Limit cycles for Singular Perturbation Problems via Inverse Integrating Factor

abstract: In this paper singularly perturbed vector fields Xε defined in R are discussed. The main results use the solutions of the linear partial differential equation XεV = div(Xε)V to give conditions for the existence of limit cycles converging to a singular orbit with respect to the Hausdorff distance.


Introduction and statement of the main results
The present work fits within the geometric study of singular perturbation problems expressed by one-parameter families of vector fields X ε : R 2 −→ R 2 where X ε (x, y) = (f (x, y, ε), εg(x, y, ε)) with ε ≥ 0, f, g ∈ C r for r ≥ 1 or f, g ∈ C for which we want to study the phase portrait, for sufficient small ε, near the set of singular points of X 0 : Special emphasis will be given on systems which the solutions of the linear partial differential equation 2000 Mathematics Subject Classification: 34C25,34C35,34E15 The system of differential equations associated to X ε is x = f (x, y, ε), y = εg(x, y, ε) with x = x(τ ), y = y(τ ).
The main trick in geometric singular perturbation (GSP) is to consider the above family in addition to the family ε ẋ = f (x, y, ε), ẏ = g(x, y, ε) with x = x(t), y = y(t) obtained after the time rescaling t = ετ .System (2) is the fast system and (3) is the slow system.
Observe that for ε > 0 the phase portrait of the fast and the slow systems coincide, but for ε = 0 the problems are completely different.
We call the slow manifold of the singular perturbation problem, and the dynamical system defined by (3) on , for ε = 0, is called the reduced problem.Combining results on the dynamics of these two limiting problems, with ε = 0, one obtains information on the dynamics for small values of ε.In fact, such techniques can be exploited to formally construct approximate solutions on pieces of curves that satisfy some limiting version of the original equation as ε goes to zero.
Let n 1 and n 2 be normally hyperbolic points on , see for a definition Section 2. A singular orbit consists of three pieces of smooth curves: an orbit of the reduced problem starting at n 1 , an orbit of the reduced problem ending at n 2 and a orbit of the fast problem connecting the two previous peaces.
For two compact sets A, B ⊆ R 2 we define the Hausdorff distance by The main question in GSP-theory is to exhibit conditions under which a singular orbit can be approached by regular orbits for ε 0, with respect to the Hausdorff distance.The most interesting question is to decide if X ε has a limit cycle approaching a singular orbit.In this case, the singular orbit should have a non normally hyperbolic point, that means there is a turning point in the usual terminology, i.e an extreme local of the function defined implicitly by f (x, y, 0) = 0. Some papers are in this direction [2,3,4,8,9,10].
In the qualitative theory of differential equations, research on limit cycles is a difficult part.Limit cycles of planar vector fields were defined by Poincaré and at the end of the 1920s van der Pol, Liénard and Andronov proved that a closed trajectory of a self-sustained oscillation occurring in a vacuum tube circuit was a limit cycle as considered by Poincaré.There are some methods for proving the nonexistence and existence of limit cycles: Bendixon-Dulac, Poincaré-Bendixson, the return map, etc.The main trick used in this paper is to use the criteria introduced in [6] to study the limit cycles of X ε , for ε 0. The main results of this paper are the following.
Corollary 1.1A.Consider X ε and V like in Theorem 1.1.If the level zero of the function V (x, y, ε) does not contain a closed curve, for 0 < ε < ε 0 , then X ε does not present a limit cycle.
We remark that Theorem 1.1 provides a necessary condition in order that a singular orbit Γ can generate, for ε > 0 sufficiently small, a limit cycle.More specifically, if are analytical in their variables, then V (x, y, ε) is analytical, and for some C 1 function ϕ, and We remark that Theorem 1.2 provides an way to compute an approximation of the solution V (x, y, ε).
In Section 2 we present basic facts of the GSP-theory and one criteria for the study the existence and nonexistence of limit cycles introduced in [6].In Section 3 we prove the main result and in Section 4 we present some examples and applications.

