On the Limit Cycles of a Class of Planar Singular Perturbed Differential Equations

Relaxation oscillations of two-dimensional planar singular perturbed systems with a layer equation exhibiting canard cycles are studied. The canard cycles under consideration contain two turning points and two jump points. We suppose that there exist three parameters permitting generic breaking at both the turning points and the connecting fast orbit. The conditions of one (resp., two, three) relaxation oscillation near the canard cycles are given by studying a map from the space of phase parameters to the space of breaking parameters.


Introduction
As we know, the second part of Hilbert's 16th Problem is related with the number and distributions of limit cycles of a general polynomial vector field of th degree. Let ( ) denote the maximum number of limit cycles of a general planar polynomial vector field of degree . As mentioned in [1,2], there are little studies on an upper bound of ( ), but there are many results on the lower bounds of ( ); for example, (3) ≥ 13, (4) ≥ 21, and (5) ≥ 28 (see [3][4][5][6][7] for more details). For the hardness of Hilbert's 16th Problem, according to Smale [8], it might be appropriate to deal with Hilbert's 16th Problem restricted to the classical polynomial Liénard equations. In 2007, Dumortier et al. [9] found 4 limit cycles in a singular perturbed Liénard equation of degree 7 by applying geometric singular perturbation theory, and this result overturns Lin de Melo and Pugh's conjecture (see [9,10] for details). The general form of planar singular perturbed differential equation can be given by as follows: where ( , , ) and ( , , ) are two smooth functions with respect to variables ( , , ) ∈ R 2 × R and is a small real number. As > 0 and small, we make the time scaling = / and get the following equivalent standard form of slowfast system which has the same phase portraits as the one of system (1) with the slow variable and the fast variable : = ( , , ) , = ( , , ) .
(2) Let = 0 in system (1) and (2), we, respectively, obtain the following limiting system (3) System (3) and (4) are, respectively, called the reduced equation and the layer equation of system (1). For problems of planar singular perturbed system (1), the reduced equation captures essentially the slow dynamics and the layer equation captures the fast dynamics. The layer equation is a onedimensional dynamical system in the fast variable with the slow variable acting as a parameter. 2 The Scientific World Journal The equation ( , , ) = 0 defines the critical manifold of the equilibrium of the layer equation (4). The reduced equation describes the dynamics on the critical manifold . Due to geometric singular perturbation theory of Fenichel (see [11] for more details), normally hyperbolic pieces of critical manifolds turn to locally invariant slow manifolds for sufficiently small. Hence under suitable assumptions, orbits of singular perturbed system (2) can be obtained as perturbations of a slow-fast orbit which consist of pieces of the reduced equation (3) and the layer equation (4). Slow-fast orbit is not orbit of system (2), and it is the limit set of system (2) as approaches 0.
If a slow-fast orbit of system (2) denoted by Γ is closed, then Γ is called slow-fast cycle. Further a slow-fast cycle Γ is called common (see [9]) if all its slow curves have the same type: attracting or repelling. A slow-fast cycle Γ is called a canard cycle if it contains both attracting and repelling slow curves. Here slow-fast cycle is not periodic orbit of system (2), but a limit periodic set as approaches 0 and the limit cycles that are close to slow-fast cycle are called relaxation oscillations. In 2007, Dumortier et al. [9] proved that at least three limit cycles of Liénard equations with wellchosen polynomial of degree 7 bifurcate from the canard cycle consisting of two jump points by analyzing the zeros of slow divergence integral of canard cycle (see Figure 1).
In 2011, de Maesschalck and Dumortier [10] showed that four limit cycles bifurcated from the canard cycle of a singular perturbed Liénard system of degree six, which consists of a single fast orbit and a single slow curve and contains one turning point (see Figure 2). It can be clearly checked that the slow curve on one side of the turning point is attracting, and the part on the other side is repelling. This kind of canard cycle is said to be a 1-breaking parameter family of canard cycle. The main method used is to study the zeros of slow divergence integral of canard cycle.  In 2007, Dumortier and Roussarie [14,15] considered two-dimensional slow-fast systems with a layer equation exhibiting canard cycle which contains a turning point and a fast orbit connecting two jump points (see Figure 3). At both the turning point and jump point, the presence of two parameters permitting generic breaking is assumed. The conditions of existing one (two or three) limit cycle in the above planar system are given by studying the fixed points of the Poincare map near canard cycles.
In this paper, we want to study the number of limit cycles near canard cycle which contains two turning points 2 , 3 and two jump points 1 , 4 that allow three breaking mechanisms and each one corresponds to one phase parameter (see Figure 4). This paper is organized as follows. Our main results will be presented in the first part. The proofs of the results are presented in the second part. In the third part, a concrete example of planar singular Liénard equation existing three limit cycles will be given.
( 3 ) When = (0, 0, 0), there exists a canard cycle containing four horizontal segments: one between the two Morse maxima = 1 , = 4 denoted by ℎ , one below the left Morse maximum value and at the height = denoted by , one below the right Morse maximum value and at the height = V denoted by (V), and one at the height = and between 2 and 3 denoted by , where ∈ ( 1 , ), V ∈ ( 2 , ), ∈ ( , 0 ), and the corresponding canard cycle is denoted by Γ V .
An essential tool to study the limit cycles bifurcated from the canard cycle is the slow divergence integral (see [9,10,[13][14][15]); the slow divergence integral of the slow curve of system (7) between 1 , 2 is defined as follows: Consider canard cycle Γ V of system (7). Let 1 ( ), 2 ( ), respectively, denote -coordinates of intersection points between and slow curve = 0 ( ), where − 0 < 2 ( ) < 1 ( ) < 2 ; let 1 ( ), 2 ( ), respectively, denotecoordinates of intersection points between and = 0 ( ), where 2 < 1 ( ) < 2 ( ) < 3 ; let 1 (V), 2 (V), respectively, denote -coordinates of intersection points between and curve applying the slow divergence integral formula introduced in (8) to canard cycle Γ V of system (7), we get the following six integrals: It is easy to check that Then, the canard cycle Γ V is associated with the above six functions: ( ), (V), ( ), (V), ( ), and ( ), which are the slow divergence integrals of six slow curves contained in Γ V . In detail, two of these curves are located on the left of 2 and their slow divergence integrals are functions of , two of them are on the right of 3 and their slow divergence integrals are functions of V, and two of them are between 2 and 3 and their slow divergence integrals are functions of (see Figure 4). Now we give the following main results.
, then for > 0 and small enough there exits (̂0( ),V 0 ( ),̂0( )) that is a generic cusp singularity of Φ . A codimension 2 relaxation oscillation bifurcates from Γ V and this degenerated limit cycle is generically unfolded by the parameter ( , , ) for > 0 and small enough, producing system (7) having three hyperbolic limit cycles in the vicinity of canard cycle Γ V .

