The focus case of a nonsmooth Rayleigh–Duffing oscillator

In this paper, we study the global dynamics of a nonsmooth Rayleigh–Duffing equation x¨+ax˙+bx˙|x˙|+cx+dx3=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{x}+a{\dot{x}}+b{\dot{x}}|{\dot{x}}|+cx+{\mathrm {d}}x^3=0$$\end{document} for the case d>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d>0$$\end{document}, i.e., the focus case. The global dynamics of this nonsmooth Rayleigh–Duffing oscillator for the case d<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d<0$$\end{document}, i.e., the saddle case, has been studied in the companion volume (Wang and Chen in Int J Non-Linear Mech 129: 103657, 2021). The research for the focus case is more complex than the saddle case, such as the appearance of five limit cycles and the gluing bifurcation which means that two double limit cycle bifurcation curves and one homoclinic bifurcation curve are very adjacent. We present bifurcation diagram, including one pitchfork bifurcation curve, two Hopf bifurcation curves, two double limit cycle bifurcation curves and one homoclinic bifurcation curve. Finally, numerical phase portraits illustrate our theoretical results.


Introduction and main results
In the middle of seventeenth century, C. Huygens built the theory of physical pendulum for the first time and created the single pendulum mechanical clock based on a simple oscillator model. In the study of modern nonlinear science, the dynamics of oscillators plays a more and more important role. A large number of researches on the dynamics of oscillators have emerged; see examples in [4,5,15,16,20,24].
Notice that the stable manifold on the right-hand side of the saddle at the origin in III∪ H 1 or IV possibly connects with equilibria at infinity from either the positive In addition, three bifurcation curves DL 1 , DL 2 and H L are very close, implying the appearance of the gluing bifurcation. We give some numerical simulations to show this bifurcation phenomenon in Sect. 5.
An outline of this paper is as follows. Local dynamics of system (1.2b) are studied in Sect. 2, such as qualitative properties of equilibria (including equilibria at infinity) of system (1.2b) and local bifurcations. In Sect. 3, we study the limit cycles and homoclinic loops of system (1.2b). Theorems 1.1 and 1.2 are proved in Sect. 4. In Sect. 5, numerical simulations illustrate our analytical results. In Sect. 6, we conclude the global dynamics for the focus cases of smooth and nonsmooth Rayleigh-Duffing oscillators and van der Pol-Duffing oscillators.

Conditions of parameters
Qualitative properties of equilibria (see [23]) will be constructed. From equations in (1.2b) with μ 1 = 0 and μ 2 > 0, it is easy to see that there is a unique vertical isocline V := (x, y) ∈ R 2 : y = 0, 0 < x < and a unique horizontal isocline where > 0 is sufficiently small. Moreover, let where 0 < α < π/2 is chosen to be arbitrarily close to 0. Obviously, H is in the fourth quadrant and above L − , as shown in Fig. 3a HE 0 L − is a generalized normal sector of Class I, as shown in Fig. 3a. Thus, by [23, Lemma 1] system (1.2b) has infinitely many orbits approaching E 0 in HE 0 L − as t → ∞. By the symmetry, E 0 is a stable node when μ 1 = 0 and μ 2 > 0.
L + E 0 H is a generalized normal sector of Class I, as shown in Fig. 3b. Thus, system (1.2b) has infinitely many orbits leaving E 0 in L + E 0 H as t → ∞. One can check thatẋ > 0 andẏ < 0 in HE 0 V. By [23,Lemma 4], there are no orbits connecting E 0 in by the symmetry E 0 is an unstable node when μ 1 = 0 and μ 2 < 0.
When μ 1 = 0 and μ 2 = 0, matrix J 0 is nilpotent. Notice that system (1.2b) can be written as Bernoulli equation which has a general solution 4x 3 sgn(y) − 6x 2 + 6xsgn(y) − 3 − Ce −2xsgn(y) + 4y 2 = 0 and C is an arbitrary constant. Then, with the initial condition y(0) = 0, the constant C is determined to be −3 and implying that there are no orbits connecting E 0 . Thus, E 0 is a center or a focus of system (1.2b). Assume that AB is the upper half part of an orbit near E 0 , and A (resp. B) is its intersection point with the negative (resp. positive) x-axis. Let E 0 (x, y) = x 4 /4 + y 2 /2. Then, implying that E 0 is a stable nilpotent focus when μ 1 = 0 and μ 2 = 0.
Based on Lemmas 2.1 and 2.2, we investigate bifurcations from finite equilibria in the following three propositions. Proposition 2.1 Consider μ 1 > 0. There is a unique limit cycle occurring in a small neighborhood of E l (resp. E r ) if μ 2 varies from μ 2 = 0 to μ 2 = − and no limit cycles in any small neighborhood of E l (resp. E r ) if 0 ≤ μ 2 < , where > 0 is sufficiently small. Moreover, the limit cycles are stable.
Similar to Proposition 2.1, a generalized Hopf bifurcation from E 0 will occur when μ 1 < 0 and μ 2 varies from μ 2 = 0 to μ 2 = − , where > 0 is sufficiently small. Notice that E 0 is a stable nilpotent focus when μ 1 = μ 2 = 0, where limit cycles may also be bifurcated. The following proposition presents a generalized Hopf bifurcation from E 0 , which is available in both cases μ 1 < 0 and μ 1 = 0.

Proposition 2.2
Consider μ 1 ≤ 0. There is a unique limit cycle occurring in a small neighborhood of E 0 if μ 2 varies from μ 2 = 0 to μ 2 = − and no limit cycles in any small neighborhood of E 0 if μ 2 ≥ 0, where > 0 is sufficiently small. Moreover, the limit cycle is stable.
By Bendixson-Dulac Criterion, system (1.2b) exhibits no limit cycles. Therefore, when μ 1 ≤ 0 system (1.2b) exhibits a unique limit cycle if − < μ 2 < 0 and no limit cycles if μ 2 ≥ 0, where > 0 is sufficiently small. Since E 0 is the unique finite equilibrium of system (1.2b), the limit cycle must surround E 0 if it exists. Notice that the vector field of system (1.2b) is rotated with respect to μ 2 by [19,27] and system (1.2b) exhibits no limit cycles for μ 2 = 0. Assume that the unique limit cycle of system (1.2b) does not lie in a small neighborhood of E 0 when μ 2 = − , where > 0 is sufficiently small. By the rotated property, the unique stable limit cycle still persists when μ 2 = 0. This is a contradiction, which completes the proof.
