# Relaxation Oscillations in Singularly Perturbed Generalized Lienard Systems with Non-Generic Turning Points.

1 Introduction

Lienard equation is a classical and important nonlinear oscillator, modeling nonlinear vibrations in the damped, one-degree-of-freedom systems. It can be viewed as a generalization of van der Pol equation, and has found numerous applications in various real world situations including electronic devices and biology, etc., see e.g. , , ,  and . For a long time, the existence, uniqueness, number and profile of limit cycles in Lienard equations are always the important and valuable topics in the field of ODEs.

[d.sup.2]x/[dt.sup.2] = [mu] f(x,b) dx/dt + g(x,a) = 0,

where f and g are sufficiently smooth functions, a [member of] [R.sup.n] and b [member of] [R.sup.m] are the bifurcation parameters, and [mu] [member of] R is related to the strength of damping. After the Lienard transformation, this equation becomes

[mathematical expression not reproducible]

where F(x, b) = [[integral].sup.x.sub.0] f (s, b) ds. If g(x, a) is linear in x, this system is called classical Lienard system, and otherwise, it is called generalized one.

When [mu] is small, i.e. when the damping is small, in this case, Lienard system can be viewed as Hamiltonian-perturbation system. By using Abel integral etc., bifurcation of limit cycles has been studied extensively, especially when f and g are polynomials, see e.g. Han et al.  and Xiong et al. .

When [mu] is large, then after a new time scale [tau] = t/[mu] is introduced, Lienard equation can be changed to

[epsilon] [d.sup.2]x/d[[tau].sup.2] + f (x, b) dx/d[tau] + g(x, a) = 0,

which is a singular perturbation equation since the highest-order derivative has been multiplied by a small parameter, where 0 < [member of] = 1/[[mu].sup.2] [much less than] 1. On the Lienard plane, this singularly perturbed equation is equivalent to a slow-fast dynamical system, namely,

[mathematical expression not reproducible]

Roussarie  pointed out that the singularly perturbed Lienard systems consists of the boundary of the so-called Lienard space.

The limit cycles occurring in singularly perturbed Lienard systems are relaxation oscillations, characterized by the presence of phases in the cycle with different time scales: A phase of slow change is followed by a short phase of rapid change where the system jumps to the next stage of slow variation. In this direction, non-standard analysis was first introduced to detect canard relaxation oscillations in van der Pol equation . Then Eckhaus  pointed out that classical asymptotic analysis method is also capable of identifying canard relaxation oscillations and regular relaxation oscillations in singularly perturbed Lienard systems. Krupa and Szmolyan distinguished the generic jump and canard points for general singular perturbation system. Now it is well known that the results obtained from classical asymptotic analysis method are closely related to which from blow-up technique, see van Gils, Krupa and Szmolyan . Based on geometric singular perturbation method and blow-up technique, Dumortier and Roussarie ,  developed a tool, namely, the slow divergence integral. By this integral, the maximal number of limit cycles is related to its non-degenerate zeroes, and more limit cycles than expected by the conjecture of Lins et al.  in the classical singularly perturbed Lienard systems had been found in  and . The cyclicity in the predator-prey models with Holling-III and IV responses had also been obtained, see Li and Zhu .

In all the above-mentioned works, it was assumed that the turning points on the critical curves are generic (non-degenerate). When the critical curve have non-generic (degenerate) turning points, the work on this topic is relatively rare. By following the method of Eckhaus , Li  detected canards with or without heads in a singularly perturbed Lienard system of the following form,

[mathematical expression not reproducible]

in which, it was assumed that x = 0 is a m-order turning point on the critical curve y = F(x), where m is an even number. Then Li  extended this approach to a more general system, namely,

[mathematical expression not reproducible]

where the parametric conditions for the existence of canards were obtained. By using also asymptotic analysis method, Chen et al.  further studied a more general singularly perturbed Lienard system, namely,

[mathematical expression not reproducible]

where it was also assumed that x = 0 is a degenerate turning point. Chen et al.  proved the existence of canard solutions, and optimized the canard curve based on a new asymptotic lemma.

In this paper, we will extend the asymptotic analysis method of Eckhaus  to study the birth of relaxation oscillations in the following singularly perturbed generalized Lienard system,

[mathematical expression not reproducible] (1.1)

in which, we assume that the critical curve admits a non-generic turning point, where 0 < [epsilon] [much less than] 1 is the singular parameter, while a [member of] R is a regular parameter (named also breaking parameter by following Dumoutier ), g, f and h are sufficiently smooth functions. We demonstrate the birth of relaxation oscillations of regular and canard types in (1.1) and give the corresponding parameter conditions.

