Abstract and Applied Analysis

Volume 2014 (2014), Article ID 819798, 14 pages

http://dx.doi.org/10.1155/2014/819798

Research Article

Limit Cycle Bifurcations by Perturbing a Compound Loop with a Cusp and a Nilpotent Saddle

Huanhuan Tian and Maoan Han

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 13 April 2014; Accepted 14 June 2014; Published 16 July 2014

Academic Editor: Tonghua Zhang

Copyright © 2014 Huanhuan Tian and Maoan Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the expansions of the first order Melnikov functions for general near-Hamiltonian systems near a compound loop with a cusp and a nilpotent saddle. We also obtain formulas for the first coefficients appearing in the expansions and then establish a bifurcation theorem on the number of limit cycles. As an application example, we give a lower bound of the maximal number of limit cycles for a polynomial system of Liénard type.

1. Introduction

Consider a planar system of the form

where is a small parameter and , , and are functions in and with bounded. For , (1) becomes

which is a Hamiltonian system. As we know, the system (1) is said to be a near-Hamiltonian system. For (1), the main task is to study the number of limit cycles which are bifurcated from periodic orbits of the unperturbed system (2). On this aspect, the first order Melnikov function of (1) plays an important role. We can use the expansions of it near Hamiltonian values corresponding to a center or an invariant loop to find its zeros and hence the number of limit cycles. See a survey article [1]. There have been many works on this topic. For the study of general near-Hamiltonian systems, see [2–12]; and especially for the system (2) with the elliptic case, one can see [13–17] and references therein. In [2–4], the number of limit cycles of the system (1) near a homoclinic loop with a cusp of order one or two or a nilpotent saddle of order one (for the definition of an order of a cusp or nilpotent saddle, see [5]) was studied. In the heteroclinic case with two hyperbolic saddles, a hyperbolic saddle and a cusp of order one, or two cusps of order one or two, the number of limit cycles of the system (1) was studied in [5, 8, 9], respectively. In this paper, we suppose that the unperturbed system (2) has a compound loop consisting of a cusp of order one, a nilpotent saddle of order one, a homoclinic loop to , and two heteroclinic orbits connecting and , as shown in Figure 1. We aim to study the number of limit cycles of (1) near the loop for small.

Figure 1: Compound loop with a cusp and a nilpotent saddle.

2. Main Results with Proof

Now consider the systems (1) and (2). Suppose that (2) has a compound loop denoted by and defined by equation , where is a cusp and is a nilpotent saddle both having order one, are heteroclinic orbits satisfying and , and is a homoclinic loop to . Then, the level curves of define two families of periodic orbits and for on one side of and a family of periodic orbits for on another side of . For the definiteness, let both and exist for and exist for . Thus, we have three Melnikov functions

Let denote a closed set with diameter and with center at , . See Figure 2(a). And further introduce

Here the Cl. denotes the closure of a set. Then by (3) and (4), for sufficiently small we can write

where

(a)

(b)

Figure 2

By [5], there exist two transformations of the form

where is a matrix satisfying such that (1) becomes

where

for near . Note that for near and for near . Then we have

where denote the image of under and , , and denote the image of , , and under , respectively. Then, by using [3, 4] we can obtain the following two lemmas, respectively.

Lemma 1. Consider system (10) with and suppose (11), (13) hold. Then there are constants satisfying

such that

for ,

for , where at with , and

where

Lemma 2. Consider system (10) with and suppose (12), (15) hold. Then we have

for ,

for , where at with , , and

where , and are constants, given by

For convenience, let

Theorem 3. Assume that system (1) has a compound loop as stated before. Then, the functions given in (3) at have the following expansions:

for , and

for , where

where . In particular,

if . Here, , are constants and are given in Lemma 1.

Proof. First, by (6), (10) with , (12), (14), (29), and Theorem 2.2 in [4], we directly obtain (27) with given by (30) and (31), respectively. Then we study the expansions of and .

By (5), (7), (29), and Lemmas 1 and 2, we have

for , and

for , where

Let

It follows further that

Then by Lemma in [5], we have

It is easy to see that

Noting , by (19), (23), and (29), we have

Then by the proof of (3.13) in [3], the following equations hold:

Here, and are constants. By a similar argument used in Theorems 2.2 and 2.4 in [4], one can obtain

Here , , , are constants. Therefore, we can obtain (31) and (32). Thus we have proved Theorem 3.

In the following we use Theorem 3 to study the problem of limit cycle bifurcation near . For the sake of convenience, we say that (1) has a distribution of limit cycles if there are and limit cycles near the inside of and , respectively, and limit cycles near the outside of . Then we can prove the following theorem.

Theorem 4. Assume that system (1) has a compound loop as stated before and (26)–(28) hold. Define ,  ,  . Let there exist , such that .

(1)If , and

 then (1) can have limit cycles near for some near , where or .(2)If , and

 then (1) can have 11 limit cycles near for some near .(3)If , and

 then (1) can have 18 limit cycles near for some near .

