GLOBAL DYNAMICS OF A SD OSCILLATOR

In this paper we derive the global bifurcation diagrams of a SD oscillator which exhibits both smooth and discontinuous dynamics depending on the value of a parameter a. We research all possible bifurcations of this system, including Pitchfork bifurcation, degenerate Hopf bifurcation, Homoclinic bifurcation, Double limit cycle bifurcation, Bautin bifurcation and BogdanovTakens bifurcation. Besides we prove that the system has at most five limit cycles. At last, we give all numerical phase portraits to illustrate our results.


Introduction and main results
In recent years SD (Smooth and Discontinuous, for short) oscillator was proposed and investigated for studying the transition from smooth to discontinuous dynamics, see for instance [2,3,4,5,16]. In those papers it is proposed the elastic beam model for studying this transition, where a ≥ 0, b and ξ can take arbitrary real values. More precisely, the smooth dynamics appears when a > 0, while the discontinuous dynamic behavior occurs at a = 0. The global dynamics was completely studied in [5] when a = 0, and in [16] when |a − 1| < ε, |ξ| < ε and ε is sufficiently small.
In this paper we call large limit cycles to the ones surrounding all three equilibria, and small limit cycles the ones surrounding a single equilibrium. For the notions and definitions which appear in the statement of Theorem 1 see its proof.
The paper is organized as follows. In section 2 we analyze the local bifurcations, namely pitchfork bifurcation, Hopf bifurcation, Bautin bifurcation and codimension 2 Bogdanov-Takens bifurcation with symmetry. In section 3 we estimate the number of limit cycles in difference parameter regions and curves. In section 4 we study the global bifurcations, namely the different kinds of homoclinic connections and the double limit cycles. In section 5 we give the numerical phase portraits in difference parameter regions.

Local bifurcations
Computing the Jacobian matrix at equilibrium E 0 , we have Then, at E 0 the determinant det(J 0 ) = 1−1/a and the trace tr(J 0 ) = −bξ, implying that E 0 is a saddle if a < 1, stable focus or node if a > 1 and b > 0, and unstable focus or node if a > 1 and b < 0. When a = 1, equilibrium E 0 is a stable node if b > 0, and unstable node if b < 0 by applying [8,Theorem 2.19]. By the symmetry of the vector field (2), equilibrium E L is of the same type as E R . The Jacobian matrix at E R is Calculating det(J R ) = 1 − a 2 and tr(J R ) = −ξ(b + 3 − 3a 2 ). Hence E R is a stable focus or node if a < 1 and b + 3 − 3a 2 > 0, and unstable focus or node if a < 1 and b + 3 − 3a 2 < 0.
2.1. Pitchfork bifurcation. From the expressions of the equilibria E L , E 0 and E R it follows immediately that a pitchfork bifurcation occurs at the origin of coordinates when a = 1; i.e. for a ≥ 1 we have a unique antisaddle, while for 0 < a < 1 from the previous antisaddle it bifurcates at a = 1 a saddle and two antisaddles. For more details on this kind of bifurcation see [7,10].

Hopf bifurcations.
There are two kinds of Hopf bifurcations, one at the equilibrium E 0 , and the other at the equilibria E L and E R , which is essentially the same bifurcation in both, because due to the invariance of system (2) with respect to the symmetry (x, y) → (−x, −y), what occurs at the equilibrium point E L occurs to its symmetric E R .

2.2.1
Hopf bifurcation at E 0 . The next result characterizes the Hopf bifurcation at the equilibrium point E 0 , it is proved using the averaging theory in this way we also can estimated the shape of the limit cycle bifurcating from E 0 , and we avoid the computation of the Liapunov constant.
Proposition 2. The following statements hold for the differential system (2).
(a) If a > 1, b = 0 and ξ > 0, then a Hopf bifurcation takes place at the equilibrium point located at the origin of coordinates, and the limit cycle γ bifurcating from this equilibrium exists for b < 0 sufficiently small.
(b) For ε > 0 sufficiently small if b = −βε 2 < 0, then the limit cycle γ passes through the point Moreover, this limit cycle is stable.