Obviously, system (1.1) is more general than those studied in the previous papers. In fact, when g(y) = y, it reduces to the systems considered before. In literatures, system (1.1) is still called generalized Lienard system. Locally, system (1.1) can be changed to the simpler systems mentioned above after a series of transformations, but globally it is impossible. So if one aims to study system (1.1) globally, we must face to system (1.1). This is the first reason why we consider system (1.1) in this article. Secondly, a great deal of biological models can be changed to the form of system (1.1) after a series of transformations involving the dependent and independent variables as well as the parameters, see e.g. De Maesschalck , Hus and Shi  and Li and Zhu . Thus, the study on the slow-fast dynamics such as the existence of relaxation oscillations in these models is reduced to study system (1.1), which is one of the main topics of the present paper.

The remaining parts of this paper are organized as follows. In section 2, several necessary assumptions and lemmas are given. In sections 3 and 4, we respectively study the existence of regular relaxation oscillations and canard relaxation oscillations in system (1.1) and give the corresponding parametric curves. In section 5, based on the conclusions obtained in the two previous sections, we consider two biological models, namely, a FitzHugh-Nagumo equation and a two-dimensional predator-prey model with Holling-II response, in which, we show the existence of regular relaxation oscillations and canard relaxation oscillations in these models and give the critical parameter values.

2 Preliminaries and assumptions

For system (1.1), the critical curve is defined by g(y) = f(x), on which, the slow flows evolve according to the following equation

g'(y)h(x,y, a) = f'(x) dx/dt.

To fix the directions of the slow flows and the fast fibers, we need the following assumptions.

(A1) x = [x.sub.1] < 0 and x = [x.sub.2] =0 are respectively the generic and nongeneric extreme points of the critical curve satisfying f'([x.sub.1]) = 0, f"([x.sub.1]) < 0, f '(0) = ... = [f.sup.(m-1)](0) = 0, and [f.sup.(m)] (0) > 0, where m > 2 is an even number.

(A2) [mathematical expression not reproducible], where [C.sub.1] and [C.sub.2] are constants chosen such that [C.sub.1] < [x.sub.1] < 0 < [C.sub.2], f ([C.sub.1]) < 0 and f([C.sub.2]) > f([x.sub.1]).

(A3) h(x, y, a) = 0 has a unique root x = [phi] (y, a) satisfying: 1) h(x, y, a) < 0 when x > [phi] (y, a), and h(x, y, a) > 0 when x < [phi](y, a), 2) when y [member of] (f (0),f ([x.sub.1])) and a [member of] ([a.sub.1], [a.sub.0]), we have [phi](y, a) [member of] ([x.sub.1], 0), when a [right arrow] [a.sub.0] and y [right arrow] f (0), we get [phi] (y, a) [right arrow] 0, and when a [right arrow] [a.sub.1] and y [right arrow] f ([x.sub.1]), we obtain [phi] (y, a) [right arrow] [x.sub.1].

(A4) g(0) = f (0) and, g'(y) > 0 for any y [member of] R.

(A5) h(0,0, [a.sub.0]) =0 and [partial derivative]h/[partial derivative]a (0,0, [a.sub.0]) [not equal to] 0.

Remark 1. Under the assumption (A1), the critical curve is S-shape (cubiclike), where the right and left branches are attracting, while the middle one is repelling, see Figure 1.

Remark 2. Under the assumptions (A1) to (A3), system (1.1) possesses an equilibrium point, namely, ([x.sub.e], [y.sub.e]) = ([phi](y, a), [g.sup.-1] f ([phi](y,a))), where [g.sup.-1] denotes the inverse function of g. When a [member of] ([a.sub.1], [a.sub.0]), this equilibrium point lies on the middle branch of the critical curve. In this case, we mark the directions of the slow flows and the fast fibers in Figure 1a.

Remark 3. When a [right arrow] [a.sub.0] or a [right arrow] [a.sub.1], the equilibrium point of the system coincides with the extreme points of the critical curve, see Figure 1b, where the directions of the slow flows and the fast fibers when a [right arrow] [a.sub.0] are stated.

In the following, two lemmas needed in the proof of our main results are given firstly. Consider

[mathematical expression not reproducible]

in which, 1) F is continuous and uniformly bound on

G = {Y | Y [member of] [bar.D]} x {[tau] | 0 [less than or equal to] [tau] < A} x {[epsilon] | 0 < [epsilon] [less than or equal to] [[epsilon].sub.0]},

where D [subset] [R.sup.n] is a bounded region; 2) F is Lipchitz-continuous with respect to Y in G.