Proof. (1) Because of the similarity in the proof, we only prove the conclusion for and omit the rest. By our assumptions, there exists such that , and

By the implicit function theorem, we can take as free parameters varying near zero. Obviously, for these parameters varying near zero we have . In the following we proceed the process by 9 steps.

Step 1. Fix and vary near .

First, for , we have by (26) for .

Let . Then for if . Thus, has a zero. Hence, for ,

(1) the condition implies a distribution of one limit cycle.

Step 2. Fix , and vary near .

First, for , we have by (26), (27), and (28) for , and for .

Let . Then for , and for if . Thus, , and each gets a zero and the zero of got in Step 1 still exists. Hence, for ,

(2) the conditions imply a distribution of 4 limit cycles.

Step 3. Fix , and vary near .

First, for , we have by (26), (27), and (28) for , and for .

Let . Then for , and for if . Thus, and each gets a new zero and the zeros got in above steps still exist. Hence, for ,

(3) the conditions imply a distribution of 6 limit cycles.

Step 4. Fix , and vary near .

First, for , we have by (26) and (28) for , and for .

Let . Then for , and for if . Thus, and each has a new zero and the zeros got in above steps still exist. Hence, for ,

(4) the conditions imply a distribution of 8 limit cycles.

Step 5. Fix , and vary near with .

First, for , we have by (26), (27), and (28) for , and for .

Let . Then

if , and

if . Thus, , and each has one more zero in the first case and and each has a new zero in the second case. And the zeros got in above steps still exist. Hence, for ,

(5i)the conditions imply a distribution of 11 limit cycles, and(5ii)the conditions imply a distribution of 10 limit cycles.

Step 6. Fix , with and vary near .

First, for , we have by (26) and (27) and for .

Let . Then and for , if . Thus, and each gets a new zero and the zeros got in above steps still exist. Hence, for ,

(6i)the conditions imply a distribution of 13 limit cycles,(6ii)the conditions imply a distribution of 12 limit cycles.

Step 7. Fix , and vary near .

First, for , we have by (26) and (28) for and for .

Let . Then

if , and

if . Thus, and each gets one more zero in the first case and only has a new zero in the second case. And the zeros got in above steps still exist. Hence, for ,

(7i)the conditions imply a distribution of 14 limit cycles,(7ii)the conditions imply a distribution of 14 limit cycles.

Step 8. Fix , and vary near .

First, for , we have by (26), (27), and (28) for , and for .

Let . Then for , and for if . Thus, , and each gets a new zero and the zeros got in above steps still exist. Hence, for ,

(8) the conditions imply a distribution of 17 limit cycles.

Step 9. Fix and vary near .

First, for , we have by (26), (27), and (28) , for , and for .

Let . Then

if ,

if and , and

if and . Thus, we have correspondingly (a) and each has a new zero, (b) and each has a new zero, or (c) and each has a new zero. And the zeros got in above steps still exist.

Hence, for ,

(9i)the conditions ,,,,, imply a distribution of 19 limit cycles,(9ii)the conditions , , ,, ,,, , imply a distribution of 19 limit cycles,(9iii)the conditions ,,,, , imply a distribution of 19 limit cycles.

Thus we get the conclusion for .

(2) By our assumptions in case (2) and the implicit function theorem we can take as free parameters varying near zero. Obviously, for these parameters varying near zero we have . By a similar argument in the above proof, we can prove that for ,

(i)the conditions ,,,, , imply a distribution of 11 limit cycles,(ii)the conditions , , , , , imply a distribution of 11 limit cycles,(iii)the conditions , ,,,, imply a distribution of 11 limit cycles.

(3) By our assumptions in case (3) and the implicit function theorem we can take as free parameters varying near zero. Obviously, for these parameters varying near zero we have . By a similar argument used in proving case (1), we can prove that for ,

(i)the conditions ,,,, imply a distribution of 18 limit cycles,(ii)the conditions ,, , , , , , , ,, imply a distribution of 18 limit cycles,(iii)the conditions , , ,, , imply a distribution of 18 limit cycles.

This completes the proof.

3. An Application

Consider a Liénard system of the form

where

System (54) is Hamiltonian with

We have the following theorem.

Theorem 5. Let denote the maximal number of limit cycles of the system (54) for small and all . Then we have , .

Proof. It is easy to verify that the unperturbed system has a compound loop with a cusp of order one and a nilpotent saddle of order one, are heteroclinic orbits satisfying and , and is a homoclinic loop to . Inside (, resp.), there is a center (, resp.).

Because of the similarity in the proof, here we only prove the case for and omit the rest of the proof.

Let . By Theorem 3, we obtain

where

Therefore,

Note that is a nilpotent saddle of order one and . By (23), we have

Making the transformation , system (54) becomes

Then we have

By (19), we have

Note that

We have if and only if , and . It implies further that

where

Similarly,

Therefore,

Let , . Furthermore, one sees that equations have the solution , which gives . And further, if . Thus, fix and take .

Then we have

Hence by Theorem 4(3), we know that there are 18 limit cycles near for some near . This ends the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The project was supported by National Natural Science Foundation of China (11271261) and FP7-PEOPLE-2012-IRSES-316338.

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