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Double canard cycles in singularly perturbed planar systems with two canard points

We consider double canard cycles including two canards in singularly perturbed planar systems with two canard points. Previous work studied the complex oscillations including relaxation oscillations and canard cycles in singularly perturbed planar systems with one-parameter layer equations, which have precisely one canard point, two jump points or one canard point and one jump point. Based on the normal form theory, blow-up technique and Melnikov theory, we investigate double canard cycles induced by two Hopf breaking mechanisms at two non-degenerate canard points. Finally, we apply the obtained results to a class of cubic Lienard equations with quadratic damping.


Introduction
Singular perturbation problems induced by multiple time scales widely appear in applied science and engineering, such as cellular physiology, fluid mechanics, population dynamics and so on [29,30,33,40]. These singularly perturbed systems are also referred to as slow-fast systems. Based on the normally hyperbolic invariant manifolds theory (see, for instance, [20,21,22,45]), Fenichel [23] in 1979 laid the foundation of geometric singular perturbation theory (abbreviated as GSPT) to investigate multiple time scales dynamics. Since then, GSPT has become an hotspot research subject in the field of dynamical systems. There is an enormous literature on this topic. We refer the readers to the excellent survey articles [8,9,10,11,19,27,28,34,41] and the references therein.
In this paper we consider a singularly perturbed planar system in the form ε dx dτ = εẋ = f (x, y, µ, ε), dy dτ =ẏ = g(x, y, µ, ε), where (x, y) ∈ R 2 , µ = (λ, η) = (λ, η 1 , ..., η m ) ∈ R × R m with m ≥ 1, the parameter ε satisfies 0 < ε ≤ ε 0 1 for some small ε 0 , and the functions f and g are C k with k ≥ 3. By a time rescaling τ = εt, system (1.1) is changed to dx dt = x = f (x, y, µ, ε), dy dt = y = εg(x, y, µ, ε). For each fixed µ, we observe that the phase state of system (1.3) is defined on the set of equilibria of system (1.4), that is, This set is called the critical set. If it is a submanifold of R 2 , then it is called the critical manifold. The branches of the set C µ,0 are called the slow curves. By the Fenichel theory [23], a normally hyperbolic submanifold M µ,0 (with or without boundary) of the critical manifold C µ,0 is perturbed to a slow manifold M µ,ε near M µ,0 for sufficiently small ε. We call M µ,0 the normally hyperbolic manifold if ∂f /∂x = 0 along M µ,0 . The points in C µ,0 with ∂f /∂x = 0 are called the contact points, where the normal hyperbolicity breaks down. The most generic contact points are jump points, for which the reduced flow (1.3) directs towards the contact points. More degenerate contact points are canard points, a simple zero of the function g in system (1.2), which leads to a possibility of periodic orbits in its neighborhood. Canard points are also called the turning points in some references. Geometric analysis of the contact points was initiated in [11], where Dumortier and Roussarie applied the blow-up technique to study the singularly perturbed van der Pol equation. Following the pioneering work [11] of Dumortier and Roussarie, many efforts have been devoted to expand the capabilities of this technique. For example, Krupa and Szmolyan used the technique provided in [11] to extend the slow manifolds of planar singularly perturbed systems near jump points and canard points, and more results on the blow-up technique and its generalizations are referred to [1,4,5,15,33].
Jump points and canard points in planar singularly perturbed systems with one-parameter layer equations can lead to relaxation oscillations and canard solutions, respectively. Relaxtion oscillation is a periodic orbit which spends a long time along the slow manifold towards a jump point, jumps from this contact point, spends a short time parallel to the fast orbits towards another stable slow manifold, follows the slow manifold again until another jump point is reached, and finally returns to its starting point via several similarly successive motions [24,32]. Canards are the orbits contained in the intersection of an attracting slow manifold and a repelling slow manifold. Canards are subject to a generic Hopf breaking mechanism, that is, the flow of the layer equation (1.4) has the same direction on a attracting slow curve and a repelling slow curve which are connected by a generic canard point. Periodic orbits containing canards are referred to as canard cycles.
Relaxation oscillations and canard cycles can be both seen as the perturbations of slow-fast cycles which are closed loops formed by a connected succession of critical manifolds and fast orbits of the layer equations. More precisely, relaxation oscillations arise from common cycles which only contain repelling critical manifolds or attracting critical manifolds (see Figure 1(a)), canard cycles from canard slow-fast cycles which contain at least one attracting and one repelling critical manifolds (see Figures 1(b), 1(c) and 1(d)). It is worth mentioning that much work on relaxation oscillations, canard cycles with head and canard cycles without head bifurcating from slow-fast cycles assumed that planar singularly perturbed systems have precisely one canard points, two jump points or one canard point and one jump point. However, there are numerous models of the form (1.1) with S-shaped critical manifolds possessing two canard points in real world applications, such as in a circadian oscillator model based on dimerization and proteolysis of PER and TIM proteins in Drosophila [2,42], a class of cubic Liénard equations with quadratic damping [13], a predator-prey model of generalized Holling type III [43] and so on. Stimulated by it, we studied the dynamics of system (1.2) with a S-shaped critical manifold, which has precisely two canard points and one saddle lying on the repelling slow curve. Under a certain condition, there is the simultaneous occurrence of two generic Hopf breaking mechanisms. We establish the existence of double canard cycles, that is, canard cycles include two canards. This canard cycle can be seen as the limit cycle bifurcating from a double canard slow-fast cycle or two-layer canard slow-fast cycles [17], that is, a slow-fast canard cycle passes through two layers of fast orbits and contains two generic Hopf breaking mechanisms at two non-degenerate canard points. See Figure   2. The proofs for these results are based on the normal forms established near the canard points, the blow-up technique and the extended Melnikov theory obtained by Wechselberger in [44].
This paper is organized as follows. In section 2 we make some hypotheses and state the main results on the existence of double canard cycles. In section 3 we give the normal forms of system (1.2) near canard points.
Section 4 is contributed to investigating the existence of the canards near canard points by the Melnikov theory. The proof for the main results is given in section 5 and then we apply the main results to cubic Liénard equations with quadratic damping in section 6. In the finial section, we make some remarks on the limit cycles arising from planar singularly perturbed systems with two canard points.