Proof By Lemmas 2.1 and 2.2, the number of equilibria at infinity varies from 3 to 1 when μ 1 is changed from a positive value to a non-positive one. Then, μ 1 = 0 is the pitchfork bifurcation curve. Thus, (a) is proved. When μ 1 > 0, by Proposition 2.1 the stable weak foci E l and E r become unstable rough foci and two limit cycles occur at the same time, one is in a small neighborhood of E l and the other is in a small neighborhood of E r , as μ 2 varies from 0 to a negative value. Then, H 1 is a generalized Hopf bifurcation curve. When μ 1 < 0 (resp. μ 1 = 0), by Proposition 2.2, E 0 becomes an unstable rough focus (resp. unstable degenerate node) from the stable weak focus (resp. stable nilpotent focus) and one stable limit cycle occurs in a small neighborhood of E 0 as μ 2 is changed to a negative value from 0. Then, H 2 is a generalized Hopf bifurcation curve and (b) is proved.
To see the behavior of orbits when either |x| or |y| is large, we need to discuss the possible equilibria at infinity. By Poincaré transformations x = 1/z, y = u/z and x = v/z, y = 1/z, system (1.2b) can be where dτ = z 2 dt. Obviously, system (2.16) has no equilibria on the u-axis and D : (0, 0) is an equilibrium of system (2.17). Moreover, D corresponds to a pair of equilibria I + y and I − y at infinity of system (1.2b), which lie on the positive y-axis and negative y-axis, respectively. The following lemma exhibits the qualitative properties of I + y and I − y .
Lemma 2.3 I + y and I − y are degenerate saddle nodes. Moreover, for system (1.2b) the direction of the orbit on the boundary of the Poincaré disc is clockwise and there are infinitely many orbits leaving I + y in the direction of the negative x-axis and infinitely many orbits leaving I − y in the direction of the positive x-axis as t → ∞.
and 0 < α < π/2 is chosen to be arbitrarily close to 0 and > 0 is sufficiently small, as shown in Fig. 4. One can check thatv > 0 andż > 0 in L + DH 1 and Since any orbit connecting D along where > 0 is sufficiently small. Obviously, V is above H 2 , as shown in Fig. 4. One can get thatv > 0 anḋ z < 0 in the open region H 2 DH 3 . Then, Since any orbit connecting D along θ = π in H 2 DH 3 is in the form of z = cv +o(v ), where > 1 and c > 0 (< 0) if is even (odd), one can compute where 0 < α < π/2 is chosen to be arbitrarily close to 0 and > 0 is sufficiently small. Notice thatv < 0, z > 0, in the open region L − DV. Then, L − DV is a generalized normal sector of Class III, as shown in Fig. 4. Moreover, let and C is in the region L − DV. Then, L − DV covers a generalized normal sector of Class I, which is L − DC, as shown in Fig. 4. Thus, there are infinitely many orbits leaving D along θ = π in the region L − DV as τ → ∞. Combining the facts that z = 0 is also an orbit of system (2.17) andv| z=0 = v 4 , there is only one orbit approaching D, which is exactly z = 0 and v < 0. Therefore, the properties of I + y and I − y for system (1.2b) are shown in Fig. 5.

Limit cycles and homoclinic loops
In this section, we investigate the existence and the number of limit cycles and homoclinic loops for system (1.2b). For simplicity, the whole parameter space is divided into the following four subsets: Lemma 3.1 When (c1) holds, system (1.2b) exhibits neither limit cycles nor homoclinic loops.
Proof When (c1) holds, one can calculate the diver- By Bendixson-Dulac Criterion, system (1.2b) has no closed orbits, implying the nonexistence of limit cycles and homoclinic loops.

Lemma 3.2 When (c2)
holds, there is a unique limit cycle for system (1.2b). Moreover, the limit cycle is stable.
For the remaining two cases, which is a division of the forth quadrant of the (μ 1 , μ 2 )-plane, by Lemma 2.1 system (1.2b) has three equilibria and E 0 is a saddle. Then, the limit cycle bifurcation or the homoclinic bifurcation may occur. Since system (1.2b) is symmetric with respect to E 0 and E 0 is a saddle, limit cycles can surround all three equilibria E 0 , E l and E r or surround one of equilibria E l and E r . In what follows, let large limit cycles be the ones surrounding all three equilibria E 0 , E l and E r , and small limit cycles be the ones surrounding E l or E r for simplicity.
To study the existence and number of closed orbits, we consider horizontal and vertical isoclines of system (1.2b). Clearly, the vertical isocline of system (1.2b) is the x-axis. The horizontal isocline μ 1 x + x 3 − (μ 2 + |y|)y = 0 depends on the parameters μ 1 and μ 2 . Lemma 3.3 Consider μ 1 > 0 and μ 2 < 0. The graphs of the horizontal isocline of system (1.2b) have the following three types: Fig. 6c.
Proof Notice that the graph of μ 1 x−x 3 −(μ 2 +|y|)y = 0 is symmetric about the origin. It is enough to investigate the graphs of It is easy to see that p( denoted by x 2 and x 3 . Without losing generality, assume x 2 < x 3 . Then, q(x, y) ≥ 0 implies x ≤ x 2 or x 3 ≤ x ≤ x 1 and the graph of q(x, y) = 0 is shown in Fig. 7a. When μ 2 = −2(12μ 3 1 ) 1/4 /3, we get p lmin = 0 and p(x) has exactly one zero point on the interval (−∞, q(x, y) ≥ 0 implies x ≤ x 1 and the graph of q(x, y) = 0 is shown in Fig. 7b. When μ 2 < −2(12μ 3 1 ) 1/4 /3, we get p lmin > 0 and p(x) has no zeros on the interval (−∞, Then, q(x, y) ≥ 0 implies x ≤ x 1 and the graph of q(x, y) = 0 is shown in Fig. 7c. By symmetry about the origin, we can get the graphs of the horizontal isocline of system (1.2b) as shown in Fig. 6.
is one of the orbits leaving from I − y , we can think that it crosses the x-axis at positive infinity, because the infinity on the positive x-axis is connected with I − y by a unique orbit. Then, denote the first intersec- Fig. 8. Moreover,
) and the positive x-axis and | | is sufficiently small. Then, increases continuously as μ 2 increases. Therefore, the statement (ii) is proved.
By Lemmas 3.3 and 3.4, we can get the nonexistence of small limit cycles and homoclinic loops when (c3) holds.
Lemma 3.5 When (c3) holds, there are neither small limit cycles nor homoclinic loops for system (1.2b).
Proof Since system (1.2b) is symmetric about E 0 and E 0 is a saddle, it suffices to prove that there are no small limit cycles and no homoclinic loops surrounding E r .