Since we want to study the Hopf bifurcation at the origin of coordinates we blow up the origin doing the scaling r = εR, then differential system (3) taking as new independent variable the θ becomes In order to apply the averaging theory described in the appendix we need that the differential equation (4) starts at least with order ε. So we do the change of variables R → ρ defined by Then differential equation (4) in the new variable ρ writes Differential equation (5) is written into the normal form (37) for applying the averaging theory summarized in the appendix, using the notation of the appendix we only need to take n = 1, x = ρ, t = θ, µ = ε 2 , F 1 (t, x) = F 1 (θ, ρ) and T = 2π, all the necessary assumptions for applying the averaging theory described in the appendix are satisfied. Then we compute Since β > 0 the averaged function f 1 (ρ) has a unique positive zero, ρ = 2 β/3 which satisfies the condition In fact this last expression is positive because a > 1 and ξ > 0, and consequently, by the results described in the appendix the differential equation (5) has a periodic solution ρ(θ, ε) satisfying that Moreover, this periodic solution ρ(θ, ε) is unstable because the derivative (6) is positive. Now we will go back through the changes of variables for obtaining the periodic solution bifurcating from the equilibrium at the origin of coordinates of the differential system (2). Thus, the periodic solution ρ(θ, ε) satisfying the initial condition (7) in the variables of the differential system (4) becomes the periodic solution This periodic solution in the differential system (3) becomes (r(t, ε), θ(t, ε)) with and it pass through the point in the coordinates (r, θ). Finally, this periodic solution in the coordinates of system (2) is the periodic solution (x(t, ε), y(t, ε)) given by ρ(θ(t, ε), ε) cos(θ(t, ε)), sin(θ(t, ε)) + O(ε 3 ), passing through the point (8) now in coordinates (x, y). Therefore, when ε → 0 such periodic solution tends to the origin, so it is a periodic solution of a Hopf bifurcation.
We remark that the periodic solution ρ(θ, ε) was an unstable limit cycle, but due to the fact thatθ is negative in a neighborhood of the origin, when we pass the unstable limit cycle R(θ, ε) to the periodic solution (r(t, ε), θ(t, ε)) it changes to a stable limit cycle.
In short, Proposition 2 shows the existence of the Hopf bifurcation surface H 1 .

Hopf bifurcation at
Proposition 3. The following statements hold for the differential system (2).
We translate the equilibrium point E R to the origin of coordinates doing the change and system (2) is transformed into Writing the differential system (9) in polar coordinates we get Again since we want to study the Hopf bifurcation now at the origin of coordinates we blow up the origin doing the scaling r = εR, then differential system (10) taking as new independent variable the θ writes Again for applying the averaging theory of the appendix we need that the differential equation (11) starts at least with order ε. Hence we do the change of variables R → ρ defined by Then differential equation (11) in the new variable ρ writes Differential equation (12) is already into the normal form (37) for applying the averaging theory of the appendix. Again using the notation of the appendix we take n = 1, x = ρ, t = θ, µ = ε, F 1 (t, x) = F 1 (θ, ρ) and T = 2π, and all the necessary hypotheses for applying the averaging theory of the appendix hold. Then we compute Since the first averaged function f 1 (ρ) is identically zero, we must compute the second one f 2 (ρ). We start calculating Then the second averaged function has a unique positive zero, ρ = 2 β/(3(3a 2 − 1)), recall that a = 1/ √ 3. This zero satisfies the condition by assumptions. Consequently, by the results described in the appendix the differential equation (12) has a periodic solution ρ(θ, ε) satisfying that Moreover, from (13) this periodic solution ρ(θ, ε) is unstable if β < 0, and stable if β > 0. Now we will go back through the changes of variables for obtaining the periodic solution bifurcating from the equilibrium E R of the differential system (2). Thus, the periodic solution ρ(θ, ε) satisfying the initial condition (14) in the variables of the differential system (11) becomes the periodic solution This periodic solution in the differential system (10) becomes (r(t, ε), θ(t, ε)) with and it pass through the point in the coordinates (r, θ). This periodic solution in the coordinates of system (9) is the periodic solution (X(t, ε), Y (t, ε)) given by passing through the point (15) now in coordinates (X, Y ). Finally, we get the periodic solution (x(t, ε), y(t, ε)) given by and passing through the point in coordinates (x, y). So, when ε → 0 such periodic solution tends to the equilibrium E R , so it is a periodic solution of a Hopf bifurcation.
Again we note that the limit cycle ρ(θ, ε) was unstable if β < 0, and stable if β > 0, but due to the fact thatθ is negative in a neighborhood of the origin, when we pass the limit cycle R(θ, ε) to the cycle (r(t, ε), θ(t, ε)) it changes its type of stability.