Lemma 1. (Eckhaus ) Let [Y.sup.(1)] and [Y.sup.(2)] be respectively the solutions of

[mathematical expression not reproducible]

If the following conditions are satisfied: (i) [mathematical expression not reproducible],

(ii) [mathematical expression not reproducible]

(iii) [Y.sup.(2)] ([tau], [epsilon]) exists for 0 [less than or equal to] t [less than or equal to] T, where T < A and [Y.sup.(2)] [member of] [D.sub.0], then [Y.sup.(1)]([tau], [epsilon]) exists for 0 [less than or equal to] [tau] [less than or equal to] T and satisfies

[absolute value of [Y.sup.(1)](]tau],[epsilon]) - [Y.sup.(2)]([tau], [epsilon])] = 0([[delta].sub.0]([epsilon])) + o([[delta].sub.f] ([epsilon])).

Lemma 2. (Eckhaus ) Let [Y.sup.(2)] ([tau], [member of]) be a valid approximation to [Y.sup.(1)](t, [tau]), i.e.

[Y.sup.(1)]([tau], [epsilon]) - [Y.sup.(2])([tau],[member of]) = 0(1), 0 [greater than or equal to] t [greater than or equal to] T,

where T is a constant which can be taken to be arbitrarily large, then there must be a [??]([epsilon]) = o(1) such that the above approximation holds for [mathematical expression not reproducible]. Similarly, if [Y.sup.(2)]([tau], [member of]) is an approximation to [Y.sup.(1)]([tau], [member of]) for 0 < d [less than or equal to] [tau] <[less than or equal to] T, where d can be chosen to be arbitrarily small, then there must be a [??]([epsilon]) = o(1) such that the approximation holds for [??]([epsilon]) = [less than or equal to] t [less than or equal to] T.

Lemma 3. (Chen et al. ) Let m be an even number, [phi](x) and [psi] (x) are both defined on [a, b]. If

1) [mathematical expression not reproducible]) is integrable on [a, b] for any n [greater than or equal to] [n.sub.0],

2) [psi] (x) has a unique maximal point [mathematical expression not reproducible]

holds on any closed subinterval [[alpha], [beta]] [subset] [a, b], [xi] [member of] [[alpha], [beta]],

3) near x = [xi], [[psi].sup.(m)](x) is continuous and satisfies [mathematical expression not reproducible]

4) [[phi](x) is continuous at x = [xi], [phi] ([xi]) [not equal to] 0, then as n [right arrow] + [infinity] we have

(i) [mathematical expression not reproducible]

(ii) [mathematical expression not reproducible]

when k is odd.

3 Regular relaxation oscillations in system (1.1)

In this section, let a [member of] ([a.sub.1], [a.sub.0]) be such that the equilibrium point ([x.sub.e], [y.sub.e]) of system (1.1) lies on the middle branch of the critical curve. We will show that only regular relaxation oscillations can occur in this case.

Let t = [t.sub.0]/[epsilon] = [tau], with which, system (1.1) becomes

[mathematical expression not reproducible]

It then follows from Lemma 1 that

[mathematical expression not reproducible], (3.1)

in which, [bar.x]([tau]) is the solution to the following system,

d[bar.x]/d[tau] = g([y.sub.0]) - f ([bar.x]), [bar.x](0) = [x.sub.0],

where ([x.sub.0], [y.sub.0]) is an initial value chosen satisfying f([x.sub.0]) [not equal to] g([y.sub.0]), inside the basin of attraction of CO, and outside the so-called singular periodic orbit OABC (see Figure 2).

The solution [bar.x]([member of]) exists for all [tau]. Hence, we can take T arbitrarily large in (3.1). It thus follows from Lemma 2 that

[mathematical expression not reproducible]

where [[tau]*([epsilon]) = [tau]/[delta]([epsilon]), [delta]([member of]) = o(1). Let [[bar.x].sub.0] be chosen satisfies g([y.sub.0]) - f ([bar.x].sub]) = 0, then we have

[bar.x]([tau[) = =[[bar.x].sub.0] + O(exp(-f'([bar.x].sub.0)[tau])).

Hence

[bar.x] ([tau] *([epsilon])) = [bar.x]0 + o(1).

