Limit Cycles for a Class of Continuous and Discontinuous Cubic Polynomial Differential Systems

We study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of isochronous cubic polynomial centers x˙=y(-1+2αx+2βx2),y˙=x+α(y2-x2)+2βxy2,α∈R,β<0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{x}=y(-1+2\alpha x+2\beta x^2),\quad \dot{y}=x+\alpha (y^2-x^2)+2\beta xy^2, \quad \alpha \in \mathbb {R},\,\beta <0, \end{aligned}$$\end{document}when it is perturbed inside the classes of all continuous and discontinuous cubic polynomial differential systems with two zones of discontinuity separated by a straight line. We obtain that this number is 3 for the perturbed continuous systems and at least 12 for the discontinuous ones using the averaging method of first order.


Introduction and Statement of the Main Results
One of the main open problems in the qualitative theory of real planar differential systems is the determination of their limit cycles. A classical way to produce limit cycles is perturbing a system which has a center. Thus the limit cycles bifurcate in the perturbed system from some of the periodic orbits of the period annulus of the center of the unperturbed system, see for instance Pontrjagin [27], the second part of the book [7] and the hundreds of references quoted there. In this paper we shall perturb isochronous centers. The perturbation of some isochronous centers have already been studied see for instance [6,17,18,20]. For a survey on isochronous centers see [5].
In [4] the authors studied some classes of isochronous cubic polynomial differential systems. In particular they obtained the familẏ x = y(−1 + 2αx + 2βx 2 ) = P(x, y), where α ∈ R and β < 0. This family has a rational first integral of degree 2, see system (iv) of Theorem 7 in [4], see also [21,22]. An open question is: What happens with the periodic orbits of the unperturb system (1) when it is perturbed inside the class of all cubic continuous and of all cubic discontinuous polynomial differential systems with two zones of discontinuity separated by a straight line? This question restricted to the continuous polynomial differential systems is related with the 16-th Hilbert problem, which ask for the maximum number of limit cycles that a polynomial differential system of a given degree can have, see for more details [14,15,17,18].
The same question restricted to the discontinuous polynomial differential systems is interesting for the engineering working in control theory, electrical circuits and mechanical problems, because this kind of differential systems appears there in a natural way, see for instance the book [8] and the hundreds of references quoted there.
More precisely, we consider the following continuous polynomial differential systemsẋ = y(−1 + 2αx + 2βx 2 ) + εp 1 (x, y), y = x + α(y 2 − x 2 ) + 2βx y 2 + εq 1 (x, y), (2) and the discontinuous polynomial differential systems where X 1 (x, y) = y(−1 + 2αx + 2βx 2 ) + εp 1 (x, y) x + α(y 2 − x 2 ) + 2βx y 2 + εq 1 (x, y) , with p 1 (x, y) = a 1 x + a 2 y + a 3 x 2 + a 4 x y + a 5 y 2 + a 6 x 3 + a 7 x 2 y + a 8 x y 2 + a 9 y 3 , 8 x y 2 + b 9 y 3 , p 2 (x, y) = c 1 x + c 2 y + c 3 x 2 + c 4 x y + c 5 y 2 + c 6 x 3 + c 7 x 2 y + c 8 x y 2 + c 9 y 3 , In this paper we study the maximum number of limit cycles of systems (2) and (3) which can be obtained using the averaging theory of first order. There are essentially four methods for determining the number of limit cycles which bifurcate from the periodic orbits surrounding a center. The first method is based in the Poincaré return map, see for instance [2,6]. The second uses the Poincaré-Pontrjagin-Melnikov integrals or the Abelian integrals. In the plane these two methods are equivalent, see for instance section 5 of Chapter 6 of [1] and section 6 of Chapter 4 of [13]. The third method uses the inverse integrating factor, see section 6 of [9] or [10,11,30]. The main tool of the last method is the averaging theory, see for example [3,28,29]. From [3] it follows easily that in the plane the method of the Abelian integrals is equivalent to the averaging method of first order. While the first two methods only provide information on the number of periodic orbits of the unperturbed system that become limit cycles after the perturbation, the last two methods also provide the shape of the bifurcated limit cycles up to the order of the perturbation parameter, see for instance [10][11][12]19].
In what follows we state our main results.
Theorem 1 For |ε| = 0 sufficiently small the maximum number of limit cycles of system (2), bifurcating from the periodic solutions of isochronous center (1), is at most 3 using the averaging theory of first order, and this number is reached.

Theorem 2
For |ε| = 0 sufficiently small the maximum number of limit cycles of system (3), bifurcating from the periodic solutions of isochronous center (1), is at least 12 using the averaging theory of first order.
Theorem 2 is proved in Sect. 4. In Sect. 2 we present the basic results that we need for proving Theorems 1 and 2.

