Bifurcation of Limit Cycles by Perturbing a Piecewise Linear Hamiltonian System

In this paper, we study limit cycle bifurcations for planar piecewise smooth near-Hamiltonian systems with nth-order polynomial perturbation. The piecewise smooth linear differential systems with two centers formed in two ways, one is that a center-fold point at the origin, the other is a center-fold at the origin and another unique center point exists. We first explore the expression of the first order Melnikov function. Then by using the Melnikov function method, we give estimations of the number of limit cycles bifurcating from the period annulus. For the latter case, the simultaneous occurrence of limit cycles near both sides of the homoclinic loop is partially addressed.


Introduction and Main Results
Recently, many scholars are interested in the study of piecewise smooth (PWS) dynamical systems due to their enormous potential applications in mechanics, electrical engineering, automatic control theory, see the works of di Bernardo et al. [1], Filippov [2], Andronov et al. [3], Kunze [4] and the references therein. In the qualitative theory of planar PWS systems, one of the important problem is to determine the number of limit cycles. So far as we know, for PWS near-Hamiltonian system, there have two basic methods to study the number of limit cycles. One is the averaging method established in [7,8] and the other is the Melnikov function method developed in [12,13]. Until now, there are many excellent results which have been achieved by using the two methods, see [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]28] and the references therein. In this paper, we will utilize the Melnikov function method to discuss the number of limit cycles for a PWS near-Hamiltonian system on the plane. Here, a widely studied system is the following near-Hamiltonian systeṁ where with H ± , p ± , q ± ∈ C ∞ on R 2 , ε > 0 is small and δ ∈ D ⊂ R m is a vector parameter with D compact. Then there are two corresponding subsystems as follows ẋ = H + y + ε p + (x, y, δ), y = −H + x + εq + (x, y, δ), and ẋ = H − y + ε p − (x, y, δ), on R 2 , which are called the right subsystem and the left subsystem respectively. Planar piecewise linear differential systems, including centers, have been discussed in [24][25][26][27]15,12]. In [24], Llibre et al. obtained the maximum number of non-sliding limit cycles that an L R−system can have. In [25], three limit cycles were obtained by perturbing a global center with a different piecewise linear system in each zone. In [26], Llibre et al. proved that three limit cycles can bifurcate from a piecewise linear centre, using only a first-order piecewise linear perturbation. In [15], Liang and Han studied the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized homoclinic or generalized double homoclinic loop. For a planar PWS near-Hamiltonian systems with nth-order polynomial perturbation, Liu and Han in [12] proved that the maximal number of limit cycles is n on any compact set for |ε| sufficiently small. They discussed a PWS near-Hamiltonian system with linear differential systems formed by two centers in the way that the origin is a center-fold. The PWS linear differential systems, with two zones with a, b ∈ R, and set q(x, y) = for n ≥ 1 and δ (a ± i j , b ± i j ) ∈ D ⊂ R 2(n+1)(n+2) with D compact. Then system (1) becomes ẋ = y + ε p + (x, y), y = −(x − a) + εq + (x, y), x ≥ 0, ẋ = y + ε p − (x, y), Noting a, b ∈ R and by symmetry, we just need to consider six cases of them as follows: (1) a = 0, b = 0, (2) a < 0, b = 0, (3) a > 0, b = 0, (4) a < 0, b > 0, (5) a > 0, b > 0, (6) a > 0, b < 0.
Without loss of generality, for b = 0, we can suppose a = ±1 in system (6) and consider respectively: ẋ = y + ε p + (x, y), y = −(x + 1) + εq + (x, y), x ≥ 0, ẋ = y + ε p − (x, y), and ẋ = y + ε p + (x, y), Note that for system (7)| ε=0 , there exist a family of periodic orbits as follows Obviously, system (7)| ε=0 has a period annulus h∈(0,+∞) L h . Let for h > 0, . Besides, let AB = L h |x ≥ 0 and B A = L h |x ≤ 0. Then by [12], for system (7) we have the first order Melnikov function below where From (9), it is easy to see that Then by (10) we have the first order Melnikov function of system (7) of the following form Let Z (n) denote the sharpest maximal number of isolated zeros of M(h) in h ∈ (0, +∞) for any choice of polynomials of degree n, p ± and q ± . Then for system (7) we have the first main theorem. Theorem 1. 1 For system (7), we have Z (n) = n + [ n+1 2 ] for all n ≥ 1. For system (8)| ε=0 there exist two families of periodic orbits L h outside L + 0 and which is tangent to y−axis at the origin, In this case, each periodic orbit L h of system (8)| ε=0 surrounds the origin and approaches L + 0 as h → 0 + . Clearly, the periodic orbits orientate clockwise. Let In addiction, denote by AB = L h |x ≥ 0 and B A = L h |x ≤ 0. Hence, we have two the first order Melnikov functions as follows and where From (13), one gets that Similar to the definition of Z (n), we let Z 0 (n) denote the sharpest maximal number of isolated zeros of I (h) in h ∈ (−1, 0) for any choice of polynomials of degree n, p ± and q ± . In addition, the notation l + k means that there are l limit cycles outside of L + 0 and k limit cycles inside of L + 0 at the same time. Then we have another two main results below. Theorem 1. 2 For system (8), we have n + [ n+1 2 ] ≤ Z (n) ≤ n + [ n+1 2 ] + 1 and Z 0 (n) = [ n− 1 2 ] for all n ≥ 1.
From Theorem 1.3, we have the following conjecture.
Then by the definition ofp(x, y) we havẽ From (18)-(20), we have Besides, from (20) we can derive where Then by (21), we obtain Next, by the same method as in [14], we can calculate I 1 (h) and I 0 (h) as follows respectively: and where and