Firstly, we claim that any small limit cycle surrounding E r must lie in the region |y| < −μ 2 when μ 1 > 0 and μ 2 = −2(12μ 3 1 ) 1/4 /3. In fact, the graph of the horizontal isocline of system (1.2b) is shown in Fig. 7b. Denote the segment of the horizontal isocline connecting E 0 and the point A : ( √ μ 1 , μ 2 ) by E 0 A, as shown in Fig. 10. Notice thaṫ Then, passing through a point in the fourth quadrant and below the line y = μ 2 means that the limit cycle cannot be a small one.
Assume that the peak point of a small limit cycle surrounding E r is above y = −μ 2 . The limit cycle will fall below y = −μ 2 and move right down in the first quadrant, then move left and stay above y = μ 2 in the fourth quadrant. Denote the rightmost intersection point of this small limit cycle and y = −μ 2 by B : (x 0 , −μ 2 ), as shown in Fig. 10. Clearly, the limit cycle On the one hand, one can check that . This is a contradiction. Secondly, we prove that there are no small limit cycles surrounding E r when μ 1 > 0 and μ 2 ≤ −2(12μ 3 1 ) 1/4 /3. Since any small limit cycle surrounding E r lies in the region |y| < −μ 2 when μ 1 > 0 and μ 2 = −2(12μ 3 1 ) 1/4 /3, it follows from (3.7) that dE/dt > 0. That means the nonexistence of limit cycles for μ 1 > 0 and μ 2 = −2(12μ 3 1 ) 1/4 /3. Thus, Moreover, we claim that , an annular region, whose ω-limit set lies in itself, can be constructed because E r is unstable. By Poincaré-Bendixson Theorem, at least one small limit cycle surrounding E r exists. From Lemma 3.4, x dÃ (μ 1 , μ 2 ) increases and x dB (μ 1 , μ 2 ) decreases as μ 2 decreases. It follows from (3.8) that when μ 1 > 0 and μ 2 < −2(12μ 3 1 ) 1/4 /3. Then, there are no limit cycles surrounding E r when μ 1 > 0 and μ 2 < −2(12μ 3 1 ) 1/4 /3. Thirdly, we prove that there are no homoclinic loops surrounding E r when μ 1 > 0 and μ 2 ≤ −2(12μ 3 Then, the existence of homoclinic loops will lead to the existence of at least one small limit cycle surrounding E r , which contradicts the conclusion of the second step. To consider the number of large limit cycles for (c3), we investigate the relation of the divergence integrals of two hypothetic large limit cycles in the following lemma, which gives the monotonicity of the divergence integrals and also can be applied for the case (c4). Lemma 3.6 When μ 1 > 0 and μ 2 < 0, if there are at least two large limit cycles for system (1.2b), the following assertion is true: where 1 , 2 are large limit cycles and 1 lies in the region enclosed by 2 .
Proof Notice that the derivative of energy function E(x, y) defined in (3.6) with respect to t is (3.7). Then, any limit cycle cannot lie in the region |y| < −μ 2 . For i = 1, 2, assume that i crosses the x-axis and y = −μ 2 /2 at A i , B i , C i and D i successively, as shown in Fig. 11. Passing through B 1 and C 1 , respectively, lines perpendicular to the x-axis cross 2 at two points, denote by E 2 and F 2 . Let x B 1 and x C 1 be the abscissas of B 1 and C 1 .
We claim that x B 1 < − √ μ 1 and x C 1 > √ μ 1 . Since 1 cannot lie in the region |y| < −μ 2 , we denote the leftmost intersection point of 1 and y = −μ 2 by P 1 , and the rightmost intersection point by P 2 . Let P 3 and P 4 be the points (− √ μ 1 , −μ 2 ) and ( √ μ 1 , −μ 2 ). We can prove x B 1 < − √ μ 1 and x C 1 > √ μ 1 by showing that P 1 is on the left side of P 3 and P 2 is on the right side of P 4 . From the graph of the horizontal isocline of system (1.2b), as shown in Fig. 6, P 1 cannot be on the right side of P 4 and P 2 cannot be on the left side of P 3 . If P 1 is between P 3 and P 4 , as shown in Fig. 12a, 1 will cross x = − √ μ 1 at a point below P 3 , denoted by That implies a contradiction. Hence, P 1 lies on the left side of P 3 . If P 2 is between P 3 and P 4 , from the graph of the horizontal isocline (see Fig. 6), P 2 must be in the second quadrant. Denote the symmetric point of P 2 about the y-axis by P 6 . Let P 7 be the point on 1 such that the line connecting P 6 and P 7 is perpendicular to the xaxis. By the derivative of E(x, y), as shown in (3.7), we get E(P 2 ) < E(P 7 ). From (3.6), we calculate E(P 2 ) = E(P 6 ) and E(P 7 ) < E(P 6 ). That is a contradiction, indicating that P 2 lies on the right side of P 4 . Therefore, on the arc A i B i and on the arc C i D i , dy/dt is signed and |x| > √ μ 1 . On the one hand, the arc A i B i can be regarded as the graph of the function Similarly, (3.10) On the other hand, the arcs B 1 C 1 and E 2 F 2 can be regarded as graphs of functions y = y 1 (x) and y = y 2 (x), respectively, where x B 1 ≤ x ≤ x C 1 . Then, Since −μ 2 − 2y < 0 on the arcs B 2 E 2 and F 2 C 2 , one can check that (3.12) It follows from (3.9)-(3.12) that Therefore, the proof is finished.
From Lemma 3.6, when (c3) holds the uniqueness, stability and hyperbolicity of large limit cycles are given in the following lemma. Combining Lemma 3.5 and the following lemma, the qualitative properties of closed orbits of system (1.2b) with (c3) will be completely obtained. Lemma 3.7 When (c3) holds, system (1.2b) exhibits a unique large limit cycle, which is stable and hyperbolic.
, by the instability of E r andẏ| x> √ μ 1 ,y>0 < 0, an annular region whose ωlimit set lies in itself can be constructed. By Poincaré-Bendixson Theorem, there is at least one small limit cycle surrounding E r , which conflicts with Lemma 3.5. By Lemma 2.3, both the equilibria at infinity are degenerate saddle-nodes and all orbits of system (1.2b) are positively bounded. Combining x 0 A (μ 1 , μ 2 ) − x 0 B (μ 1 , μ 2 ) > 0, an annular region whose ω-limit set lies in itself, can be constructed. By Poincaré-Bendixson Theorem, we get the existence of large limit cycles of system (1.2b).
