On the uniqueness of limit cycles in discontinuous Liénard-type systems

In this paper, we investigate the uniqueness and stability of limit cycles for a nonlinear Liénard-type differential system with a discontinuity line. By employing a transformation technique and considering the characteristic exponent of the periodic orbit, we give several criteria for the discontinuous planar nonlinear Liénard-type system. An example with different nonlinear functions H(y) is presented to illustrate the obtained results.


Introduction
As is well known, the Liénard system is widely used to describe the dynamics appearing in various models (mathematical, physical and mechanical engineering models etc.).Many nonlinear systems can be transformed into the Liénard form by suitable changes [6,10].So investigation for the Liénard system is significant from both application and theoretical point of view.Up to now, there have been many achievements on the existence, uniqueness and the number of limit cycles for continuous or even smooth differential system, especially for the Liénard system, see for example [1,3,6,13,14,17] and references therein.
In addition, much progress has been made in studying the existence and uniqueness of limit cycles for discontinuous planar differential system, see for example [2, 4, 5, 7-9, 11, 12, 15, 16] and references therein.However, most of the existing papers focus on the investigation for the discontinuous planar piecewise linear differential system [5,[7][8][9]15].For the discontinuous planar nonlinear Liénard system, there are only a few papers.In [11], the authors studied the nonexistence and uniqueness of limit cycles for a discontinuous nonlinear Liénard system.In [12], the number of limit cycles for a discontinuous planar generalized Liénard polynomial differential equation was studied.In [16], the authors studied the number of limit cycles bifurcating from the origin for a class of discontinuous planar Liénard systems.However, on the discontinuous planar nonlinear Liénard-type system, the relevant problems are more complicated which are not easy to be handled due to the nonlinearity of function H(y).To F. F. Jiang and J. T. Sun the best of our knowledge, there has been no result on the nonlinear Liénard-type system allowing discontinuities.
In this paper, we investigate the uniqueness and stability of limit cycles for a nonlinear Liénard-type differential system with a discontinuity line.We first give some geometrical properties for the discontinuous system.Then by taking a change of variable and considering the characteristic exponent of the periodic orbit, we obtain that the discontinuous planar nonlinear Liénard-type system has at most one stable limit cycle.
The paper is organized as follows.In the next section, we present some preliminaries and geometrical properties for the discontinuous system.In Section 3, we first give several relevant lemmas, then under different hypotheses of the function H(y) we provide several criteria on the uniqueness and stability of limit cycles for the discontinuous planar nonlinear Liénardtype system.In Section 4, an example with different nonlinear functions H(y) is presented to illustrate the obtained results.Conclusion is outlined in Section 5.

Preliminaries
Consider the following Liénard-type differential system with a discontinuity line where x 0 f (s) ds with F(0) = 0, H(y) ∈ C(R, R), yH(y) > 0 for y = 0 and H(+∞) = +∞, and functions f (x), g(x) are given by For system (2.1) with (2.2), the corresponding vector field is as follows where V i (x, y) = (F i (x) − H(y), g i (x)) T and F i (x) = x 0 f i (s) ds for i = 1, 2. In this paper, we assume that the following hypotheses hold for system (2.1) with (2.2).
Obviously, the origin O(0, 0) is a unique equilibrium point of (2.1).From (H2) we obtain that the isocline curve H(y) = F(x) is passing through the origin and F(x) ≥ 0 for x ∈ R. By (H3) and the inverse function theorem, the derivative dH −1 (F(x)) dx = f (x) H (y) has the same sign as x for y > 0, so the isocline curve H(y) = F(x) passing through the origin is increasing for x > 0 and decreasing for x < 0 on the (x, y) plane.
Moreover, since F 1 (0) = F 2 (0) = 0 it follows that the horizontal component of the vector field (2.3) is continuous.By Filippov's first order theory [4,5] then the origin O is a unique sliding point on Σ 0 (for any (0, y) ≤ 0 then we speak of the point (0, y) as a sliding point).Therefore, there exists no sliding limit cycle (isolated periodic orbit which has some points in the sliding set (a set of sliding points)) for system (2.1), and then we focus our attention on the crossing limit cycle (isolated periodic orbit which does not share points with the sliding set).
Proof.It is obvious that the origin O is a unique equilibrium point of (2.1).Since x = −H(y) for x = 0 and yH(y) > 0 for y = 0, it follows that the periodic orbit goes around the origin counterclockwise.
Let Γ be the periodic orbit surrounding the origin, A(x A , y A ) and B(x B , y B ) are two points on Γ such that the x-exponent x A and x B are the minimum and maximum values, then we have that x A < 0 < x B .when x = 0 one has that By the vector field of system (2.1), the derivative (2.4) vanishes at the points A(x A , y A ) and B(x B , y B ), i.e., F(x A ) = H(y A ), F(x B ) = H(y B ).Moreover, along the curve H(y) = F(x) it follows from (H1) and (H3) that the second derivative d 2 x dy 2 = − H (y) g(x) has opposite sign as x for x = 0.So (2.4) vanishes only once for x > 0 (x < 0).Correspondingly, the periodic orbit Γ intersects the curve F(x) = H(y) only once for x > 0 (x < 0).