Lemma 2.1
(i) For r = 2l, l ≥ 0, it holds that (ii) For r = 2l + 1, l ≥ 0, one can have where Using the formula we have from (30) It follows that where Then Further, by using the formula one gets that It follows that where Also let Then By (30) Then, by (31), (32) we have for l ≥ 1, k ≥ 1 Therefore, for l ≥ 0, k ≥ 0 This ends the proof.
From Lemma 2.1 and with a similar method as in [19], we can derive (27) as follows where Substituting (34) into (26) gives where Then, substituting (25) and (35) into (24), we obtain Theorem 2.1 The first order Melnikov function of system (7) given by (12) has the following form Furthermore, the n + [ n+1 2 ] + 1 coefficients can be taken as free parameters.
where i j = 0 for j odd and i j > 0 for j even. Noting that the curve , then by (11) and the formula (2.18), (2.19) in [14], one gets where Combing (37) and (39), we have Note that where E m×m represents the m × m unit matrix. Furthermore Then, according to the rule for derivatives of composite functions, we have It is direct that Moreover, by (40), we can get that (e 1 , e 3 , · · · , e 2[ n−1 . By (40) and (41), (23) and (36), we can see that . Therefore, from the above discussion, we can get that all of the coefficients of (38) are linearly independent and can be taken as free parameters. This ends the proof.

The Melnikov Function M(h) of System (8)
In this section, we explore the expression of the first order Melnikov function M(h) on both sides of the generalized homoclinic loop L + 0 of system (8). Firstly, we give the expression of the first order Melnikov function I (h) for h ∈ (−1, 0). (14), then by (4), (5) and simple calculation, we have from (14) cos 2s θ sin 2m θ dθ.
Secondly, we give the expression of the first order Melnikov function M(h) for h ∈ (0, +∞). For h > 0 denote by γ h the circle defined by H + (x, y) = h 2 . Let On the other hand Since γ − h can be represented as x = 1 − 1 + h − y 2 , then for I − (h), similar to (18)- (25), and by (26) we have where and where S ik (h) is given by (34). Then, after the same series of calculations as in Sect. 2, we get the following formula where Substituting (48) and (50) into (47), we obtain Combine (45), (46) and (52), we have Now we can prove the following theorem.