Denote by the innermost large limit cycle for (1.2b). Since x 0 If div(y, μ 1 x − x 3 − (μ 2 + |y|)y) = 0, is externally unstable. By Lemma 3.4 and [27, Theorem 3.4 of Chapter 3.4], at least two limit cycles will be bifurcated from , including a stable inner limit cycle and an unstable outer one, when μ 2 varies to a larger value, which contradicts the conclusion of Lemma 3.6. Thus, Moreover, is stable and hyperbolic by (3.13). Notice that any two adjacent closed orbits cannot stable simultaneously and is the innermost large limit cycle. It follows from Lemma 3.6 that is the unique large limit cycle for system (1.2b).
When (c4) holds, the number of large limit cycles is different for different parameters. In the following lemma, we first give an upper bound on the number of large limit cycles and discuss the stabilities of large limit cycles. Lemma 3.8 When (c4) holds, there are at most two large limit cycles for system (1.2b), and (a) the inner limit cycle is unstable and the outer one is stable if there are two limit cycles; (b) the limit cycle is stable or semi-stale (internally unstable and externally stable) if there is a unique limit cycle.
That means 1 is unstable, 3 is stable and 2 is internally stable and externally unstable. By Lemma 3.4 and [27, Theorem 3.4 of Chapter 3.4], a stable inner limit cycle and an unstable outer one will be bifurcated from 2 when μ 2 varies to a larger value, which is contradictory with the conclusion of Lemma 3.6.
Thus, 1 is unstable, which conflicts with the fact that 1 is in the region enclosed by 2 and they are adjacent to each other. Therefore, 2 is stable.
The stability of 2 implies that 1 is externally unstable. If 1 is externally unstable and internally unstable, by Lemma 3.4 and [27, Theorem 3.4 of Chapter 3.4], a stable inner limit cycle and an unstable outer one will be bifurcated from 1 . That contradicts the conclusion of Lemma 3.6. Then, 1 is unstable, yielding that the statement (a) is proved.
Thirdly, we prove the statement (b). Assume that there is a unique large limit cycle for system (1.2b), denoted by . By Lemma 2.3, is externally stable. Then, is stable or semi-stale (internally unstable and externally stable), implying that the statement (b) is proved.
Denote points on the positive orbit and negative orbit passing through (d, 0) of system (3.18) with have the similar expressions with (3.19), (3.20), (3.21) and (3.22), respectively, except that the lower limit integration with respect to τ is d rather than 0.
As in Lemma 3.9 for large limit cycles, we also give a region where only stable small limit cycles may exist. Proof Assume that there is a small limit cycle γ in the region {(x, y) : x ≥ √ μ 1 /3} for system (1.2b).
For the limit cycles passing through the region {(x, y) : 0 < x < √ μ 1 /3}, the number is difficult to be gained theoretically. In Sect. 6, numerical simulations show that small limit cycles surrounding the same equilibrium are gluing together and cannot be distinguished. Based on the number of small limit cycles surrounding one equilibrium in focus case of smooth Rayleigh-Duffing oscillator [6] and the numerical simulations of system (1.2b), we conjecture that there are at most two limit cycles surrounding E r for system (1.2b). All our later conclusions are based on the conjecture. If the conjecture is invalid, system (1.2b) will exhibit more complex dynamical behavior. But the closed orbits and bifurcation curves in the following conclusions remain while more limit cycles and bifurcation curves appear which are very close to the original ones.

Lemma 3.12 When (c4) holds, there are at most two small limit cycles surrounding E r for system (1.2b), and (a) the inner limit cycle is stable and the outer one is unstable if there are two limit cycles. (b) the limit cycle is stable or semi-stable (internally stable and externally unstable) if there is a unique limit cycle.
Proof Firstly, consider that there are two small limit cycles surrounding E r of system (1.2b). If only one of two limit cycles is semi-stable, by Lemma 3.4 and [27, Theorem 3.4 of Chapter 3.4], at least one stable limit cycle and one unstable limit cycle will be bifurcated from the semi-stable one and the limit cycle of multiple-odd does not disappear as μ 2 increases. If both two limit cycles are semi-stable, since they are adjacent to each other, at least four limit cycles will be bifurcated from them. Then, one limit cycle is stable and the other one is unstable. By Lemma 2.1, E r is unstable when (c4) holds. Thus the inner one is internally stable, indicating the inner limit cycle is stable and the outer one is unstable. Secondly, if there is a unique small limit cycle surrounding E r , we can only obtain it is internally stable. Then, the proof is finished. Now we are ready to prove the existence of homoclinic loops and give the number of limit cycles in (c4). We first get a homoclinic bifurcation curve in Proposition 3.1. Then, a double large limit cycle bifurcation curve and a double small limit bifurcation curve are obtained in Propositions 3.2 and 3.3, respectively. Proposition 3.1 There is a decreasing C ∞ function ϕ(μ 1 ) such that −2(12μ 3 1 ) 1/4 /3 < ϕ(μ 1 ) < 0 and (a) system (1.2b) exhibits one figure-eight loop if and only if μ 2 = ϕ(μ 1 ); (b) when μ 2 = ϕ(μ 1 ), system (1.2b) exhibits three limit cycles, where two limit cycles are stable and small, another one is stable and large; (c) when μ 2 = ϕ(μ 1 ) − , system (1.2b) exhibits five limit cycles, where four limit cycles are small (two of them surround E r , the inner limit cycle is stable and the outer one is unstable), another one is stable and large; (d) when μ 2 = ϕ(μ 1 ) + , system (1.2b) has four limit cycles, where two limit cycles are stable and small, another two are large (the inner limit cycle is unstable and the outer one is stable), where > 0 is sufficiently small.
Notice that tr J 0 = μ 2 > 0, where J 0 is the Jacobian matrix at E 0 and defined in (2.1). By [7, Theorem 3.3], the homoclinic loop of E 0 is asymptotically unstable. Then, if there are limit cycles for system (1.2b), the outermost small limit cycle must be externally stable and the innermost large limit cycle must be internally stable.
By Lemma 2.1, E r is unstable when μ 2 = −2 (12μ 3 1 ) 1/4 /3 < 0. Combining the instability of the homoclinic loop, there is at least one small limit cycle surrounding E r . Moreover, since the small limit cycle which is closest to the homoclinic loop is externally stable, by Lemma 3.12 there is a unique small limit cycle surrounding E r , which is stable. By the symmetry, there is a unique small limit cycle surrounding E l , which is stable.