This says that, after starting from ([x.sub.0], [y.sub.0]), the trajectory of system (1.1) evolves near the following orbit

x = [bar.x] ([tau]), y = [y.sub.0],

and then penetrates into a small neighborhood of the curve g(y) = f (x).

To see the dynamics near the critical curve, let

g(y) = f (x) + [epsilon][[phi].sub.0](x, y) + [epsilon] [[phi].sub.0] (x, y) [[epsilon].sup.2] [[phi].sub.1] (x,y) + ...,

and then differentiating this formula with respect to t yields

g'(y) dy/dt = f'(x) dx/dt + [epsilon]d [[phi].sub.0]/dt + ...,

After substituting system (1.1) into this equality, to leading-order one gets

[mathematical expression not reproducible]

i.e.,

[[phi].sub.0] = g'(y)h(x, y, a)/f'(x). (3.2)

Thus

g(y) = f (x) + [epsilon][[[phi].sub.0] + o(1)].

One can see from (3.2) that, the slow flow evolves well (which means that the flow of system (1.1) evolves also well) near the critical curve until it enters into a small neighborhood of the turning point [x.sub.2] =0 where f/(0) = 0. Since g/(0) > 0 and h(0,0, a) [not equal to] 0 when a [member of] ([a.sub.1],[a.sub.0]), so the slow flow [[phi].sub.0] becomes unbounded in this small neighborhood. The ill-posedness of the slow flow results that the flow of system (1.1) may blow up. In this sense, we say that

h(0, 0, a) [not equal to] 0, a [member of] ([a.sub.1], [a.sub.0]) (3.3)

is a necessary condition for the occurrence of blow-up of the flow of system (1.1).

To see whether the flow of system (1.1) really blows up when it enters the small neighborhood around the turning point x = 0, let us write

f/(x) = [x.sup.m-1]p(x), p(x) > 0,

which yields

f (x) = [x.sup.m] (1/m f p(0) + O(x))

and then introduce the following blow-up transformations,

[mathematical expression not reproducible], (3.4)

where [[delta]* ([epsilon]) = o(1) will be determined later, and [t.sub.0] denotes the moment when the flow enters into an O([epsilon])-neighborhood of the turning point, after substituting (3.4) into equation (1.1), one gets

[mathematical expression not reproducible]

Let [mathematical expression not reproducible], with which, we get

[mathematical expression not reproducible]

which can be approximated by

[mathematical expression not reproducible] (3.5)

with an [mathematical expression not reproducible] accuracy. Equation (3.5) is solvable whose solutions are given by

[mathematical expression not reproducible]

where ([[xi].sup.0], [[eta].sup.0]) is an initial value.

Clearly, when [tau]* reaches [mathematical expression not reproducible] explodes, i.e., the flow of system (1.1) blows up there. Since then the flow leaves the neighborhood of the turning point (0, 0) along the fast fiber y = 0 with an [mathematical expression not reproducible] distance.

Summarizing, after starting from the initial point ([x.sub.0], [y.sub.0]) with g([y.sub.0])--f([x.sub.0]) [not equal to] o(1) and inside the basin of attraction of CO, the flow of system (1.1) quickly enters into an O ([member of])-neighborhood of the stable branch CO, and then stays near CO until it reaches near a small neighborhood of the turning point O where the flow blows up and leaves along the fast fiber y = 0 with an [mathematical expression not reproducible] distance. Afterward, the flow approaches to AB and stays near AB until it reaches near the turning point B, where it blows up again and jumps to CO along the fast fiber BC with an O([[epsilon].sup.2/3]) distance. That is to say, the flow of system (1.1) spiraling around the singular periodic orbit OABC asymptotically, see Figure 2.

On the other hand, it can be proved that the equilibrium point ([x.sub.e], [y.sub.e]) is an unstable node by following Shen . The flow of system (1.1) starting from near ([x.sub.e], [y.sub.e]) will spiral out and approaches to the singular periodic orbit OABC gradually. Till now, the existence of at least one relaxation oscillation near the singular periodic orbit OABC follows directly from the Poincare-Bendixson Theorem .

Theorem 1. If (A1) to (A4) and (3.3) hold, then when a [member of] ([a.sub.1], [a.sub.0]) and [epsilon] is sufficiently small, system (1.1) admits at least one relaxation oscillation near the singular periodic orbit OABC. More precisely, the relaxation oscillations are near the slow branches CO and AB with an O([member of]) distance, and near the fast fibers BC and OA with an O([[epsilon].sup.2/3]) and an [mathematical expression not reproducible] distances respectively.

