutions which tend to 0 as , for a forced generalized Liénard system . 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.
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  considered a nonautonomous Hamiltonian system
where and is a differentiable function such that , 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  considered (1) with a bounded time-dependent force i.e.
where and and , and found a solution which satisfies
In addition, they also found in  such a kind of special solutions for a similar equation to (2). As pointed in , the limit 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  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 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 . When , the equation becomes the equation in .
Equation (4) is equivalent to the system
There are some results of boundedness and the oscillation of the solutions to Equation (4) [24–30]. In 2007, Hesaaraki and Moradifam  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. 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 ) 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: (H1) = p is nonzero and continuous bounded function on R and , where , such that , .; (H2) = For each , there exists a constant such that () and , where , .; (H3) = , , and for each ..
Our main result will be given as follows.
Theorem 1.1 Suppose that conditions (H1)-(H3) hold. Then system (4) has a nontrivial homoclinic solution, which satisfies thatas.
When , the conditions of Theorem 1.1 are not equivalent to the conditions of the results in . Obviously, let
Then we get
and the conditions (H1)-(H3) hold.
2 Proof of theorem
Lemma 2.1 Suppose that the conditions (H1)-(H3) 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 . For every , , we claim that for all . We only need to prove this claim for every solution starting with the cylindrical surface .
If , by (H2), we have
Then the curve cannot leave Ξ from for .
If , since , by (H2), we have . Then
Then the curve cannot leave Ξ from for .
If , from , so the curve cannot leave Ξ from for .
If , by (H2),(H3), we have
Then the curve cannot leave Ξ from for .
Since the region D is symmetrical, we can prove that the curve cannot leave Ξ from for . Then we complete the proof of the claim and this lemma. □
From the combination of Lemma 2.2 and the Massera’s theorem (see ), we have the following lemma.
Lemma 2.2 Suppose that (H1)-(H3) hold, and p is a T-periodic function. Then (4) has a T-periodic solutionsatisfying, where B depends on D.
Remark 2.3 From the construction of D, we can take as a value of B in Lemma 2.2.
Now we consider the following two periodic equations
where p is -periodic function and ϕ is -periodic function. Suppose that all conditions in Theorem 2.1 hold and
Then, we obtain two periodic solutions of (7) and of (8), from the construction of D in Theorem 2.1, we can obtain two Jordan domains of (7) and of (8) such that . The trajectory of is contained in and the trajectory of is contained in . Hence, we can find a region which is independent of and contains both the two trajectories.
For each , let be a 2k-periodic function such that for all . Since as indicated in (H1), we see that is continuous on the whole R. Now, we consider a series of periodic equations
or periodic systems
Noting that for every , and from the discussion above we can get results as follows.
Lemma 2.4 Suppose that conditions (H1)-(H3) hold. Then system (9) possesses a 2k-periodic solutionfor every. Moreover, there exists a constant B independent of k such that.
Now, for the sequence of periodic solutions determined in Lemma 2.4, we obtain the following lemma by using the Ascoli-Arzela theorem.
Lemma 2.5 There exist a subsequenceofand continuous functionsandsuch that, , as, in.
Proof For each , , by Lemma 2.4, there exists a constant such that . We note that
For , since is bounded, we can define
By the conditions (H2), we can get the estimation as follows.
Moreover, by (H3), we have
Since p is continuous and bounded, there is a constant such that
Hence, we obtain
So and are both equicontinuous. On the other hand, from Lemma 2.4, is bounded uniformly, and so does . Hence we obtain the existence of a subsequence convergent to a certain in by Ascoli-Arzela theorem. □
Remark 2.6 We cannot conclude that the convergence of the subsequence is uniform for . However, for every , , the uniform convergence holds on .
To be convenient, we still represent by . For each , let , denote the Banach space of continuous 2k-periodic functions on R with values in R under the norm
Now, we prove the main result that is the desired homoclinic solution of (5).
Lemma 2.7 The functiondetermined 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 is a solution of (9). For every and we have
For each fixed , , since and on uniformly, we have
on uniformly. Hence there exists a constant such that (10) can be transformed into
for each and . So is continuous on for . Note the fact that is a derivative of in for every and converges to uniformly in by Lemma 2.5. Since (11) holds and uniformly on , we have in . Because a and b are arbitrary, we conclude that in R and satisfies (5). Moreover, we have actually proved that converges to in .
Step 2. We prove that , as . We note that
by Lemmas 2.1 and 2.2, we have
from (H2), we obtain
Integrating (12) from −k to k, we have
Hence, by (H1), there is a constant independent of k such that
On the other hand, we have
From Lemmas 2.1 and 2.2, there exists a constant independent of k such that
Let , since , we have and . Integrating (14) from −k to k, we have
Then there exists a constant independent of k such that
We note that
holds for every and , . Since on uniformly. Let , we obtain
and let , we have
This implies that as . Similarly, as .
Step 3. implies that 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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This work was supported by the Sichuan Provincial Department of Education Fund (12ZA068), and the projects of Leshan Normal University (Z1164, Z1006).
The author declares that he has no competing interests.
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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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