Bifurcation diagrams for singularly perturbed system: the multi-dimensional case.

We consider a singularly perturbed system where the fast dynamics of the unperturbed problem exhibits a trajectory homoclinic to a critical point. We assume that the slow time system admits a unique critical point, which undergoes a bifurcation as a second parameter varies: transcritical, saddle-node, or pitchfork. We generalize to the multidimensional case the results obtained in a previous paper where the slow-time system is 1-dimensional. We prove the existence of a unique trajectory (˘ x ( t; "; (cid:21) ) ; ˘ y ( t; "; (cid:21) )) homoclinic to a centre manifold of the slow manifold. Then we construct curves in the 2-dimensional parameters space, dividing it in diﬀerent areas where (˘ x ( t; "; (cid:21) ) ; ˘ y ( t; "; (cid:21) )) is either homoclinic, heteroclinic, or unbounded. We derive explicit formulas for the tangents of these curves. The results are illustrated by some examples.


Introduction
In this paper we consider the following singularly perturbed system: {ẋ = εf (x, y, ε, λ) y = g(x, y, ε, λ) (1.1) where x ∈ R, y ∈ R n and (x, y) ∈ Ω, Ω ⊂ R 1+n is open, λ and ε are small real parameters and f (x, y, ε, λ), g(x, y, ε, λ) are C r , bounded with their derivatives, r ≥ 3. We suppose that the following conditions hold: (i) for all x ∈ R, we have g(x, 0, 0, 0) = 0, (ii) the infimum over x ∈ R of the moduli of the real parts of the eigenvalues of the jacobian matrix ∂g ∂y (x, 0, 0, 0) is greater than a positive number Λ g .
(iii) the equationẏ = g(0, y, 0, 0) has a solution h(t) homoclinic to the origin 0 ∈ R n (iv)ḣ(t) is the unique bounded solution of the linear variational system: up to a scalar multiple.
According to condition (ii), for any x ∈ R, the linear systemẏ = ∂g ∂y (x, 0, 0, 0)y has an exponential dichotomy on R with exponent Λ g > 0 and projections, say, P 0 (x). For simplicity we set P 0 (0) = P 0 . Let rank[P 0 (x)] = p, p being the number of eigenvalues of ∂g ∂y (x, 0, 0, 0) with positive real parts: we stress that p is constant. From assumptions (ii) and (iii) it follows that the linear system (1.2) and its adjoinṫ have exponential dichotomies on both R + and R − ; i.e. there are projections P ± and k > 0 such that where Y (t) is the fundamental matrix of (1.2), and the analogous estimate hold for (1.3). Here and later we use the shorthand notation ± to represent both the + and − equations and functions. Observe that rank(P + ) = rank(P − ) = p and the projections of the dichotomy of (1.3) on R ± are I − [P ± ] * . Moreover from (i)-(iv) it follows that (1.3) has a unique bounded solution on R, up to a multiplicative constant. We denote one of these solutions by ψ(t). Note that ψ := ψ(0) satisfies N [P + ] * ∩ R[P − ] * = span(ψ) = [RP + ∩ N P − ] ⊥ ; we assume w.l.o.g. that |ψ(0)| = 1. As a second remark we observe that condition (i) implies the existence of a so called local "slow manifold" M c . I.e. there is a function v(x, ε, λ) defined for x, ε, λ small enough, such that v(x, 0, 0) ≡ 0 and the manifold y = v(x, ε, λ) is an invariant centre manifold for the flow of (1.1) (see for example [2,11]). Moreover v(x, ε, λ) is C r−1 , bounded with its derivatives. Using the flow of (1.1) we can pass from the local manifold y = v(x, ε, λ) to a global slow manifold for system (1.1) which will be denoted by M c = M c (ε, λ). Note that if we assume (as in the examples in section 4) that g( Let x c (t, ξ, ε, λ) be the solution of the scalar ODE: ) describes the flow on the slow manifold M c , and (1.5) is the so called "slow time" system. The behavior of homoclinic and heteroclinic trajectories subject to singular perturbation has been studied in several papers, see e.g. [1,2,4,5,6,11,13]. In particular in [6] the authors built up a theory to prove the existence of solutions homoclinic to M c , for the perturbed problem (1.1) assuming conditions (i)-(iv) and giving transversality conditions of several different types. They refine previous results obtained in [4]. This paper is thought as a sequel of [6]. Here we assume that the "slow time" system (1.5) is one-dimensional so that there is a unique solution (x(t, ε, λ),ỹ(t, ε, λ)) homoclinic to M c . Moreover we assume that (1.5) undergoes a bifurcation for ε = 0 as λ changes sign. We mainly focus on the transcritical and saddle-node case, i.e. we assume f has one of the following form: where a(ε) and b(ε) are positive C r−1 functions and the terms contained in O(x 3 ) are C r−1 in x, ε and C r−2 in λ. The aim of this paper is to derive further Melnikov conditions which enable us to divide the ε, λ space into different sets in which (x(t, ε, λ),ỹ(t, ε, λ)) has different behavior: it is homoclinic, heteroclinic or it does not converge to critical points either in the past or in the future. We stress that we have explicit formulas for the derivatives of the curves defining the border of these sets. This is the content of Theorems 3.2, 3.5 which regard respectively the case where (1.5) undergoes a transcritical or a saddle-node bifurcation.