Remark 4. Theorem 1 shows that, under (A1) to (A4) and (3.3), when a [member of] ([a.sub.1], [a.sub.0]), the flow of system (1.1) leaves the turning point O near the fast fiber y = 0 with an [mathematical expression not reproducible] distance. This result can be seen as a partial generalization of Theorem 2.1 in . For the following general singular perturbation system,

[mathematical expression not reproducible]

Krupa and Szmolyan  proved via blow-up technique that, when the turning point (0, 0) is generic, the flow of this system will leave this turning point near the fast fiber y = 0 with an O([[epsilon].sup.2/3]) distance.

4 Canard relaxation oscillations in system (1)

This section will consider the case a [right arrow] [a.sub.0], under which, the equilibrium point coincides with a critical point of the critical curve.

Let g(y) - f (x) = [epsilon][phi], then one gets from system (1.1) that

[mathematical expression not reproducible] (4.1)

where [mathematical expression not reproducible], then system (4.1) becomes

[mathematical expression not reproducible]

i.e.

[mathematical expression not reproducible] (4.2)

Formally, the solution to equation (4.2) are given by

[mathematical expression not reproducible]

Denote

dQ/dx = [x.sup.m-1] [p.sup.2] + [[epsilon].sup.2] p [[phi].sub.1]/dx,

then

[mathematical expression not reproducible]

where p/ = dp/d[xi] and

[mathematical expression not reproducible]

Accordingly we can write

[mathematical expression not reproducible]

where

[mathematical expression not reproducible]

and

[mathematical expression not reproducible]

Let [x.sub.1] < 0, [x.sub.0] > 0, x [member of] [[x.sub.1], [x.sub.0]] and U be a set of continuous functions u(x) on [[x.sub.1], [x.sub.0]]. Then we define an operator L on U by

Lu = [[phi].sub.1] (x, u) + [[phi].sup.2] (x, u), x [member of] [[x.sub.1], [x.sub.0]].

When [mathematical expression not reproducible]

[mathematical expression not reproducible]

Accordingly we can obtain

[mathematical expression not reproducible]

in which,

[mathematical expression not reproducible]

where [mathematical expression not reproducible] is determined by

[mathematical expression not reproducible].

So far we know that if [??] (x) is bounded on [[x.sub.1], [x.sub.0]], then according to the contraction mapping principle, the operator L admits a unique fixed point, which corresponds to the solution to equation (4.2) with

[mathematical expression not reproducible].

Hence, if [??](x) is bounded, then equation (4.2) has bounded solutions on [[x.sub.1], [x.sub.0]], which states that canard solutions in system (1.1) exist. In the following, we show that, under certain conditions, [??] (x) is bounded.

Since h(0, 0, [a.sub.0]) = 0, so by Taylor expansion and Lemma 3, it can be deduced that

[mathematical expression not reproducible]

where l is an even number, [g.sup.-1] is the inverse function of g, all the partial derivatives take their values at [mathematical expression not reproducible] denotes the Lagrange remainder term, and [GAMMA] is the Gamma function. Thus,

[mathematical expression not reproducible]

Denote

[mathematical expression not reproducible]

thus,

[mathematical expression not reproducible]

Since [partial derivative]h/[partial derivative] (0,0, [a.sub.0]) [not equal to] 0, it thus can be verified that [mathematical expression not reproducible]. By implicit function theorem, there exists a unique function [a.sub.c] = [a.sub.c]([epsilon]) satisfying [a.sub.c](0) = [a.sub.0] such that G([a.sub.c] ([epsilon]), [epsilon]) = 0.

Let us take the breaking parameter a as

[mathematical expression not reproducible]

where [sigma] and k are two constants, it then follows from equation (4.3) that

[mathematical expression not reproducible] (4.4)

Let ([x.sub.0], [y.sub.0]) be an initial point near the right branch of the critical curve, [x.sup.*.sub.0] < 0 be a root of the equation [mathematical expression not reproducible], and k be an arbitrary constant such that [x.sub.k] > 0 and [x.sup.*.sub.k] [member of] (A, 0) are respectively the roots solved from the equations [mathematical expression not reproducible], then we can obtain from (4.4) that [[phi].sub.1](x) is bounded provided max{[x.sup.*.sub.k], [x.sup.*.sub.0]} [less than or equal to] x [less than or equal to] min{[x.sub.0], [x.sub.k]}. Accordingly, system (1.1) has a canard solution satisfying

g(y) = f (x) - [epsilon]/p(x) + [[epsilon].sup.2] [[phi].sub.1] (x).