Theorem 3.1
The first order Melnikov function of system (8) given by (12) has the following form Moreover, the n + [ n+1 2 ] + 1 coefficients can be taken as free parameters.
Proof By calculation, we can get that b 0 = −π v * 0 . Thus, we only consider the linear dependence of the following parameters: By (12), (39) and (53) we have and By (40) and (56), we have By (40) and (57), it is easy to get that By (23) and (51), we can obtain that Then by the derivative rule for composite function, we have Hence In addition, by (40) and (56), we can get that ). By (40) and (57), ,0 ), and by (23), (51), we can see that ). Therefore, from the above discussion, we can get that the coefficients (55) can be taken as free parameters. This ends the proof.

Proof of the Main Results
For system (7), letting ρ = √ h, then (38) can be rewritten as follows Note that where Theorem 4.1 For ρ ∈ (0, +∞), the functions are linearly independent.
In order to obtain an exact upper bound of Z (n), we use the same method as Lemma 3.2 and Lemma 3.3 in [19]. Then we present two Lemmas as follows:

Lemma 4.2 Let f (n) (h) represent the nth-order derivative of f (h). (i) ([19]) For any m-degree polynomial P m (h), it holds
whereP m (h) is a polynomial of degree no more than m. (ii) For n + 1 − m ≥ 1 and n, m ∈ N, one gets where P n+m (h) is a polynomial of degree n + m.
Proof Since conclusion (i) has been proved in [19], now we just need to provide a proof for (ii). Note that Then, in accordance with Hence, for n + 1 − m ≥ 1, This ends the proof.
From [28], we can have Lemma 4.3 as follows (12) has at most k zeros in h for all δ ∈ D on the interval (0, +∞) when M(h) ≡ 0, taking into account the multiplicity, then there exist at most k limit cycles of system (1) that bifurcate from the period annulus ∪ h∈(0,+∞) L h .

Lemma 4.3 If the function M(h) defined in
Proof of Theorem 1.1 Note that where ω(h) = arctan √ h. Then we can rewrite the first order Melnikov function M(h) given by (38) as follows According to Lemma 4.2, we take the ([ n+1 2 ] + 1)th-order derivative of M(h), and then Therefore, (M(h)) ( Combining with (65), we have Z (n) = n + [ n+1 2 ] for n ≥ 1. Then by Lemma 4.3, after perturbation for system (7)| ε=0 , the sharpest maximum number of limit cycles, for any polynomials of degree n, p ± and q ± , which bifurcating from the period annulus This ends the proof.
Before we start the proof of Theorem 1.3, we first expand the first order Melnikov function of system (8) into a power series at h = 0. In terms of then, (33) can be rewritten as Substituting (69) into (54), we obtain where Set 2 + e m−1 , m even, m = 1, 2, · · · , n + 1, (72) Then we can rewrite (70) as Next, we explore the linear correlation of the coefficients in (74). Obviously, from Theorem 3.1, we can get that the coefficients are linear independent. By Lemma 3.1, we know that is full rank. Besides, (43) can be rewritten aŝ where p + i j is as before satisfying (23). Then by (77), we have Hence, by (76) and (78) we can get that In addiction, note that where G 5 is given in Theorem 3.1. Then, by (79), (80) and (71), we can obtain that where L( ) denotes by a linear combination. Further, by (73) and (81), we have Proof of Theorem 1. 3 We only choose n = 3, 5, 8 to prove the theorem, and the other cases are similar. For n = 3, together with (44), Theorem 3.1 and above discussion, we can obtain expansions of M(h) and I (h) as follows we have ρ 5 = 2 3π b 1 > 0. Then we can choose b 0 , ρ 1 , ρ 2 , ρ 3 , ρ 4 as free parameters and change them in two ways as follows.