By Lemma 2.3, all orbits of system (1.2b) are positively bounded. Combining the instability of the homoclinic loop, we get the existence of large limit cycles of system (1.2b). Since the large limit cycle closest to the homoclinic loop is internally stable, by Lemma 3.8 there is a unique large limit cycle of system (1.2b), which is stable. The statement (b) is proved.
Since the homoclinic loop of E 0 is asymptotically unstable when μ 2 = ϕ(μ 1 ), there exists a d ∈ (0, √ μ 1 ) such that x dÃ (μ 1 , ϕ(μ 1 )) − x dB (μ 1 , ϕ(μ 1 )) < 0. By continuous dependence of the solution on parameters, for sufficiently small > 0. On the one hand, by Lemma 3.4, (3.27) Then, by the continuous dependence of the solution on initial values, there exists a d 1 ∈ (0, d) such that On the other hand, since E r is unstable when μ 2 = ϕ(μ 1 )− < 0, by Poincaré-Bendixson Theorem there exists a d 2 ∈ (d, √ μ 1 ) such that That implies at least two small limit cycles surround E r . By Lemma 3.12, there are exactly two small limit cycles surrounding E r , the inner limit cycle is stable and the outer one is unstable. By the symmetry, there are also two small limit cycles surrounding E l , the inner limit cycle is stable and the outer one is unstable. The existence of large limit cycles comes from (3.27) and the positive boundedness of all the orbits. Moreover, (3.27) also implies that the innermost large limit cycle is internally stable. By Lemma 3.8, the large limit cycle is unique and stable. The statement (c) is proved.
By the instability of the homoclinic loop of E 0 when μ 2 = ϕ(μ 1 ), there exists a c > 0 such that x c A (μ 1 , ϕ(μ 1 )) − x c B (μ 1 , ϕ(μ 1 )) > 0. By continuous dependence of the solution on parameters, On the one hand, by Lemma 3.4, Then, by the continuous dependence of the solution on initial values, there exists a c 1 ∈ (0, c) such that On the other hand, since all the orbits are positively bounded by Lemma 2.3, by Poincaré-Bendixson Theorem there exists a c 2 ∈ (c, ∞) such that That implies at least two large limit cycles exist for system (1.2b). By Lemma 3.8, there are exactly two large limit cycles, the inner limit cycle is unstable and the outer one is stable.
The existence of small limit cycles surrounding E r comes from (3.28) and the instability of E r . Moreover, (3.28) also implies that the outermost small limit cycle surrounding E r is externally stable. By Lemma 3.12, the small limit cycle surrounding E r is unique and stable. By symmetry, there is also a unique small limit cycle surrounding E l , which is stable. The statement (d) is proved.
For the statement (b), from Lemma 3.4 it is easy to check that That means there are at least two small limit cycles surrounding E r when 2 (μ 1 ) < μ 2 < ϕ(μ 1 ). By Lemma 3.12, system (1.2b) exhibits two small limit cycles surrounding E r when 2 (μ 1 ) < μ 2 < ϕ(μ 1 ), where the inner one is stable and the outer one is unstable. The statement (b) is proved.
For the statement (d), by Lemma 3.4 when −2(12μ 3 1 ) 1/4 /3 < μ 2 < ϕ(μ 1 ). Then, similar as in the proof of the statement (c) of Proposition 3.1, the existence of large limit cycles comes from (3.39) and the positive boundedness of all the orbits. Moreover, (3.39) also implies that the innermost large limit cycle is internally stable. By Lemma 3.8, the large limit cycle is unique and stable. The statement (d) is proved.

Proofs of Theorems 1.1 and 1.2
In this section, we prove Theorems 1.1 and 1.2 under the assumption that there are at most two limit cycles surrounding E r . Proof of Theorem 1.2 From Lemma 2.3, system (1.2b) has two infinite equilibria I + y and I − y . Moreover, there are infinitely many orbits leaving I + y in the direction of the negative x-axis and infinitely many orbits leaving I − y in the direction of the positive x-axis as t → ∞. When (μ 1 , μ 2 ) ∈ I or (μ 1 , μ 2 ) ∈ P , by Lemma 2.2 system (1.2b) has a unique finite equilibrium E 0 which is unstable. From Lemma 3.2 there is a unique limit cycle for system (1.2b), which is stable. Thus, the limit cycle is the ω-limit set of all the orbits except E 0 .
When (μ 1 , μ 2 ) ∈ III or (μ 1 , μ 2 , μ 3 ) ∈ H 1 , by Lemma 2.1 system (1.2b) has three finite equilibria E 0 , E l and E r . Moreover, E 0 is a saddle, E l and E r are stable. It follows from Lemmas 3.1 that there are no small limit cycles for system (1.2b). Thus, except the stable manifolds of E 0 , all the other orbits approach E l or E r as t → ∞.
When (μ 1 , μ 2 ) ∈ IV, by Lemma 2.1 system (1.2b) has three finite equilibria E 0 , E l and E r . Moreover, E 0 is a saddle; E l and E r are unstable. Proposition 3.2 (c) and (d) shows that system (1.2b) exhibits two limit cycles, which are small and stable. Thus, except the stable manifolds of E 0 , all the other orbits approach the small limit cycles surrounding E l or E r as t → ∞.
When (μ 1 , μ 2 ) ∈ DL 1 , E 0 is a saddle, E l and E r are unstable by Lemma 2.1. From Proposition 3.2 (a) and (d), there are three limit cycles for system (1.2b), where two small ones are stable, the large one is internally unstable and externally stable. Thus, the unstable manifolds of E 0 approach small limit cycles and the stable manifolds of E 0 leave from the large limit cycle as t → ∞.
When (μ 1 , μ 2 ) ∈ V, E 0 is a saddle, E l and E r are unstable by Lemma 2.1. By Proposition 3.2 (b) and (d), there are four limit cycles for system (1.2b), where two small ones are stable, the inner large one is unstable and the outer large one is stable. Thus, the unstable manifolds of E 0 approach small limit cycles and the stable manifolds of E 0 leave from the inner large limit cycle as t → ∞.
When (μ 1 , μ 2 ) ∈ H L, E 0 is a saddle, E l and E r are unstable by Lemma 2.1. According to Proposition 3.1 (a) and (b), system (1.2b) exhibits one figure-eight loop and three limit cycles, where two limit cycles are stable and small and another one is stable and large. Moreover, the figure-eight loop is unstable as in the proof of Proposition 3.1.