In particular Proposition 3 shows the existence of the Hopf bifurcation surfaces H 2 and H 3 .

2.2.3
The Bautin bifurcation curve B 0 . The standard or classical Hopf bifurcation in a 2-dimensional differential system, i.e. that a limit cycle bifurcates from an equilibrium point, takes place in an equilibrium point with purely imaginary eigenvalues which is not a center because the first Liapunov constant at that equilibrium is not zero. These are the Hopf bifurcations studied in subsection 2.2.2. But when the first Liapunov constant is zero, also can bifurcate a limit cycle of the equilibrium point if the second Liapunov constant is not zero, such more degenerate Hopf bifurcation is called for some authors a Bautin bifurcation. See for more details about these Hopf bifurcations [11,Chapter 8].
where, for simplicity, we still use the variables (x, y, t) instead of the new ones (x 1 , y 1 , τ ). From [10, p. 156] we compute the first Liapunov constant at the origin of system (16) and we getĝ 1 = 3ξ( . From the expressions of g 1 , we can confirm that the classical Hopf bifurcation happens when a = 1/ √ 3 and b = 3a 2 − 3. Clearly, when a = 1/ √ 3 and b = −2 we haveĝ 1 = 0. By [11,Chapter 8], we can obtain that the second Liapunov constantĝ 2 = 5 √ 6 ξ/32 > 0 at those values of a and b. In short, system (2)  2.4. The dynamics near infinity. In this subsection we will discuss the qualitative properties of the equilibria at infinity, which describe the behavior of the orbits of system (2) when x and y are sufficiently large.
Proposition 4. As shown in Figure 3 the differential system (2) with ξ > 0 has four equilibria at infinity I ± A , I ± B , where I ± A are the two endpoints of the x-axis, and I ± B are the two endpoints of the y-axis. The equilibria I ± A are unstable star Proof. Doing the Poincaré transformation x = 1/z, y = u/z, system (2) becomes where dt = z 2 dτ . Obviously, this system has a unique equilibrium A : (0, 0) on the u-axis (the infinity), where A is an unstable star node. Doing the other Poincaré transformation x = v/z, y = 1/z, system (2) writes as where dt = z 2 dτ . In this local chart we only need to study the equilibrium B : (0, 0) of system (17), which corresponds to two equilibria I B + and I B − at infinity of the system (2) at the endpoints of the positive and negative y-semiaxes, respectively. By Lemmas 1 and 3 of [14, Chapter 2] we only need to discuss the orbits along characteristic directions of system (17) at B.
Hence a necessary condition for θ = θ 0 to be an characteristic direction is G(θ 0 ) = 0, which has exactly two roots 0 and π. Except these two directions, there are no directions along which system (17) has orbits connecting B.
Notice that the vector field (17) is symmetric with respect to the v-axis. Thus, we only need to discuss the orbits connecting the origin B of (17) in the half plane z ≥ 0. We will construct some related open quasi-sectors to determine how many orbits of (17) connecting B in the first and the second quadrants.
Observing that system (17) has four horizontal isoclines: the v-axis, the z-axis and where > 0 is a sufficiently small constant. Set The possible vertical isocline is Obviously, the isocline V is tangent to the v-axis at the origin. Set where σ > 0 is a small constant. Hence, if there exist orbits of system (17) connecting B along the direction of the v-axis in the first and the second quadrants, then near the origin must lie in the sector regions ∆ The directions of vector field of (17), i.e., the directions of arrows, and the positions of the isoclines are shown in Figure 4. Firstly, we consider the case a < 1. We can check thatv > 0 andż > 0 in ∆ VBH + ;v < 0 andż > 0 in ∆ V + BV; andv > 0 andż < 0 in ∆ V − BH − . Lemma 4 in [15] guarantees that no orbits connect B in ∆ V + BV. There are also no orbits connecting B in the interior of ∆ is not equal to the slopes of the curves tangent to the v-axis. On the other hand, we compute that (∂/∂v)(Ψ 1 (v, z)/Ψ 2 (v, z)) < 0 in the generalized normal sector ∆ VBH + of class II, i.e.ṙ > 0 in ∆ VBH + and all positive semi-orbits starting from the curves BV and BH + go into ∆ VBH + . The definition of generalized normal sectors can be seen in [15,Section 2]. Therefore, there exists a unique orbit leaving from B in ∆ VBH + by Lemma 2 and Lemma 5 in [15].
