- Open Access
Homoclinic solutions for a forced generalized Liénard system
© Zhang; licensee Springer 2012
- Received: 9 February 2012
- Accepted: 1 June 2012
- Published: 29 June 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.
- bounded solution
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.
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]).
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 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 .
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.
and the conditions (H1)-(H3) hold.
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.
Let . For every , , we claim that for all . We only need to prove this claim for every solution starting with the cylindrical surface .
Then the curve cannot leave Ξ from for .
Then the curve cannot leave Ξ from for .
If , from , so the curve cannot leave Ξ from for .
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.
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.
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.
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 .
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.
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 .
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.
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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