When (μ 1 , μ 2 ) ∈ VI, E 0 is a saddle, E l and E r are unstable by Lemma 2.1. On the base of Proposition 3.3 (b) and (d), there are five limit cycles for system (1.2b), where the large one is stable, the inner small ones are stable and the outer small ones are unstable. Thus, the unstable manifolds of E 0 approach the large limit cycle and the stable manifolds of E 0 leave from the outer small limit cycles as t → ∞.
When (μ 1 , μ 2 ) ∈ DL 2 , E 0 is a saddle, E l and E r are unstable by Lemma 2.1. By Proposition 3.3 (a) and (d), there are three limit cycles for system (1.2b), where the large one is stable, two small ones are internally stable, externally unstable. Thus, the unstable manifolds of E 0 approach the large limit cycle and the stable manifolds of E 0 leave from small limit cycles as t → ∞.
When (μ 1 , μ 2 ) ∈ VII, E 0 is a saddle, E l and E r are unstable by Lemma 2.1. From Lemmas 3.5, 3.7 and Proposition 3.3 (c) and (d), system (1.2b) exhibits a unique limit cycle, which is large and stable. Thus, the unstable manifolds of E 0 approach the large limit cycle as t → ∞.

Numerical examples and discussions
The phase portraits and bifurcations of (1.2b) by numerical simulations are shown in this section. The following numerical simulations are obtained by pplane8 function of MATLAB which displays the orbits of a planar autonomous differential equation in a specified rectangular bounded region. For the qualitative properties at infinity of system (1.2b), Lemma 2.3 tells us that the orbits near infinity are positively bounded, which can also be verified by dE dt | (1.2b) = −(μ 2 + |y|)y 2 < 0 when y is sufficiently large, where E is defined in (3.6). Moreover, by the transformation x = r cos θ and y = r sin θ , system (1.2b) can be written as its polar forṁ r = − cos 3 θ sin θ r 3 − sin 2 θ | sin θ | r 2 +((μ 1 + 1) cos θ sin θ − μ 2 sin 2 θ) r, θ = − cos 4 θ r 2 − cos θ sin θ | sin θ | r +μ 1 cos 2 θ − μ 2 cos θ sin θ − sin 2 θ, implying that the orbits are rotating clockwise when r is sufficiently large. However, these qualitative properties cannot be reflected in the numerical simulations unless they are compacted into a closed region. P4 program (P4 is short for polynomial planar phase portraits, see [8,Chapter10]) can give numerical global phase portraits in the Poincaré disk instead of in a rectangular bounded region. But P4 program is not available for non-polynomial system. How to simulate nonsmooth planar phase portraits in a compact region is still a challenging and significant problem. We first consider the case |μ 1 | = 1. When μ 1 = −1, by Lemma 2.2, system (1.2b) exhibits a unique equilibrium E 0 .
When μ 1 = −1, by Lemma 2.1, system (1.2b) exhibits three equilibria E 0 , E l and E r . We fix μ 1 = 1 and see how the phase diagram changes as μ 2 increases.
Example 2 When μ 1 = 1 and μ 2 = 1, E 0 is a saddle, E l and E r are stable foci, as shown in Fig. 17a. When μ 1 = 1 and μ 2 = −0.5, E 0 is a saddle; E l and E r are unstable foci. There are two limit cycles for system (1.2b), which are small and stable, as shown in Fig.  17b.
We continue to reduce μ 2 from −0.5 and find a series of changes in the topological structure of the phase diagram near μ 2 = −0.55. In the following examples, E 0 is a saddle; E l and E r are unstable foci. Then, we draw stable manifolds and unstable manifolds at E 0 , so as to infer the existence and stability of limit cycles.
Example 3 When μ 1 = 1 and μ 2 = −0.553949, the ω-limit sets of unstable manifolds at E 0 are small limit cycles, which are very close to E 0 , as shown in Fig. 18a. When μ 1 = 1 and μ 2 = −0.554552, the α-limit set of stable manifolds at E 0 is a large limit cycle, which is very close to E 0 , as shown in Fig. 18b.
When μ 1 = 1 and μ 2 varies from −0.553949 to −0.554552, double large limit cycle bifurcations, Example 4 When μ 1 = 1 and μ 2 = −0.554022, the ω-limit sets of unstable manifolds at E 0 are small limit cycles and the α-limit set of stable manifolds at E 0 is a large limit cycle, which are all very close to E 0 , as shown in Fig. 19a. On the region near E 0 in Fig.  19a, we can see its topological structure, as shown in Fig. 19b. When μ 1 = 1 and μ 2 = −0.554205, system (1.2b) exhibits a figure-eight loop, a large limit cycle and two small limit cycles, as shown in Fig. 19c, and the local topological structure near E 0 is shown in Fig.  19d. When μ 1 = 1 and μ 2 = −0.554539, the ω-limit sets of unstable manifolds at E 0 are a large limit cycle and the α-limit set of stable manifolds at E 0 are small limit cycles, which are all very close to E 0 , as shown in Fig. 19e. On the region near E 0 in Fig. 19e, we can see its topological structure, as shown in Fig. 19f.
In Fig. 19a, b, we can get that the large limit cycle is internally unstable and the two small limit cycles (surrounding E l and E r , respectively) are external stable when μ 1 = 1 and μ 2 = −0.554022. Then, sys- tem (1.2b) exhibits a semi-stable large limit cycle or two large limit cycles which are very close to each other. That implies 1 (1) ≈ −0.554022. In Fig. 19c, d, we can see a large limit cycle and two small limit cycles coexist with the figure-eight loop and ϕ(1) ≈ −0.554205. In Fig. 19e, f, we can get that the large limit cycle is internally stable and the two small limit cycles (surrounding E l and E r , respectively) are external unstable when μ 1 = 1 and μ 2 = −0.554539. Then, one semi-stable small limit cycle or two small limit cycles which are very close to each other surround E r . That implies 2 (1) ≈ −0.554539. Thus, the bifurcation curves DL 1 , H L and DL 2 are very close to each other.
If we continue to reduce μ 2 when μ 1 = 1, the large limit cycle still exists and is expanding.
Example 5 When μ 1 = 1 and μ 2 = −1, E 0 is a saddle; E l and E r are unstable foci. System (1.2b) exhibits a unique limit cycle, which is large and stable, as shown in Fig. 20.

Example 7
When μ 1 = 10 and μ 2 = 5, E 0 is a saddle; E l and E r are stable foci, as shown in Fig. 22a. When μ 1 = 10 and μ 2 = −3, E 0 is a saddle; E l and E r are unstable foci. There are two limit cycles for system (1.2b), which are small and stable, as shown in Fig.  22b. When μ 1 = 10 and μ 2 = −4.915, the stable and unstable manifolds at E 0 are shown in Fig. 22c, implying the coexistence of large limit cycles and small limit cycles. When μ 1 = 10 and μ 2 = −6, E 0 is a saddle, E l and E r are unstable foci. System (1.2b) exhibits a unique limit cycle, which is large and stable, as shown in Fig. 22d.