Main results
In this section, we first introduce some essential hypotheses and state the main results on double canard cycles arising from system (1.1) with an S-shaped critical manifold possessing two canard points. We assume that for a fixed µ = µ 0 = (λ 0 , η 1,0 , ..., η m,0 ), the function f satisfies the following hypotheses: (H1) For a fixed µ = µ 0 , there exists a smooth function φ having precisely two different extreme points α 1 and α 2 with α 1 < α 2 such that the critical manifold C 0 := C µ0,0 is represented by (H2) At the points (α j , ω j ) := (α j , φ(α j )), j = 1, 2, the functions f satisfies the singularities: and the following non-degenerate conditions: (H3) Along the critical manifold C 0 , the function f satisfies that We remark that by (H2) and the Implicit Function Theorem, there exist an open neighbourhood U µ (µ 0 ) ⊂ R 1+m of µ = µ 0 and exactly four C k functions We can choose an appropriate set U µ (µ 0 ) such that (x j (µ),ỹ j (µ)) satisfy the non-degenerate conditions Then we get two manifolds L 0 µ and R 0 µ , which are parameterized by µ and given by We refer to L 0 µ and R 0 µ as the contact point manifolds. Without loss of generality, assume that U µ (µ 0 ) = R 1+m .
Under the above hypotheses, we see that (α j , ω j ) are both contact points. Our interest is to study the case that they are both canard points. Then we further make the following hypotheses on the function g.
(H4) For µ = µ 0 , the function g satisfies that and in the extended space {(x, y, λ) ∈ R 3 }, the curves L 0 µ and R 0 µ transversely intersect the manifold given by g(x, y, λ, η 0 , 0) = 0 at (α j , ω j , 0), that is, (H5) For µ = µ 0 , system (1.1) has precisely one equilibrium E 0 := (x m , y m ) on the section M which is a saddle, the function φ respectively reaches its minimum and maximum values at x = α 1 and x = α 2 , and the slow motions governed by Under the hypotheses (H1)-(H5), for µ = µ 0 both contact points are canard points. The dynamics of the limiting systems (1.3) and (1.4) are shown in Figure 3(a). Our goal is to study the double canard cycles arising from system (1.1) with (H1)-(H5). To establish the existence of double canard cycles, we construct double canard slow-fast cycles in the following way. For each s ∈ (0, by (see Figure 3 The limit cycles bifurcating from Γ(s 1 , s 2 ) is closely related to slow divergence integrals. Here for any x 1 and along the critical manifold is in the form (see, for instance, [6,7,18,32,35,36]) Four slow divergence integrals associated with Γ(s 1 , s 2 ) are given by Before stating the main results, we define some important constants which are useful in the subsequent proof.
Let the constants d j,r2 , d j,λ2 and d j,ηi be in the form where a j,i are given by and φ j,i = φ i whose lengthy expressions are given as in Lemma 3.1 for (α, ω) = (α j , ω j ). The main theorem are stated in the following.
Furthermore, there exist three hyperbolic limit cycles arising from Γ ε (s 1 , s 2 ) under some suitable conditions.

Normal forms near canard points
In this section, we consider the normal forms of system (1.2) near the canard points. Let the functionsg j associated with (α j , ω j ) be defined bỹ By (H4) we obtain that the functionsg j satisfy the following: Whenever there is no confusion, the zero vector is denoted by 0. Then by the Implicit Function Theorem, there exists an open neighbourhood U ε (0) ⊂ R of ε = 0 and exactly two C k functions Without loss of generality we assume that U ε (0) = R. The normal forms of system (1.2) near the canard points are given in the next lemma.

13)
where ζ = ±1, the C k functions φ i are defined by the functionφ j have the form and D j denotes the j-th partial derivative with respect to the j-th variable, and D ij with i = 1, j = 1, 2, 4 + m, Proof. It suffices to study the following system dx dt = x =f (x, y, µ, ε), dy dt = y = εḡ(x, y, µ, ε). Thus, the functionf can be written as the form whereφ i are defined as in this lemma.

