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Research Article | Open Access
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.
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, , , and (see [3–7] for more details). For the hardness of Hilbert’s 16th Problem, according to Smale , it might be appropriate to deal with Hilbert’s 16th Problem restricted to the classical polynomial Liénard equations. In 2007, Dumortier et al.  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 and is a small real number. As and small, we make the time scaling and get the following equivalent standard form of slow-fast system which has the same phase portraits as the one of system (1) with the slow variable and the fast variable :
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 one-dimensional dynamical system in the fast variable with the slow variable acting as a parameter.
The equation 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  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 .
At contact points that are isolated points on the critical manifold where normal hyperbolicity is lost, the blow-up method pioneered by Dumortier and Roussarie  is a powerful geometric tool in the analysis of orbits of system (2) near nonhyperbolic points. According to the paper , we know that the admitted contact points of system (2) have been divided into two classes: a generic jump point and a generic turning point. If, after translation, rescaling of the variables , and rescaling of time, system (2) can, locally near , be, respectively, written as or then correspondingly nonhyperbolic point of system (2) is called a generic jump point or a generic turning point of system (2), where functions and are smooth and and is smooth at in case of a generic turning point.
If a slow-fast orbit of system (2) denoted by is closed, then is called slow-fast cycle. Further a slow-fast cycle is called common (see ) if all its slow curves have the same type: attracting or repelling. A slow-fast cycle 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 and the limit cycles that are close to slow-fast cycle are called relaxation oscillations. In 2007, Dumortier et al.  proved that at least three limit cycles of Liénard equations with well-chosen 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  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 , and two jump points , 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.
Consider the following smooth slow-fast Liénard system : where is near .
In the following, we assume that the smooth functions , fulfill the following conditions.()On the interval , as , the function has five singular points: two Morse maximum points at , , two Morse minimum points at , , and one Morse maximum at with . The parameter is just the difference . Let , , , , and . Then the values are assumed to be below the minimum value .()Suppose that , , , , and , but , , and , for or , for or (see Figure 4).()When , there exists a canard cycle containing four horizontal segments: one between the two Morse maxima , denoted by , one below the left Morse maximum value and at the height denoted by , one below the right Morse maximum value and at the height denoted by , and one at the height and between and denoted by , where , , , and the corresponding canard cycle is denoted by .
An essential tool to study the limit cycles bifurcated from the canard cycle is the slow divergence integral (see [9, 10, 13–15]); the slow divergence integral of the slow curve of system (7) between , is defined as follows:
Consider canard cycle of system (7). Let , , respectively, denote -coordinates of intersection points between and slow curve , where ; let , , respectively, denote -coordinates of intersection points between and , where ; let , , respectively, denote -coordinates of intersection points between and curve , where . By applying the slow divergence integral formula introduced in (8) to canard cycle of system (7), we get the following six integrals:
It is easy to check that
Then, the canard cycle is associated with the above six functions: , , , , , and , which are the slow divergence integrals of six slow curves contained in . In detail, two of these curves are located on the left of and their slow divergence integrals are functions of , two of them are on the right of and their slow divergence integrals are functions of , and two of them are between and and their slow divergence integrals are functions of (see Figure 4).
Now we give the following main results.
Theorem 1. Consider a general slow-fast Liénard system (7). Suppose that the smooth functions , fulfill the conditions , , . Let denote the total slow divergence integral of of system (7).(1)If there exist , , and such that , then for and small enough is a regular point of , whose explicit expression will be given in Section 2 and system (7) has a hyperbolic relaxation oscillation which is near .(2)If there exist , , and such that , , and , then for and small enough there exits , , that is a generic fold singularity of . A relaxation oscillation bifurcates from and this semistable limit cycle is generically unfolded by the parameter for and small enough, producing a pair of hyperbolic limit cycles of system (7).(3)If there exist , , and such that , , , and , then for and small enough there exits , , that is a generic cusp singularity of . A codimension 2 relaxation oscillation bifurcates from and this degenerated limit cycle is generically unfolded by the parameter for and small enough, producing system (7) having three hyperbolic limit cycles in the vicinity of canard cycle .
2. The Proof of Main Results
To study the limit cycle of the near the canard cycle , we choose one vertical section at , cutting the segment , section transversal to the turning point , and section transversal to the turning point . The parameter is the breaking parameter for the section , a rescaling of , given by , is the breaking parameter at , and a rescaling of , given by , is the breaking parameter at (see  for more details). In the following, we denote by . Let , , and be three sections which are transverse to the horizontal segments , parameterized, respectively, by , , .
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 and .
Lemma 3. For small enough, a limit cycle of system (7) cuts in , in , and in if and only if , where is given by the following: where , , , , , and are -regularly smooth in , and, respectively, equal to , , , , , and for .
Proof. From [13–15], due to the chosen orientation on the , the transitions have the following expressions:(1)from to : ,(2)from to : ,(3)from to : ,(4)from to : ,(5)from to : ,(6)from to : ,where functions , , , , , are -regularly smooth in , and functions , , , , , are -regularly smooth in and satisfy that , , , , , and .