This canard solution will form a canard relaxation oscillation finally.

Moreover, it can be seen from (4.4) that the first term is dominant if [x.sub.0] < [x.sub.k], and otherwise the second one will be dominant if [x.sub.0] > [x.sub.k].

For the former case, after starting from the basin of attraction of the right branch of the critical curve, all the trajectories of system (1.1) enters into an O([epsilon])-neighborhood of the critical curve g(y) = f (x), after going through the turning point, they stay near the repelling branch for a constant distance until it jumps off at x = [x.sup.*.sub.0]. The trajectories jump to the right if [[phi].sub.1] ([x.sub.0]) > 0 and to the left if [[phi].sub.1] ([x.sub.0]) < 0.

For the latter case, after starting from the basin of attraction of the right branch of the critical curve, all trajectories enter into an O([epsilon])- neighborhood of g(y) = f(x) at x = [x.sub.0] > [x.sub.k] and then stay near the attracting and repelling branches of the critical curve until they jump off at x = [x.sup.*.sub.k] < 0. The trajectories jump off to the right and the left respectively when [sigma] [partial derivative]h/[partial derivative]a (0,0, 0) > 0 and [sigma] [partial derivative]h/[partial derivative]a (0, 0, 0) < 0, which respectively yields the birth of canard relaxation oscillations with or without head, see Figure 3.

Summing up, we have the following conclusion.

Theorem 2. If (A1) to (A5) hold, when [epsilon] is sufficiently small and a = [mathematical expression not reproducible], where a and k are two constants, then system (1.1) has at least one canard relaxation oscillation with or without head depending on the breaking parameter and the initial values (see Figure 3).

5 Examples

5.1 A FitzHugh-Nagumo equation

Consider a FitzHugh-Nagumo (FHN) equation as follows,

[mathematical expression not reproducible] (5.1)

where 0 < [epsilon] [much less than] 1, p, s, r and a are the parameters of the model. System (5.1) is a prototypical FitzHugh-Nagumo equation describing the biological phenomena with neuronal excitability and spike generating mechanism.

By the translation as follows, u = x + 1, v = y, the extreme point (-1, 0) of the critical point is then transferred to the origin, and accordingly we have

[mathematical expression not reproducible], (52)

Therefore, in this example we have

g(y) = y, f (x) = [(x + 1).sup.3]/3 - x - 1 - p

and

h(x, y, a) = - 1/s (y + r(x +1) - a).

For system (5.2), it can be observed that:

1) When p = - 2, the critical curve y = f (x) passes through the origin, and there are two generic extreme points x = -2 and x = 0 with f'(-2) = 0, f" (-2) < 0, f'(0) = 0 and f"(0) > 0;

2) Let s > 0, r < -3 be fixed. If a < r, then inside the singular periodic orbit formed by two slow orbits and two fast fibers (bounded by four points: (0,0), (-3,0), (-2, 4/3) and (1, 4/3)), system (5.2) has a unique equilibrium ([x.sub.e], [y.sub.e]) on the middle branch of the critical curve. In this case, the turning point (0, 0) is a jump point.

3) Let s > 0, r < -3 be fixed. If a = r, then system (5.2) has a unique equilibrium ([x.sub.e], [y.sub.e]) coinciding with the extreme point (0, 0). In this case, the turning point (0,0) is a canard point.

4) h(0, 0, r, a) = 0, [partial derivative]h (0,0,r,a)/[partial derivative]a [not equal to] = 0.

By following Theorems 1-2, we have

Theorem 3. Let p = =2/3, s > 0 and r < -3 be fixed, and [epsilon] is sufficiently small, we then have the following conclusions:

1) If a < r, then system (5.2) has at least one regular relaxation oscillation near the singular periodic orbit described above;

2) If [mathematical expression not reproducible], and a, k are two constants, then system (5.2) has at least one canard relaxation oscillation with or without head depending on the sign of the quantity as follows,

[sigma] [partial derivative]h/[partial derivative]a (0, 0, r) = [sigma]/s,

where [a.sub.c]([epsilon]) is determined by the following equation

[mathematical expression not reproducible]

5.2 A predator-prey model with Holling-II response

Consider the following classical two-dimensional predator-prey model with Holling-II response,

[mathematical expression not reproducible] (5.3)

in which, 0 < [lambda] < 1, [mu] > 0 are two regular parameters, and 0 < [epsilon] [much less than] 1 is the singular parameter. The existence of the regular and singular parameters depends on the specific biological settings.

