Limit cycle bifurcations for piecewise smooth integrable differential systems

In this paper, we study a class of piecewise smooth integrable non-Hamiltonian systems, which has a center. By using the first order Melnikov function, we give an exact number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n .

1. Introduction and the main results. Limit cycle theory of piecewise smooth differential systems which raised from nonlinear oscillations, mechanics, electrical engineering and control systems [1,2,8] has been extensively studied for many years and many methodologies have been developed. In particular, it can be seen as an extension of the Hilbert's 16th problem, which has not been solved since Hilbert proposed the 23 mathematical problems in 1990 [6]. For more details on the Hilbert's 16th problem, see for instance [7,9,10,17,18] and the references therein.
In this paper, we first give the first order Melnikov function of system (1.3). Hence, we make the following assumptions for system (1.
Each of the closed curves L h is piecewise smooth, in general. Further, without loss of generality, suppose that L h has a clockwise orientation.
From Theorem 1.1 in Liu and Han [14], we get the first order Melnikov function M (h) of system (1.3) as follow.
(1.4) Remark 1. By Lemma 2.2 in [12], we have In the following, we consider the following piecewise smooth integrable differential system System (1.5) ε=0 has the first integral H ± (x, y) = x 2 + y 2 with respect to x ≥ 0 and x < 0. The origin is a center. It is worth noting that system (1.5) ε=0 has an invariant straight line ax 2 + 1 = 0 (resp. bx 2 + 1) for a < 0 (resp. b < 0). Denote Let H(n) denote the maximum number of limit cycles of (1.5) bifurcated from the period annulus 0<h<h0 L h for all possible f ± (x, y) and g ± (x, y) satisfying (1.6) up to the first order Melnikov function in ε, where h 0 = min{h 1 , h 2 }. Our main result is the following theorem.
2. The first order Melnikov function. In this section, we give an expression of the first order Melnikov function M (h) of system (1.5) for 0 < h < h 0 . Note that H + y (0, y) ≡ H − y (0, y). Then by Proposition 1.1, we have the first order Melnikov function of system (1.5) satisfying where 0 < h < h 0 , and cos i θ sin j θ 1 + bh cos 2 θ dθ, 1,j = 0 and σ 0,0 = τ 0,0 = 0. Since the coefficients a ± i,j and b ± i,j are arbitrary fir i, j ∈ N, σ i,j and τ i,j are also arbitrary. Furthermore, for the sake of convenience, we denote Lemma 2.1. Suppose that ab = 0, the following equalities hold:
Integrating the above equality from − π 2 to π 2 , we obtain the conclusion (iii). Conclusion (iv) can be proved similarly.

Remark 2.
Since σ i,j and τ i,j are arbitrary, S i,j and T i,j are also arbitrary. It is easy to get that N (i) = 0. So a ibi can be chosen arbitrary. In a similar way, we can prove that c i ,b i are arbitrary.
In the following, we show that the functions defined in (2.6) and (2.7) are linearly independent and suppose that h ∈ C. From Lemma 2.1 (v), we have I 0,0 (h) = π √ 1+ah with a > 0. Therefore, I 0,0 (h) can be analytically extended to the complex domain D 1 = C\{h ∈ R, h ≤ −h 1 } with a > 0. In a similar way, J 0,0 (h) can be analytically extended to the complex domain D 2 = C\{h ∈ R, h ≥ h 2 } with b > 0. For h < −h 1 , we denote I ± 0,0 (h) by the analytic continuation of I 0,0 (h) along an arc such that Im(h)> 0 (resp. Im(h)< 0). For other functions, we will use similar notations.