Example 9
When μ 1 = 20 and μ 2 = 22, E 0 is a saddle; E l and E r are stable nodes, as shown in Fig. 24a. When μ 1 = 20 and μ 2 = −8, E 0 is a saddle; E l and E r are unstable nodes. There are two limit cycles for system (1.2b), which are small and stable, as shown in Fig. 24b. When μ 1 = 20 and μ 2 = −9.0888, the stable and unstable manifolds at E 0 are shown in 24c, implying the coexistence of large limit cycles and small limit cycles. When μ 1 = 20 and μ 2 = −18, E 0 is a saddle, E l and E r are unstable foci. System (1.2b) exhibits a unique limit cycle, which is large and stable, as shown in Fig. 24d.

Conclusions
In this section, we compare the global dynamics for the focus case of nonsmooth Rayleigh-Duffing oscillator (1.2b) with smooth Rayleigh-Duffing oscillator, smooth van der Pol-Duffing oscillator and nonsmooth van der Pol-Duffing oscillator.

Nonsmooth and smooth Rayleigh-Duffing oscillator
The bifurcation diagram and global phase portraits in the Poincaré disk of a smooth Rayleigh-Duffing oscillatoṙ where parameters α and β are real, are given by [6]. The two systems (1.2b) and (6.1) are very similar at infinity for both qualitative properties of equilibria and bifurcations of closed orbits. In the case of having unique equilibrium, they both exhibit at most one limit cycle bifurcated from their origins. In the case of having three equilibria, a homoclinic bifurcation, a double small limit cycle bifurcation and a double large limit cycle bifurcation will occur in both systems (1.2b) and (6.1). It is worth mentioning that the gluing bifurcation which happens for (6.1) also appears for nonsmooth Rayleigh-Duffing oscillator (1.2b) from numerical simulations.
However, there are two kinds of differences in the analysis methods of equilibria and closed orbit bifurcations. One is that many classical theories, such as Hopf bifurcation, cannot be applied directly because the vector field of system (1.2b) is only C 1 . In order to overcome these difficulties, some measures like generalized Hopf bifurcation are used for system (1.2b).
The other one is that the upper bound of the number of small limit cycles for system (1.2b) cannot be (a) µ 2 = −5 (b) µ 2 = 5 obtained so that part of our conclusion is based on an assumption. We did not compare divergence integrals of two closed orbits as in [6] to (6.1), because there is an annular region in the phase diagram such that the monotonicity of divergence integrals is uncertain. However, the complete conclusion for system (6.1) can be given by investigating its Abelian integrals for small parameters and properties of the rotated vector field for general parameters. Since the Abelian integral of system (1.2b) is a combination of an elliptic integral and an elementary function, it is hard to consider the number of zeros. Then, there is a theoretical gap in global phase portraits for system (1.2b) though it cannot be reflected in the numerical simulation. The dynamics of systems (1.2b) and (6.1) are different for equilibria at infinity. Due to [6], system (6.1) has four equilibria at infinity. At each equilibrium, there is a parabolic sector surrounding it. For system (1.2b), it has two equilibria at infinity and at each equilibrium there is an elliptical sector and a parabolic sector surrounding it. In conclusion, the bifurcation diagrams of the two oscillators are similar, but the analysis of system (1.2b) is indeed more complex than system (6.1), and all global phase portraits in the Poincaré disc are different. The van der Pol-Duffing oscillatoṙ where a 2 a 4 = 0 is investigated in [11,12] for sufficiently small |a 1 |, |a 3 |. When a 4 < 0, by (x, y, t) → ( √ −a 4 x/a 2 , (−a 4 ) 3/2 y/a 2 2 , −a 2 t/a 4 ), system (6.2) can be simplified into its focus casė Global dynamical behaviors of system (6.3) are given in [3] for general a, b. Systems (1.2b) and (6.3) are both symmetric about the origin. They both have one or three equilibria at infinity, depending on different parameters, and pitchfork bifurcations occur. With the unique equilibrium, one limit cycle is bifurcated from the origin as the changes of parameters for both system (1.2b) and (6.3). With three equilibria, homoclinic bifurcations appear for some parameters in both system (1.2b) and (6.3). But the number of limit cycles coexisting with the figure eight loop is different for system (1.2b) and (6.3). For system (1.2b), there are at least three limit cycles coexisting with the figure eight loop, one is surrounding all the equilibria, the others are surrounding one equilibrium and symmetrically distributed about the origin. For system (6.3), there is one limit cycle coexisting with the figure eight loop, which is surrounding all the equilibria. As a result, with the rupture of figure eight loops, system (1.2b) can generate at least five limit cycles, while system (6.3) can only generate three limit cycles.
Due to [3], system (6.3) has four equilibria at infinity. System (1.2b) has two equilibria at infinity by Lemma 2.3. However, all the orbits are positively bounded for both system (1.2b) and (6.3).
The nonsmooth van der Pol-Duffing oscillatoṙ is studied recently by [25] and its bifurcation diagram and global phase portraits are given. The nonsmooth van der Pol-Duffing oscillator and smooth one have similar behaviors at infinity. Except that the number of limit cycles are different between system (1.2b) and system (6.4), a bifurcation for equilibria at infinity for system (6.4) cannot occur in system (1.2b). In addition, under external force (such as periodically driving), a periodically driven nonsmooth Rayleigh-Duffing oscillator can be rewritten as ẋ = y, y = −cx − dx 3 − (a + b|y|)y + εF(t), (6.5) where F(t) is a periodic function and > 0 is small. Then, the 2-saddle loop and figure-eight loop will break, and we need to consider whether the stable man-ifold and the unstable one of system (6.5) may transversely intersect with each other. In other words, system (6.5) has chaotic dynamics by Birkhoff-Smale Theory if the stable manifold and the unstable one of system (6.5) may transversely intersect with each other. Notice that the chaotic dynamics of smooth Rayleigh-Duffing oscillators have been shown in [18,21]. However, since system (1.1) is nonsmooth, the theoretical and numerical investigation of chaotic behavior of the periodically driven nonsmooth Rayleigh-Duffing oscillator will be an interesting and more difficult topic.