Similarly, in case a = 1, we can also prove that exactly one orbit connects B along the v-axis, which lies in ∆ VBL + .

Limit cycles
Lemma 5. Assume that a ≥ 1. System (2) has no limit cycles if b ≥ 0, and a unique limit cycle if b < 0.
When b < 0, the following conditions are satisfied.
(iv) f (x) and g(x) satisfy the Lipschitz condition in any bounded interval. Then, by Theorem 4.1 of [18, Chapter 4], the system (2) has a unique limit cycle, which is stable.
Note that the phase portraits (b) and (c) of Figure 2 in Theorem 1 are obtained from Lemma 5 and from the properties of equilibria.
Since the phase portrait of system (2) is symmetric with respect to the point E 0 , the small limit cycles surrounding E L are of the same type as that surrounding E R . Hence, in what follows we only consider the small limit cycles around E R . Lemma 6. If 0 < a < 1 and b ≥ a 2 − 1, then system (2) has no limit cycles.
Since the proof is similar to Lemma 4 of [5], we omit it. Lemma 6 shows that φ 1 (a, ξ) < a 2 − 1 and the phase portrait (a) of Figure 2 in Theorem 1 is obtained.
Consider equation where bothF (z) andF (z) are continuous in [0, z 0 ), andF (0) = 0. Let L J denote the integral curve of (18) passing through the point P (z J ,F (z J )) on the curve y =F (z). Also, let y = ϕ J (z) and y =φ J (z) represent the orbit segments of L J below and above the curve y =F (z). When 0 < z < z J , we clearly have ϕ J (z) <F (z) <φ J (z) and ϕ J (z) > 0 >φ J (z). Moreover, we introduce the symbol Then, we have for some a 0 .
Lemma 7 will be applied in the following lemma.
Lemma 8. If 0 < a < 1 and b < a 2 − 1, then system (2) has at most two large limit cycles. Proof. Assume that system (2) has at least two large limit cycles surrounding the three equilibria E L , E 0 and E R , and that L 1 and L 2 are the most external limit cycles, where L 2 denotes the outer one. We first consider 3a 2 − 3 ≤ b < a 2 − 1. The corresponding phase portrait is shown in Figure 5(a). By the Bendixson criterium, each L i has two intersection points, denoted by B i and C i (i = 1, 2) with the straight line x = x 0 , where x 0 is the abscissa of the equilibrium E R , as shown in Figure 5 for i = 1, 2. On the arcs A 1 B 1 and A 2 B 2 , let y = y 1 (x) and y = y 2 (x), respectively. In fact, for each i = 1, 2, we have Similarly we obtain that By Lemma 7, in order to prove the inequality , we only need to prove thatF (z(x 0 )) = 0,F (z) > 0 andF (z)F (z) is nondecreasing for z > z(x 0 ). Clearly we haveF (z(x 0 )) = 0 and F (z) > 0 for z > z(x 0 ). Note that .
where the three factors of the last line are positive and increasing. Therefore, is positive and increasing. By Lemma 7, we have that Now we consider b < 3a 2 − 3. The corresponding phase portrait is shown in Figure 5(b). By the Bendixson criterium again, each L i has two intersection points with the straight line x = x 0 , being x 0 the unique positive zero of F (x) when x > 0, and the intersection points are denoted by B i and C i (i = 1, 2). Then the inequality (22) can be proved in a similar way to the case 3a Similarly we obtain that Now, using again Lemma 7, we only need to prove thatF (z(x 0 )) = 0, andF (z) > 0, where all of the factors of the last two lines are positive and increasing. Therefore is positive and increasing. By Lemma 7, we have and therefore (22) follows. However, it is impossible to have two attracting (repelling) limit cycles surrounding the same equilibrium (equilibria) adjacent one to the other. So, from the inequality (22) and the repelling of the infinity, we obtain that system (2) has at most three large limit cycles, where the outer one is stable, the middle one is semistable, the inner one is stable. Clearly, for fixed values a and ξ, system (2) is a family of generalized rotated vector fields with respect to the parameter b. Assume that system (2) has exactly three large limit cycles. By Theorem 3.5 of [18,Chapter 4], the outer limit cycle and the inner one neither split, nor disappear as b varies monotonically. By Theorem 3.4 of [18, Chapter 4], the middle limit cycle will bifurcate into at least one stable and one unstable cycle when b varies in the suitable direction. This is a contradiction. Therefore, system (2) has at most two large limit cycles. If the two large limit cycles exist, we can obtain that the outer limit cycle is stable and the inner one is unstable.