Now we consider a change of variable as follows
(2.5) By (H2) then P(x) ≥ 0 for x ∈ R and P (x) > 0 (< 0) for x > 0 (< 0).So there exist inverse functions x 2 (P) for x ≥ 0 and x 1 (P) for x ≤ 0 as follows Moreover, for x = 0 it follows from (2.5) that the system (2.1) is transformed into the following differential systems For simplicity, denote by e i (P) = g(x i (P)) f (x i (P)) then the systems (2.6) can be written as F. F. Jiang and J. T. Sun satisfying e i (P) > 0 for P > 0, i = 1, 2. By the inverse function theorem then the isocline curve P = H(y) is increasing on the positive half (P, y) plane with P > 0.
Note that the systems (2.7) can be continuously extended to P = 0 if we let e i (0 In this case, the hypothesis (H4) becomes 0 ≤ e 2 (0) ≤ e 1 (0) and e 2 (P) < e 1 (P) for 0 < P sufficiently small.

Main results
Lemma 3.1.Let (H3) hold and consider the following differential systems and let the functions There are three possible cases as follows.
It is a contradiction and so the statement (ii) holds.
By a similar analysis for the case P − H(y) < 0, the statements hold.
Lemma 3.2.Assume that (H1)-(H3) hold for system (2.1) and let Γ be a periodic orbit surrounding the origin.Then one has that where ∆ denotes a region surrounded by Γ and divV denotes the divergence of the vector field V.Moreover, if Γ divVdt < 0 (> 0) then Γ is a stable (an unstable) limit cycle.
Proof.Let ∆ − , Γ − and ∆ + , Γ + be parts of ∆ and Γ contained in x < 0 and x > 0 respectively.M(0, y M ) and N(0, y N ) denote two intersections between Γ with the discontinuity line By Green's formula, we obtain that where MN denotes an oriented segment from the point M to N and N M is similar.The proof of the stability is similar to the one in [17].
We first show that the trajectory arc Γ 1 (P) intersects with Γ 2 (P).Consider the following differential systems where the right system is a symmetry system of dP dt = P − H(y), dy dt = e 1 (P) for P > 0. It is easy to see that the systems (3.4) have a counterclockwise periodic orbit Γ, which is constituted by trajectory arcs Γ 2 (P) and Γ 1 (P), where Γ 1 (P) denotes the symmetry trajectory arc of Γ 1 (P) with respect to the discontinuity line Σ 0 .
This implies that Γ 1 (P) does not intersect with Γ 2 (P), it is a contradiction.So the equation e 2 (P) = e 1 (P) has at least one solution P 0 for P 0 ∈ (0, min{F(a), F(b)}).
Theorem 3.4.Let (H1)-(H4) hold.Assume that the equation e 1 (P) = e 2 (P) has a unique zero P 0 with P 0 ∈ (0, min{F(a), F(b)}) and positive function e 1 (P) P is decreasing for P ∈ (0, F(a)).Then the system (2.1) has at most one periodic orbit, and it is a unique stable limit cycle if it exists.
In Figure 3.1, M 2 (s 2 , y M 2 ) and N 2 (s 2 , y N 2 ) denote two intersections between Γ with the line x = s 2 , K 1 (s 1 , y K 1 ) and L 1 (s 1 , y L 1 ) denote two intersections between Γ 0 with the line x = s 1 .Now for the purpose of the uniqueness, we compute the characteristic exponent ρ of the periodic orbit Γ as follows where the integral is counterclockwise.Denote by ρ = Γ f (x(t)) dt = I + J with We first compute the integral I = I 1 + I 2 + I 3 + I 4 , where , then it follows from (3.8) that I 1 < 0 and I 2 < 0. Similarly, by (3.9) then Furthermore, we have that ) dt, we only consider J 1 , J 2 is similar and so omitted.It follows that It is easy to see that the discontinuity line Σ 0 = {(x, y) : x = 0, −∞ < y < ∞}, functions f (x) and g(x) are given by f So the hypotheses (H1)-(H2) hold.Case 1.The function H(y) in (4.1) is given by for y ≤ −1.

Conclusion
In this paper, we have investigated the uniqueness and stability of limit cycles for a nonlinear Liénard-type differential system with a discontinuity line.Firstly, we have given some geometrical properties for the discontinuous system.Secondly, by taking a change of variable and verifying the characteristic exponent of the periodic orbit, we have obtained that the discontinuous planar nonlinear Liénard-type system has at most one stable limit cycle.Finally, we have given an example with different nonlinearity functions H(y) to illustrate the obtained results.This implies that the hypothesis (H3) does not contain (H3) * and vice versa.

Γ 2 (Figure 3 . 1 :
Figure 3.1: Left graph shows the periodic orbit of (2.1) on the (x, y) plane, right graph is the corresponding trajectory arcs on the (P, y) plane with P > 0.