Let (d i , 0) be the intersection point of γ i and the x-axis, where 0 < d i < 1 and i = 1, 2, 3. Since the vector filed of system (6.7) is C 1 and γ 1 , γ 2 , γ 3 are all simple, for sufficiently small |δ| or | | there are also at least three limit cycles when (α, β) → (α + δ, β) or (α, β) → (α, β + ), in which the three innermost ones are stable, unstable and stable. Denote the intersection points of the positive x-axis and the three innermost limit cycles bifurcated from perturbation of α (resp. β) by d δ 1 , d δ 2 and d δ 3 (resp. d 2 , d 2 and when δ > 0 and In other words, for system (6.7), the stable limit cycles expand and the unstable limit cycle compresses when α increases for a fixed β, the stable limit cycles compress and the unstable limit cycle expands when β decreases for a fixed α.
Fix β and increase α < 0 until γ 1 and γ 2 overlap or γ 3 disappears by coinciding with the outer closed orbit. If the increase of α stops after it reaches α * , system (6.7) has at least three limit cycles for α = α * , which conflicts with the proven fact. If the increase of α stops before it reaches α * , denote the stop point of α by α (1) and the limit cycle generated by expansion or compression of γ i by γ We claim that this cyclic process will stop in finite steps. In fact, from the proof of [2, Lemma 3.2] the solutions of system (6.7) depend on α and β in the same degree. If n goes to infinity, we have lim n→∞ (d (n) 1 −d (n) 3 ) = 0, implying lim n→∞ (α (2n+1) − α (2n−1) ) = 0. Thus, after finite steps the increase of α stops after it reaches α * , Therefore, system (6.7) has at least three limit cycles for α = α * , which is a contradiction. Furthermore, when 0 < β < −α system (6.6) has at most two limit cycles for general α < 0.
Finally, assume that there are two limit cycles for system (6.6). If only one of two limit cycles is semistable, by [2, Lemma 3.2] and [27, Theorem 3.4 of Chapter 3.4], at least two limit cycles will be bifurcated from the semi-stable one and the other one will not disappear. If both two limit cycles are semi-stable, since they are adjacent to each other, at least four limit cycles bifurcated from them. Then one limit cycle is stable and the other one is unstable. By [2, Lemma 2.1], the origin is unstable when α < 0. Thus, the inner one is internally stable, indicating the inner limit cycle is stable and the outer one is unstable. Moreover, if there is a unique small limit cycle, we can only obtain it is internally stable, which completes the proof.
Remark that the methods of the proof in Lemma 6.1 can also be used to prove [6,Proposition 5.3] in the focus case of a Rayleigh-Duffing oscillator, which shows that at most two limit cycles surround the equilibrium E R and a double limit cycle bifurcation happens.
Proof When 0 < β = ϕ(α) and α < 0, the existence of limit cycles comes from the internal instability of the 2-saddle loop and instability of the origin. By Lemma 6.1, there is a unique limit cycle coexisting with the 2saddle loop. The statement (a) is proved. By [2, Lemma 3.2] and [27, Theorem 3.4 of Chapter 3.4], there exists a > 0 such that system (6.7) exhibits two limit cycles when β = ϕ(α) + , where the inner one is stable and the outer one is unstable. Moreover, the stable limit cycles expand and the unstable limit cycle compresses when β increases for a fixed α. Due to [2, Lemma 3.3], there are no limit cycles for system (6.7) when β = −α.
Then there exists a function ψ(α) for α < 0 such that the stable limit cycles and the unstable limit cycle overlap if and only if β = ψ(α). The statements (b)-(d) are proved.
For any α * < 0 and sufficiently small > 0, system (6.6) exhibits two limit cycles when α = α * − and β = ψ(α * ) by [2, Lemma 3.2] and Lemma 6.1. Moreover, the inner limit cycle is stable and the outer one is unstable. To keep the existence of the semi-stable limit cycle, we need to increase β until the two limit cycles coincide. Then, ψ(α * − ) > ψ(α * ), which means that ψ(α) is decreasing. Moreover, from the proof of [2, Lemma 3.2] the ordinates of the intersections of limit cycles with y-axis continuously depend on α and β, which implies the continuity of ψ(α). It completes the proof.
In addition, numerical simulations in [2, Fig. 12(d)] and [2, Fig. 13(b)] show that system (6.6) exhibits a unique limit cycle near the saddles. That implies DL and H L are very close to each other.

Fig. 25
Bifurcation diagram and revised global phase portraits of system (6.6) Lemma 3.5] the origin is unstable and the 2-saddle loop is internally unstable if it exists. Then system (6.14) exhibits at least two limit cycles for some parameters.
Unlike smooth system (6.6), Abelian integrals cannot be used to prove that there are at most two limit cycles for nonsmooth system (6.14) because it is hard to determine the sign of the derivative of the ratio of an elliptic integral and an elementary function. Efforts are also made to rewrite system (6.14) as a piecewise Liénard system x + f (x, sgn(y))ẋ + g(x, sgn(y)) = 0 and study the number of limit cycles. Although this method can be applied to smooth system (6.6), it also fails to nonsmooth system (6.14). That is because many mature theories about the number of limit cycles in Liénard system cannot be used when f is not an even function of x. However, we can prove that there is a region where only stable limit cycles may exist. Lemma 6.3 When μ 1 < 0 and −(−4μ 1 /3) 3/4 < μ 2 < 0, system (1.2b) exhibits at most one limit cycle Proof Assume that there is a limit cycle in the region {(x, y) : |x| ≤ √ −μ 1 /3} for system (6.14).
Proof The conclusion (a) is exact the same as [26, Proposition 3.1 (a)], which comes from the rotation of vector field and is not affected by the number of limit cycles. Moreover, the 2-saddle loop is internally unstable since the tr J R = tr J L = −μ 2 > 0 when μ 2 = ϕ(μ 1 ), where Jacobian matrices J R and J L are defined in [26,Equation (2.3)]. Since the origin of system (6.14) is unstable by [26, Lemma 2.2], at least one limit cycle exists. Denote the outmost limit cycle by . Clearly, is externally stable. When μ 2 > ϕ(μ 1 ), the existence of limit cycles can be obtained by the same methods in [26, Proposition 3.1 (b)]. When μ 2 = ϕ(μ 1 ) − ε, by [26, Lemma 3.2], at least one limit cycles will be bifurcated from the 2-saddle loop, which is unstable and at least one limit cycles will be bifurcated from , which is stable. Then, the conclusion (c) is proved.
Numerical simulations in [2, Fig. 16(b)] and [2, Fig.  18(b)] show that system (6.14) exhibits a unique limit cycle near the saddles, which provide evidence that DL and H L are very close to each other.