Then system (24) has at most one closed orbit in the region {(x, y) ∈ R 2 : α < x < β}. The closed orbit is simple and unstable (resp. stable) if it exists.
Proof. Assume that system (24) exists a limit cycle γ, as shown in Figure 6(a). In the following we will ascertain the sign of Clearly w =F (x) has two inverse functions, x 1 (w) (respectively x 2 (w)), on the right (resp. left) side of the origin. The functionsλ(x i (w)) will be denoted simply by λ i (w). By w =F (x), we rewrite system (24) intȯ Let y 1 (w) and y 2 (w)(resp. z 1 (w) and z 2 (w)) be functions determined by the orbits of (26) below (resp. above) the line y = w, which correspond the parts of the trajectories of system (24) below (resp. above) the curve y =F (x) and depend whether they are to the left or right of the origin.
So in the region {(x, y) ∈ R 2 : α < x < β} γ is unstable and simple if it exists. Moreover, it is impossible to have two attracting (repelling) limit cycles surrounding the same equilibrium adjacent one to the other. Therefore, the uniqueness has also been proved. For the case λ 1 (w) < λ 2 (w) as w is small, we can prove that γ f (x)dt > 0 in a similar way to the case λ 1 (w) > λ 2 (w). So, in the region {(x, y) ∈ R 2 : α < x < β}, the limit cycle γ is stable and simple if it exists. Therefore, we have completed this proof.
The proof of Proposition 9 gives the following corollary directly.
Under the preparations of above Proposition 9 and Corollary 10, we obtain the existence of small limit cycles on the parameter surfaces H 2 and H 3 as follows.
Proof. If 1/ √ 3 ≤ a < 1 and b = 3a 2 − 3, i.e., on the parameter curve H 2 we obtain Figure 7 by Proposition 4 and Lemma 11, which shows the existence of a Poincaré-Bendixson annulus, i.e., any trajectory starting at a point of the boundary curves of the annulus enters (or leaves) the annulus, and inside the annulus there is no equilibrium points. So, the existence of some large limit cycles is obtained. Assume that system (2) has two large limit cycles. Let the outer limit cycle and inner one denoted by γ 2 and γ 1 . Therefore γi div(y−F (x), −g(x))dt ≤ 0 for i = 1, 2. By (22), γ1 div(y − F (x), −g(x))dt > γ2 div(y − F (x), −g(x))dt. However, it is impossible to have two attracting (repelling) limit cycles surrounding the same equilibrium (equilibria) adjacent one to the other. Therefore So γ 1 is a semistable limit cycle. Since the vector field of (2) is rotating with respect to b, by Theorem 3.4 of [18, p.211] there is a stable limit cycleγ 2 near γ 2 for a perturbation of b, and two limit cyclesγ 1 ,γ 1 (γ 1 is smaller thanγ 1 ) near γ 1 . By (22) and stabilities of equilibria, we obtain γ1 div(y − F (x), −g(x))dt ≤ 0, γ1 div(y − F (x), −g(x))dt ≥ 0, and γ2 div(y − F (x), −g(x))dt < 0, GLOBAL DYNAMICS OF A SD OSCILLATOR 23 which contradicts γ1 div(y − F (x), −g(x))dt > γ1 div(y − F (x), −g(x))dt. Therefore, system (2) has at most one large limit cycle. Thus, the uniqueness of limit cycle is proved and the phase portrait (d) of Figure 2 in Theorem 1 is obtained.
Moreover, we can get the phase portraits (i), (e) and (n) of Figure 2 in Theorem 1 by Proposition 3, Lemmas 8, 11, 12 and the continuity of the vector fields.