Our purpose is to find trajectories of (1.1) which are close for any t ∈ R to the homoclinic trajectory (0, h(t)) of the unperturbed system. We use the implicit function theorem to construct Melnikov conditions which ensure the existence of such trajectories, and which allow to say if they are homoclinic, heteroclinic or unbounded. The techniques can be applied also to bifurcations of higher order, i.e. when the first nonzero term of the expansion of f in x has degree 3 or more (obviously in this case we need to assume f at least C 4 or more in the x variable). However in such a case to obtain a complete unfolding of the singularity more parameters are needed. In fact we just sketch the case of pitchfork bifurcation (which however appears frequently when f , for some physical reasons, is odd in x for any ε and λ). Again, following subsection 11.2 of [12], we see that, up to changes in variables and parameters, we may reduce to f of the form where a(ε) and b(ε) are C r−1 positive functions and the o( The assumptions used in the main Theorems are the following: The paper is divided as follows. In section 2 we briefly review some facts, proved in [6]: we construct the solutions asymptotic to the slow manifold M c , then we match them via implicit function theorem, to construct a solution (x(t, ε, λ),ỹ(t, ε, λ)) homoclinic to M c . In section 3 we show which is the behavior of (x(t, ε, λ),ỹ(t, ε, λ)) as ε and λ varies, in the transcritical and in the saddle-node case (subsections 3.1 and 3.2 respectively). So we give sufficient conditions in order to have homoclinic, heteroclinic or no bounded solutions close to (0, h(t)), as the parameters vary: this is the content of Theorems 3.2 and 3.5. We emphasize that condition (v) is needed to construct (x(t, ε, λ),ỹ(t, ε, λ)), condition (vi) is needed just in the (1.6) case, to understand the behavior of such a trajectory: no further condition is needed for (1.7). Finally we explain how the same methods can be extended to describe pitchfork and higher degree bifurcations in subsection 3.3. We illustrate our results drawing some bifurcation diagrams. Finally in section 4 we construct examples for which we can explicitly compute the derivatives of the bifurcation curves appearing in the diagrams. We conclude the introduction with a remark concerning the regularity of the functions used and constructed. Remark 1.1. We stress that the loss of two orders of regularity just depends on the following facts: one order is due to the construction of the slow manifold, the other to the change of parameters that drives (1.5) either in the form (1.6) or (1.7). If we assume g(x, 0, ε, λ) ≡ 0 so that the slow manifold reduces to y = v(x, ε, λ) ≡ 0, and we assume that f satisfies directly either (1.6) or (1.7), there is no loss of regularity and we may always assume f and g just C r with r ≥ 1, and all the functions introduced would be C r as well.

The centre-stable and centre-unstable manifolds
In this section we define the local centre-stable and centre-unstable manifolds and we recall their smoothness properties. These manifolds are (locally) invariant manifolds of solutions that approach the slow manifolds y = v(x, ε, λ) at an exponential rate. In [5,6] the following result has been proved.