Let [epsilon]t = [tau], one gets the associated slow system,

[mathematical expression not reproducible] (5.4)

By introducing the following new time scale, [tau] = ([lambda] + x) [??], by which, equation (5.4) can be changed to a polynomial system, namely

[mathematical expression not reproducible] (5.5)

Let

[mathematical expression not reproducible]

then system (5.5) becomes

[mathematical expression not reproducible]

i.e.

[mathematical expression not reproducible]

where

[a.sub.1] = 1/[mu], [a.sub.2] = 1 - [lambda]/1 - [mu], [a.sub.3] = [lambda][mu]/(1 - [mu]).sup.2].

Let us further set [mathematical expression not reproducible], then we get

[mathematical expression not reproducible]

which has been written in a standard singular perturbation generalized Lienard systems, where we still use x, y and t to denote the dependent and independent variables for simplicity. Let

[mathematical expression not reproducible]

by which, we obtain

[mathematical expression not reproducible] (5.6)

where, for simplicity, the bars have been omitted.

In this example, we have

[mathematical expression not reproducible]

For system (5.6), one can see that

1) x = 0 is a generic extreme point of the following critical curve

[mathematical expression not reproducible]

with

f'(0) = 0, f" (0) = - [a.sup.2.sub.2]/[2a.sub.3] < 0.

Therefore, near x = 0, we can write f'(x) = xp(x), where p(x) > 0.

2) g(0) = f(0) = [a.sup.2.sub.2]/4[a.sub.3] and g/(y) = ([a.sub.1] + [a.sup.2.sub.2]/4[a.sub.3] [e.sup.y] > 0.

3) System (5.6) has a singular point

[mathematical expression not reproducible]

4) As [a.sub.2] [right arrow] [2a.sub.3], ([x.sub.e], [y.sub.e]) [right arrow] (0, 0), which is a canard point. When the flow of system (5.3) gets through a small neighborhood of the canard point, it displays the feature of delay loss of stability. By section 3, the position where the flow blows up is at x = [x.sub.k], which can be solved from the following equation

[mathematical expression not reproducible]

If the flow extends to the right, it forms a canard relaxation oscillation without head, and if to the left, it will stay near x = 0 and display the feature of delay loss of stability again when it passes through the transcritical bifurcation point, and finally forms another canard relaxation oscillation, see Figure 4.

5) Let [mathematical expression not reproducible], then it can be verified that

[mathematical expression not reproducible]

From Section 3, we have the following conclusion:

Theorem 4. When [epsilon] is sufficiently small and

[mathematical expression not reproducible]

where [a.sub.c](0) = 1 and [sigma], k are two constants, then system (5.3) has at least one canard solution with or without head, see Figure 4, where [a.sub.c]([member of]) is determined by the following equation,

[mathematical expression not reproducible]

https://doi.org/10.3846/13926292.2017.1315344

Acknowledgements

This work was supported by NSFC (No.11201072, 11401229), NSF of Fujian Province (No. 2015J01004, 2014J01002) and Nonlinear Analysis Innovation Team (IRTL 1206) funded by FJNU. The authors would also like to thank the referees for the proposed improvements.

References

 E. Benoit, J.L. Callot, F. Diener, M. Diener et al. Chasse au canard (premiere partie). Collectanea Mathematica, 32(1):37-76, 1981.

 X. Chen, P. Yu, M. Han and W. Zhang. Canard solutions of two-dimensional singularly perturbed systems. Chaos, Solitons & Fractals, 23(3):915-927, 2005. https://doi.org/10.1016/S0960-0779(04)00339-X.

 P. De Maesschalck. Planar canards with transcritical intersections. Acta Applicandae Mathematicae, 137(1):159-184, 2015. https://doi.org/10.1007/s10440014-9994-9.

 P. De Maesschalck and F. Dumortier. Classical Lienard equations of degree n [greater than or equal to] 6 can have [n-1/2] + 2 limit cycles. Journal of Differential Equations, 250(4):21622176, 2011. https://doi.org/10.1016/j.jde.2010.12.003.

 F. Dumortier. Slow divergence integral and balanced canard solutions. Qualitative Theory of Dynamical Systems, 10(1):65-85, 2011. https://doi.org/10.1007/s12346-011-0038-9.

 F. Dumortier, J. Llibre and J.C. Artes. Qualitative theory of planar differential systems. Springer, 2006.

 F. Dumortier, D. Panazzolo and R. Roussarie. More limit cycles than expected in Lienard equations. Proceedings of the American Mathematical Society, 135(6):1895-1904, 2007. https://doi.org/10.1090/S0002-9939-07-08688-1.