The Proof of Main Results
To study the limit cycle of the , near the canard cycle Γ V , we choose one vertical section 1 at = 0, cutting the segment ℎ , section 2 transversal to the turning point 2 , and section 3 transversal to the turning point 3 . The parameter is the breaking parameter for the section 1 , a rescaling of , given by = −1/2 , is the breaking parameter at 2 , and a rescaling of , given by = −1/2 , is the breaking parameter at 3 (see [13] for more details). In the following, we denote ( , , ) by . Let ∑ , ∑ , and ∑ be three sections which are transverse to the horizontal segments , , , parameterized, respectively, by , V, .
To study the fixed points of the obtained Poincaré map, first we give the following definition of -regularly smooth function and the following lemma by introducing the relationship between intermediate variables , V, and , , .
Definition 2 (see [9,13]). A function ( , ), with ∈ R for some ∈ N, is called -regularly smooth in , if is continuous and all partial derivatives of with respect to exist and are continuous in ( , ).  Proof. From [13][14][15], due to the chosen orientation on the ( = 1, 2, 3), the transitions have the following expressions: (1) from ∑ to 1 : → exp( ( , , )/ ) + ( , ), By using the same analysis as [14,15], we get that the system of equations for the existence of limit cycles of system (7) where new functions , , , , , and differ from the previous ones in terms of order ( ) and are -regularly smooth in ( , V, , ).

The Proof of the First Part of Theorem 1.
We take small enough and we view Φ as a map from ( , V, ) to ( , , ). First by direct computation, we get the Jacobian matrix of map Φ as follows: where , are the partial derivatives with respect to , , are the partial derivatives with respect to V, and , are the partial derivatives with respect to . Let It follows from (10) that the term in the logarithm in (17) is strictly positive. Under the conditions of the first part of Theorem 1, we can get that ̸ = 0 as > 0 and small enough by noting -regularity of function ( , V, , ). That means that the map Φ is nondegenerated at the point ( 0 , V 0 , 0 ), so from Lemma 3, we get that system (7) has one limit cycle near Γ 0 V 0 0 . So the conclusion of the first part of Theorem 1 follows.

The Proof of the Second Part of Theorem 1.
In this subsection, we give the proof of the second part of Theorem 1.
First, we present the following lemma.

Proof. From
where the map Φ is given in Lemma 3. Denote  The Scientific World Journal Next, we compute the first derivative of with respect to as (20) Then, as Δ( , V, , ) = 0 we compute the second derivative of with respect to as From (10), we get that is a fold point of map Φ . So from Lemma 3 and fold bifurcation of scalar map = ( , , , ) (see [16]), we get that system (7) has two limit cycles near Γ V by unfolding the parameters , , .
The conclusion of the second part of Theorem 1 follows.

The Proof of the Third Part of Theorem 1.
In this subsection, we give the proof of the third part of Theorem 1. First, we present the following lemma.   Consider the map Φ given in (19). By applying a similar process to the one in the proof the second part of Theorem 1, we get = ( , , , ), / = −Δ/ V , and ( 2 / 2 )| Δ=0 = −( Δ/ )/ V . Next, under the assumptions given in the third part of Theorem 1, we compute the third derivative of with respect to at̂0( ) and get From Lemma 5, we get grad( Δ/ )|̂0 ( ) =̂⋅ ( ( 0 ) + ( ), (V 0 ) + ( ), ( )).
From (19) and (26), we get that Then for > 0 and small enough we get that = 0; then we get that̂0( ) is a cusp point of map = ( , , , ) for > 0 and small enough. From Lemma 3 and cusp bifurcation of scalar map [16], we get that the above degenerated limit cycle is generically unfolded by the parameters ( , , ), for > 0 and small enough, producing system (7) having three hyperbolic limit cycles near Γ V .
The proof of the third part of Theorem 1 is completed.

Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.