Basic facts of GSP-theory and inverse integrating factor
2.1.The GSP-theory.The foundation of GSP-theory, which is briefly summarized here, was laid by Fenichel in [5].We consider only planar problems but remember that in [5] one can check the general case.
Let X ε (x, y) = (f (x, y, ε), εg(x, y, ε)) with (x, y) ∈ R 2 and the slow manifold Σ given implicitly by f (x, y, 0) = 0. We say that p = (x 0 , y 0 We assume that, for every normally hyperbolic p ∈ Σ, ∂f ∂x (p, 0) has k s eigenvalues with negative real part and k u eigenvalues with positive real part.
Theorem 2.1.Let n ∈ Σ be a hyperbolic singular point of the slow flow with j sdimensional local stable manifold W s and a j u -dimensional local unstable manifold W u .Then there exists an ε-continuous family n ε such that n 0 = n and n ε has a (j s + k s )-dimensional local stable manifold W s ε and a (j u + k u )-dimensional local unstable manifold W u ε .For a proof see [5].The importance of this theorem is that every structure of the slow system which persists under regular perturbation also persists under singular perturbation.The next step is to decide if a singular orbit can be approached by regular orbits.
Theorem 2.2.If n, m ∈ Σ, like in Theorem 2.1, are connected by an orbit of the fast problem then there exists an orbit of X ε connecting n ε and m ε .
For a proof see [11].Combining Theorem 2.1 and Theorem 2.2 one can see that if a singular orbit Γ is composed by orbits of the reduced problem on the normally hyperbolic part of the slow manifold and connected by orbits of the fast problem, then there are regular orbits Γ ε of X ε , such that Γ ε −→ Γ, for ε 0, according Hausdorff distance.To analyse the non-normally hyperbolic case there is a new technique introduced by Dumortier and Roussarie in [4] which is based on the blow up techniques.Another approach can be obtained in [3] for the same problems by assuming that the systems are time reversible.

The inverse integrating factor.
Let U be the domain of definition of the vector field X(x, y) = (p(x, y), q(x, y)) and let W be an open subset of U .A non-zero function V on W that satisfies the linear partial differential equation XV = div(X)V, is called an inverse integrating factor of the vector field X.
This function V is important because (i) R = 1/V defines on W \ {V = 0} an integrating factor of the differential system associated to the vector field.
(ii) {V = 0} contains the limit cycles of the phase portrait of the vector field X.This fact allows to study the limit cycles which bifurcate from periodic orbits of a centre ( Hamiltonian or not) and compute their shape.For doing that we develop the function V in power series of the small perturbation parameter.
A remarkable fact is that the first term of this expansion coincides with the first non-identically zero Melnikov function.
(iii) There are a great number of examples of vector fields with an inverse integrating function V being an easier function than their first integrals.

Proof of the main results
In this section we shall prove the results state in the introduction.
Proof of Corollary 1.1A: Since the set {V = 0} contains the limit cycles of X ε in U and it has no closed curve, X ε can not have limit cycles.
Proof of Theorem 1.2: We deal with planar systems of the form (2) where f (x, y, ε) and g(x, y, ε) depend analytically on their variables in an open subset U. Assume that ε is a small parameter.We look for an analytic solution of the linear partial differential equation It is known that V is analytic in the variables x, y, ε (see for instance [7]).From equation ( 5) we deduce the zero-order equation with respect to ε At k-th order with respect to ε we obtain For any value of k, the homogeneous partial differential equation for V k is the same.So, the way to solve (7) is recursive.Since equation ( 6) becomes ∂ ∂x for some C 1 function ϕ depending of the variable y.
or equivalently

Examples and applications
In the following examples we compute the inverse integrating factor of some vector fields singularly perturbed using the partial differential equations states in Theorem 1.2.
Example 4. Let X ε be the vector field given by The slow manifold is given by Σ = {(x, y) : x 2 + y 2 = 1} and the function V ε (x, y) = V 0 (x, y) = x 2 + y 2 − 1 is an inverse integrating factor of X ε .In fact we can apply Proposition 4.1 with λ = ε and g(x, y) = εx + y.The slow system associated to the vector field is and the reduced problem is The fast and slow dynamics are illustrated in Figure 4.According Theorem 1.2 the only singular orbit which can be approached by limit cycles is the slow manifold x 2 + y 2 − 1 = 0.
Therefore the closed orbit defined by V 0 = 0 is a limit cycle of X ε for ε 0.
In our last example, we consider a vector field which do not have a polynomial inverse integrating factor.Example 5.The vector field X ε,a (x, y) = (y − x 3 /3 + x, ε(a − x)) was considered in [1].It is an example of the canard phenomenon.It is known that for |a| < 1 the vector field has a unique limit cycle, which is stable, and for |a| ≥ 1 does not present limit cycles and it has a singular point (a, a 3 /3 − a) which is the -limit of any orbit.For |a| = 1 occurs an Andronov-Hopf bifurcation.When |a| 1 the amplitude of the limit cycles tends to zero.