 F. Dumortier and R. Roussarie. Bifurcation of relaxation oscillations in dimension two. Proceedings of the American Mathematical Society, 19(4):631-674, 2007.

 F. Dumortier and R.H. Roussarie. Canard cycles and center manifolds, volume 577. American Mathematical Soc., 1996.

 W. Eckhaus. Relaxation oscillations including a standard chase on French ducks. In Asymptotic Analysis II, pp. 449-494. Springer, 1983. https://doi.org/10.1007/BFb0062381.

 J. Grasman. Relaxation oscillation. In Robert A. Meyers(Ed.), Mathematics of Complexity and Dynamical Systems, pp. 1475-1488. Springer, 2012. https://doi.org/10.1007/978-1-4614-1806-1_93.

 M. Han, Y. Tian and P. Yu. Small-amplitude limit cycles of polynomial Lienard systems. Science China Mathematics, 56(8):1543-1556, 2013. https://doi.org/10.1007/s11425-013-4618-9.

 S.-B. Hsu and J. Shi. Relaxation oscillator profile of limit cycle in predator-prey system. Disc. Cont. Dyna. Syst.-B, 11:893-911, 2009.

 M. Krupa and P. Szmolyan. Extending geometric singular perturbation theory to nonhyperbolic points--fold and canard points in two dimensions. SIAM Journal on Mathematical Analysis, 33(2):286-314, 2001. https://doi.org/10.1137/S0036141099360919.

 C. Li. Duck solutions: A new kind of bifurcation phenomenon in relaxation oscillations. Acta Mathematica Sinica, 12(1):89-104, 1996.

 C. Li. Duck solutions problem in singularly perturbed problems. Sci. China: Series A, 29(12):1084-1093, 1999.

 C. Li and H. Zhu. Canard cycles for predator-prey systems with Holling types of functional response. Journal of Differential Equations, 254(2):879-910, 2013. https://doi.org/10.1016/j.jde.2012.10.003.

 A. Lins, W. De Melo and C.C. Pugh. On Lienard's equation. In Geometry and, topology, pp. 335-357. Springer, 1977. https://doi.org/10.1007/BFb0085364.

 W. Liu, D. Xiao and Y. Yi. Relaxation oscillations in a class of predatorprey systems. Journal of Differential Equations, 188(1):306-331, 2003. https://doi.org/10.1016/S0022-0396(02)00076-1.

 R. Roussarie. Putting a boundary to the space of Lienard equations. Discrete and Continuous Dynamical Systems, 17(2):441, 2007. https://doi.org/10.3934/dcds.2007.17.441.

 J. Shen. Canard limit cycles and global dynamics in a singularly perturbed predator-prey system with non-monotonic functional response. Nonlinear Analysis: Real World Applications, 31:146-165, 2016. https://doi.org/10.1016/j.nonrwa.2016.01.013.

 S Van Gils, M Krupa and P Szmolyan. Asymptotic expansions using blow-up. Zeitschrift fur angewandte Mathematik und Physik ZAMP, 56(3):369-397, 2005.

 Y. Xiong and M. Han. New lower bounds for the Hilbert number of polynomial systems of Lienard type. Journal of Differential Equations, 257(7):2565-2590, 2014. https://doi.org/10.1016/j.jde.2014.05.058.

Huan Hu, Jianhe Shen, Zheyan Zhou and Zhonghui Ou

School of Mathematics and Computer Science, Fujian Normal University 350007 Fuzhou, China

E-mail: jhshen@fjnu.edu.cn

Received April 20, 2016; revised March 28, 2017; published online May 15, 2017

Caption: Figure 1. The directions of the slow flows (single arrows) and the fast fibers (double arrows). In the figure, the large dots denote the equilibriums. (a): The case when a [member of] ([a.sub.1], [a.sub.0]); (b): The case when a = [a.sub.0].

Caption: Figure 2. Schematic plots of two representative trajectories starting respectively inside and outside the singular periodic orbit OABC. In the figure, the large dot stands for the equilibrium.

Caption: Figure 3. Schematic plots of canard relaxation oscillations with or without head. (a): [sigma] [partial derivative]h/[partial derivative]a (0, 0, 0) < (b): [sigma] [partial derivative]h/[partial derivative]a (0, 0, 0) > 0.

Caption: Figure 4. Schematic plots of canard relaxation oscillations in system (5.3). (a): [sigma] < 0; (b): [sigma] > 0.