Existence of canards
In this section, we establish the existence of canards near canard points based on the blow-up technique and the Melnikov theory. In order to study double canards near canard points (α j , ω j ), we take a quasihomogeneous blow-up transformation Π in the form x =rx, y =r 2ȳ , ε =r 2ε , λ =rλ, η i =rη i , i = 1, ..., m.
Roughly speaking, for the extended system of (4.2), which is in the form if all solutions of the extended system (4.5) near the heteroclinic orbit γ are of at most algebraic growth for t → ±∞. Then the distance function D(r 2 , λ 2 , η 2,1 , ..., η 2,m ) can be similarly obtained as in the classical case.
More precisely, we have the following lemma.
Furthermore, the distance function D is C k smooth.
Recall that the heteroclinic orbit γ has the form (4.4), then the variational equation along γ is in the form u = tu − v and v = u, whose adjoint equation (see [26, p.80]) is given by u = −tu − v and v = u. Note that the collection of the solutions of this adjoint equation decaying exponentially in forward and backward time is one-dimensional space, which is spanned byγ of the form γ(t) = −te −t 2 /2 , e −t 2 /2 , t ∈ R. and the vectorγ(0) = (0, 1), then by applying Theorem 1 and Proposition 1 in [44], the distance function D(r 2 , λ 2 , η 2,1 , ..., η 2,m ) is C k and has the expansion where the coefficients d r2 , d λ2 and d η2,i are given by the vector field X 2 is given by (4.5), the symbol ·, · denotes the inner product of two vectors. Therefore, the proof is now complete.
Thus, the proof is now complete.

Proof of Theorem A
In this section we give the proof for Theorem A by Lemma 4.1 and [37, Theorem 12] (or [16, Proposition 2]).
We will see that there exists a codimension 2 limit cycle bifurcating from a double canard slow-fast cycle. We refer the readers to [17,37] for the precise definition of the codimension of a limit cycle.
Proof of Theorem A. The proof for this theorem is divided into two steps.

Applications to Liénard equations with quadratic damping
Consider a singularly perturbed Liénard equation of the form where (x, y) ∈ R 2 , the function F is a cubic polynomial with respect to x, the parameters η, λ and ε are real and ε > 0 is sufficiently small. Liénard equation (6.1) is equivalent to the following equation where the derivative F of the cubic polynomial F and the function P defined by P (x) := −ε(η +λx−F (x)) are polynomials of degree two and three respectively. Then system (6.1) is always called a cubic Liénard equation with quadratic damping (see, for instance, [12,13]).
We can compute that for each (x, y) ∈ R 2 , If the cubic polynomial F satisfies that either F (x) ≥ 0 or F (x) ≤ 0 holds for all x ∈ R, then by Bendixson's Theorem (see, for instance, [14, Theorem 7.10, p.188]), we obtain that (6.1) has no limit cycles in R 2 . Thus one essential assumption for the existence of limit cycles is that the cubic polynomial F has precisely two different extreme points. This assumption implies that the critical manifold associated with singularly perturbed Liénard equation (6.1) is S-shaped. In this case, Liénard equation (6.1) can be normalized into a simpler form, the results are summarized in the following lemma.
Lemma 6.1 Assume that the cubic polynomial F in (6.1) has precisely two different extreme points. Then (6.1) can be changed into where the function F satisfies the following: Proof. By a translation we can move the left extreme point to the origin. Then we assume that the cubic polynomial F satisfies that where the parameters u and ν respectively satisfy u = 0 and ν > 0. We see that (6.1) is equivalent to Let x = νx, z = νz, t =t, u = −ν −2ū , ν =ν, ε =ε, η = νη and λ =λ. After dropping the bars, we can verify that (6.1) is equivalent to By taking the changes x = x, z = u z, t = t/u, ε = u ε, η = u( η − λ/2), λ = u λ and u = u in the above system, and then dropping all tildes, the parameter u in the above system is changed to u = 1. Then (6.1) is equivalent to (6.3) with F given by (6.5). Thus, the proof is finished.
We claim thatη 2 (r 2 ) ≡ 0 for each r 2 ∈ U r2 (0). Note that the normal forms (6.9) of system (6.3) near the points (0, 0) and (1, 1/6) have the only difference in the coefficient ofη. Then by this symmetry we obtain thatη 2 (r 2 ) = −η 2 (r 2 ) for each r 2 ∈ U r2 (0). This implies that the claim holds. Recalling the transformations (6.8) and (6.10), and applying Theorem A, we obtain this proposition. Thus, the proof is finished. where I i are similarly defined as in section 2. The detailed study of them will be given in the future work.
Numerical simulation shows that there exists a big double canard cycle enclosing two small limit cycles, each of which surrounds an stable focus. See Figure 7. Based on Proposition 6.1 and Figure 7, we conjecture that there exist three big limit cycles enclosing two small limit cycles under some suitable conditions. If this conjecture is right, then the maximal number of equation (6.1) is at least five. We hope that the results obtained in this paper are useful to explain this conjecture and the phenomenon in Figure 7.