Proposition 13. System (2) has at most two small limit cycles around the equilibria E R or E L for the values of the parameters in Proof. By the homeomorphism (x, y) → (x, y + F (x)), system (2) can be rewritten asẋ = y, In x > 0 we do the change of variables w = √ x 2 + a 2 and the time scaling dt = 1 + a 2 /x 2 dτ to system (33), and we geṫ w = y, When ξ = 0 this system is the Hamiltonian systeṁ with the first integral Its level curves Γ h := {(w, y) : H(w, y) = h, −1/2 ≤ h < a 2 /2 − a} are shown in Figure 8. Of course H = −1/2 corresponds to the center (1, 0), and for the values of h such that −1/2 < h < a 2 /2 − a, the curve H(w, y) = h corresponds to a periodic orbit of Hamiltonian system (35) surrounding the point (1, 0), which intersects the positive half w-axis inside the interval (a, 1). Now we consider system (34) as a perturbation of system (35) for small ξ. Here we only will discuss how many small limit cycles surround the equilibrium point (1, 0) of system (34) when ξ is sufficiently small. On the other hand, for every h ∈ (−1/2, a 2 /2−a) the orbit Γ h intersects the the segment L 1 : (a, 1) of the w-axis at exactly one point Q h (x(h), 0). Therefore, the segment L 1 can be parameterized by h ∈ (−1/2, a 2 /2 − a).
For every h ∈ (−1/2, a 2 /2 − a), we consider the trajectory of system (33) passing through the point Q h (x(h), 0) ∈ L 1 . This trajectory goes forward and backward until it intersects the positive w-axis at points Q 1 and Q 2 , respectively, as in Figure  9. We denote the piece of trajectory from Q 2 to Q 1 by γ(h, ξ, a, b).  Therefore γ(h, ξ, a, b) is a periodic orbit if and only if F (h, ξ, a, b) = 0. We consider F (h, ξ, a, b) as a perturbation of F (h, 0, a, b). The function F (h, 0, a, b) is given by The orientation of Γ h is determined by the direction of the vector field (35). By the Green's formula where D(h) is the region surrounded by Γ h .
H 3 , which leads to a contradiction. Therefore system (2) has at most one small limit cycle in G 2 . We have completed the proof.
By a similar discussion as that in the proof of Lemma 14, the phase portraits (f)-(h) and (j)-(m) of system (2) in Figure 2 of Theorem 1 are obtained from the properties of rotational vector fields, continuity, Lemma 14 and the results of section 3.
Remark 1. The surface DL 2 of double small limit cycles is the graph of a function b = ϕ 2 (a, ξ).
From [5] we can obtain that a pair of grazing loop are stable for a = 0 if they exist. However, the pair of homoclinic loops have to be unstable if they exist when a = 0. In fact, HL, DL 1 , DL 2 have a common intersection point for the limit value a = 0, i.e., ϕ(0, ξ) = φ 1 (0, ξ) = φ 1 (0, ξ) < −3. Since system (33) is a rotational vector field with respect to the parameter b, the manifolds of E 0 move monotonically as a, ξ are fixed and b increases, see [5,6]. Therefore, it is worthwhile to note that HL, DL 1 and DL 2 have no intersection points except at endpoints. Summarizing the previous results, we can obtain Theorem 1, as shown in Figure 1. In this section we give several numerical examples of previous results. Example 1. Let a = √ 2 and ξ = 1. When b = 1 the system has a unique equilibrium (0, 0), which is a sink, and no limit cycles, as shown in Figure 10(a).
However, when b = −1 the system has a unique equilibrium, the origin (0, 0) which is a source. Furthermore, from Lemma 5, there is a unique limit cycle, which is stable, as shown in Figure 10(b).
Example 2. Let ξ = 1. When a = √ 2/2 and b = −2 the system has three equilibria and exactly one large limit cycle, as shown in Figure 11(a).
When a = 0.3 and b = −2.77 the system has three equilibria and exactly two small limit cycles, as shown in Figure 11(b).
When a = √ 2/2 and b = −0.5 the system has three equilibria and no limit cycles, as shown in Figure 11(c). Example 3. Let ξ = 0.1. When a = 0.9 and b = −0.46 the system has three equilibria and exactly two large limit cycles, as shown in Figure 12(a).
When a = 0.9 and b = −0.555 the system has three equilibria, exactly two small limit cycles and one large limit cycle, as shown in Figure 12 When a = 3 √ 2/10 and b = −2.461 the system has three equilibria, exactly two small limit cycles and two large limit cycles, as shown in Figure 12(c).
When a = √ 5/5 and b = −2.41 the system has three equilibria, exactly four small limit cycles and one large limit cycle, as shown in Figure 12(d).