By using the same analysis as [14, 15], we get that the system of equations for the existence of limit cycles of system (7) is where , , and . Rewrite the above equations into the following form: where new functions , , , , , and differ from the previous ones in terms of order and are -regularly smooth in .
We can solve this system in , , and because the partial derivations of the left hand term with respect to , , and are flat in . So one can solve (13) to obtain , , and , such that the functions , , and have the same form as the left hand term of (13) except that the functions , , , , , are replaced by new functions not depending on and which are -flat perturbations of the previous ones. We will continue to call them , , , , , , and , , , , , .
The proof of Lemma 3 is completed.
2.1. The Proof of the First Part of Theorem 1
We take small enough and we view as a map from to . First by direct computation, we get the Jacobian matrix of map as follows: where , are the partial derivatives with respect to , , are the partial derivatives with respect to , and , are the partial derivatives with respect to .
Let ). To find the singular points of map , by direct computation we get the following formula about the determinant :
Let ; then is exactly the complete slow divergence integral computed along the canard cycle . With the function , we can rewrite into
For and small, the equation for the singular points of the map given by is equivalent to the equation
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 as and small enough by noting -regularity of function . That means that the map is nondegenerated at the point , so from Lemma 3, we get that system (7) has one limit cycle near .
So the conclusion of the first part of Theorem 1 follows.
2.2. 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.
Lemma 4. As and small enough, if there exists that satisfies , then it holds that where is the gradient of function and is nonzero function at .
Proof. From (10) in Section 1, we get that equation determines a surface in the neighborhood of point for and small. From the fact that is equivalent to , we get that the equation also determines the surface . For are both the normal vectors of surface at the point , then we get (18).
The proof of Lemma 4 is completed.
Denote by , from assumptions in the second part of Theorem 1; then we get and . For and small, by applying Implicit Function Theorem, we can find surface which satisfies the equation . On surface , we choose point denoted by such that , ; that is, .
From and Implicit Function Theorem, we can get from the first equation of (19). Also, from , we can get from the third one. Substituting , into the second one, we get .
Next, we compute the first derivative of with respect to as
Then, as we compute the second derivative of with respect to as
From (10), we get that , , and . So , , and .
From Lemma 4, we get .
From (19), we get that
Noticing that , , we get that , and , .
Then for and small enough we get that
Because , is equivalent to . So we get that is a fold point of map . So from Lemma 3 and fold bifurcation of scalar map (see ), we get that system (7) has two limit cycles near by unfolding the parameters , , .
The conclusion of the second part of Theorem 1 follows.
2.3. 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.
Lemma 5. As and small enough, if there exists that satisfies , , and , then it holds that where , is the gradient of function , and is nonzero function at .
Proof. From the proof of the second part of Theorem 1, as we get
From , we get that is equivalent to
So rewrite (26), and we get that the equation is equivalent to under assumptions , for and small. That means that both equations and determine the same surface in the neighborhood of . For , are both the normal vectors of surface at the point , then we get (24).
The proof of Lemma 5 is completed.
It is easy to check that , and surfaces and transversely intersect along a curve which passes through point . Therefore, for and small enough, we choose point on the curve . Then , .
Consider the map given in (19). By applying a similar process to the one in the proof the second part of Theorem 1, we get , , and . Next, under the assumptions given in the third part of Theorem 1, we compute the third derivative of with respect to at and get
From Lemma 5, we get .
Then for and small enough we get that
For , is equivalent to ; then we get that is a cusp point of map for and small enough. From Lemma 3 and cusp bifurcation of scalar map , we get that the above degenerated limit cycle is generically unfolded by the parameters , for and small enough, producing system (7) having three hyperbolic limit cycles near .
The proof of the third part of Theorem 1 is completed.
3. Application to Polynomial Liénard Equation
In this section, we will apply Theorem 1 to the following polynomial Liénard equation:
For this system, . Here we use symbols , defined in Section 2. By direct computation we get that , ; the values of at these two points are both and the values of at the points , are both and .
For a given , the equation will have three roots on the right side of the -axis. For the roots, the formula in the above equation contains complex number, so we choose the smallest one of the above three roots and denote it by the variable , where . By noting that relation between and is one to one relation, we express the other two roots in the form , , where , . Using the same way, we take , ; then the equation will have three roots on the left side of the -axis; we choose the biggest one of the above three roots and denote it by the variable ; we express the other two roots in the form , and , . Also is symmetrical about 0, so the straight line , , between and will intersect the curve with two points; then we, respectively, denote their -coordinate by , , where . The corresponding canard cycle can be seen in Figure 5.
Firstly, consider ; that is,
When , from the fact that the function is monotone on the interval we get . Let and substitute it into (34); by setting the coefficients of and of obtained equation to equal zeros, we get , . Therefore, we get
Secondly, consider ; that is,
By noticing that the even function is strictly monotone on the interval and , we get as .
Let and put it into (36); then by setting the coefficients of and of obtained equation to equal zeros, we get
Thirdly, consider function ; that is,