Preliminaries
In this section we give some known results that we shall need for proving our results.
The following theorem provides periodic solutions of a periodic continous differential system. See [29] for a proof. Consider the differential equatioṅ with x ∈ D, where D is an open subset of R n , and t ≥ 0. Moreover we assume that F(t, x) is T -periodic in t. Separately, we consider in D the averaged differential equationẏ where Theorem 3 Consider the two initial value problems (4) and (5). Suppose: Then the following statements hold.
If p is an equilibrium point of the averaged Eq. (5) and The stability or instability of the periodic solution x(t, ε) is given by the stability or instability of the equilibrium point p of the averaged system (5). In fact the equilibrium point p has the stability behavior of the Poincaré map associated to the periodic solution x(t, ε).
The following theorem is a discontinuous version of previous theorem which provides periodic solutions of a periodic discontinuous differential system. See [23] for a proof.
Theorem 4 Consider the following discontinuous differential systeṁ with Define the averaged function f : D → R n as Assume the following three conditions.
the Brouwer degree of f at a is not zero).
If the function f of Theorem 4 is of class C 1 , then it is sufficient to see that the Jacobian of the function f evaluated at a is non-zero for showing that d B ( f, V, a) = 0. For more details see Theorem 1.1.2 of [26].
Consider a planar systemẋ where P, Q : R 2 → R are continuous functions. Suppose that system (7) has a continuous family of ovals where H is a first integral of (7). Consider the following perturbations of system (7) x = P(x, y) + εp(x, y),ẏ = Q(x, y) + εq(x, y), where p, q : R 2 → R are continuous functions.
The next theorem (see Theorem 5.2 of [3] for a proof) provides a tool for transforming the perturbed system (8) in the standard form of the averaging theory given in Theorem 3.

Theorem 5
Consider system (7) and its first integral H . Assume that x Q(x, y) − y P(x, y) = 0 for all (x, y) in the period annulus formed by the ovals for all R ∈ ( √ h 1 , √ h 2 ) and all ϕ ∈ [0, 2π). Then the differential equation which describes the dependence between the square root of the energy R = √ h and the angle ϕ for system (8) is where μ = μ(x, y) is the integrating factor of system (7) corresponding to the first integral H , x = ρ(R, ϕ) cos ϕ and y = ρ(R, ϕ) sin ϕ.
We recall that μ is the integrating factor corresponding to the first integral H of system (7) if Let I be a real interval and let f 0 , . . . , f n : I → R be functions. We say that f 0 , . . . , f n are linearly independent functions if and only if The next result well-known can be found in Proposition 1 of [24].
Proposition 6 If f 0 , . . . , f n are linearly independent then there exist s 1 , s 2 , . . . , s n ∈ I and λ 0 , . . . , λ n ∈ R such that for every j ∈ {1, . . . , n} we have The functions ( f 0 , f 1 , . . . , f n ) defined on I form an Extended Chebyshev system or ET-system on I , if and only if any nontrivial linear combination of these functions has at most n zeros counting their multiplicities and this number is reached. We say that F is an Extended Complete Chebyshev system or an ECT-system on I if and only if for any k ∈ {0, 1, . . . , n}, ( f 0 , f 1 , . . . , f k ) form an ET-system. For proving that ( f 0 , f 1 , . . . , f k ) is an ECT-system on I is sufficient and necessary to prove that the Wronskian of the functions ( f 0 , . . . , f k ) with respect to s. We remember that the definition of the Wronskian is .
For more details on ECT-system see [16].
Since R ∈ (0, √ −1/(2β)) the previous two equations do not have solutions. So x Q(x, y) − y P(x, y) = 0 in the period annulus of the unperturbed center (1). Now we compute the averaged function f : (0, By computing the previous integral, we obtain that f (R) The previous computations were verified using the software Mathematica. The zeros of the function f correspond to zeros of the function N . In order to find the maximum number of zeros of f , we have to prove that (g 0 , g 1 , g 2 , g 3 ) is an ECT-system, and this is the case if W (g 0 , . . . , g k )(R) = 0, for 0 ≤ k ≤ 3, where W (g 0 , . . . , g k )(R) denotes the Wronskians of the functions (g 0 , . . . , g k ). More precisely we have Since β < 0 and R ∈ (0, √ −1/(2β)), the first three Wronskians are nonzero. Now, to get the zeros of W (g 0 , . . . g 3 )(R) is equivalent to solve the following equation We take the square in both sides of the previous equation and after some simplifications we get which is it impossible because R ∈ (0, √ −1/(2β). Thus, the Wronskian W (g 0 , . . . g 3 ) (R) is non-zero in (0, √ −1/(2β)). Hence, since (g 0 , g 1 , g 2 , g 3 ) is an ECT-system f has at most 3 zeros and they are reached. System (11) is analytic and satisfies the assumptions of Theorem 3. Therefore the zeros of f correspond to periodic orbits of perturbed system (2) and Theorem 1 follows.
We note that a simple zero of a function of one variable always has non-zero Brouwer degree, for more details see [25]. So by (ii) we only need to look for the simple zeros of the function f given in (13).
The function h(ϕ, R) = ρ(ϕ, R) sin ϕ is equal to zero if and only if ϕ = 0 or ϕ = π . Moreover we can check that (dh/dϕ)(0, R) = 0 if and only if (2β R 2 + 1)(a 2 R 2 + 2bR 2 + 1) Again, we take the square in both sides of the equation obtained passing the second member to the right hand side of the previous equality and we obtain after some simplifications (2β R 2 + 1) 3 = 0, and as before it is not possible. In a similar way it can be proved that (dh/dϕ)(π, R) = 0. Thus the assumption (iii) is satisfied.
Since the assumptions of Theorem 4 are satisfied the simple zeros of the function (13) provide 2π -periodic solutions of system (12). Finally, by Proposition 6 it follows that the function f given by (13) can be have at least 12 simple zeros. So Theorem 2 is proved.