Theorem 2.1. [6] Let f and g be bounded C r functions, r ≥ 2, with bounded derivatives, satisfying conditions (i)-(iv) of the Introduction and let the numbers β and σ satisfy 0 < rσ < β < Λ g . Then, given suitably small positive numbers µ 1 and µ 2 , there exist positive numbers of (1.1) defined respectively for t ≥ 0 and for t ≤ 0 such that for t ≥ 0, and for t ≤ 0, and ε, λ) and for k ≤ r−1 their k th derivatives also satisfy the estimate (2.2) with β replaced by β − kσ and µ 1 and µ 2 replaced by possibly larger constants. Also there is and for t ≥ 0 Following section 2.1 in [6], using Theorem 2.1 we define the local centreunstable and centre-stable manifolds near the origin in R n+1 as follows In [6] it has been proved that M cu loc and M cs loc are respectively negatively and positively invariant for (1.1). Thus, going respectively forward and backward in t, we can construct from M cu loc and M cs loc the global manifold M cu and M cs , see Lemma 2.3 in section 2.2 in [6]. Therefore M cu and M cs are respectively p + 1 and n − p + 1 dimensional immersed manifolds of R n+1 , made up by the trajectories asymptotic to M c resp. in the past and in the future.
In the next section we need the derivatives ofξ ± (ε, λ) with respect to ε and λ. For this purpose we evaluate first the derivatives ofξ ± (ξ, ε, λ); in fact it is easy to check thatξ ± (ξ, 0, λ) = ξ for any λ so the derivative with respect to λ is null. Following again section 3.1 in [6] we see that (2.19)

Existence of Homoclinic and Heteroclinic solutions.
In this section we state and prove our main results. Since (x(t, ε, λ),ỹ(t, ε, λ)) is constructed via implicit function theorem we have local uniqueness, see the explanation just after theorem 2.2. Our purpose is to divide the parameters space in different subsets in which the solution (x(t, ε, λ),ỹ(t, ε, λ)) has a different asymptotic behavior. Note that U (ε, λ) and S(ε, λ) change their stability properties when they cross the lines lines λ = 0 and ε = 0, thus we need to argue separately in the 4 different quadrants. At this point we need to distinguish between f satisfying (1.6) and (1.7).
To complete the picture we need to repeat the analysis in the other quadrants. When λ ≤ 0 < ε the critical point u is stable and s is unstable with respect to the flow of (1.5). So we define Thus, if (vi) holds, we can apply the implicit function theorem and construct the curves λ ± 4 (ε) such that J ± 4 (ε, λ ± 4 (ε)) = 0. Moreover we find Obviously Remark 3.1 holds also in this setting with trivial modifications (and when ε < 0 as well, see below). Reasoning as above and using a Taylor expansion analogous to (3.4), we can draw a detailed bifurcation diagram (see again figures 1, 2, 3). When ε < 0 we have an inversion in the stability properties of the critical points of (1.1) with respect to the stability properties of (1.5). Once again we assume (vi) and we distinguish between negative and positive values of λ. When λ > 0 we use again the functions J ± 4 defined in (3.1) and we extend the curves λ ± 4 (ε) to ε < 0; similarly for λ < 0 we use J ± 1 defined in (3.5) and we extend the curves λ ± 1 (ε). Note that also in these cases the derivatives of λ ± 1 and λ ± 4 are the ones given in (3.3) and in (3.6) so the curves are C 1 in the origin.
The bifurcation diagram changes according to the signs of the nonzero computable constants ∂J ± i ∂λ (0, 0) and of the following computable constants which may be zero possibilities can be obtained similarly (not all the combinations are effectively possible). In section 4 we construct a differential equation for which the values of these constants are explicitly computed.
For completeness we observe that, when ε = 0 (1.1) reduces tȯ From Hypothesis (v) it follows that there are δ > 0 and a uniqueξ ∈ (−δ, δ) (which is not necessarily a critical point for (1.5)) such that y(ξ, t) is a homoclinic trajectory. When the computable constants given in (3.7) are null we cannot draw the bifurcation diagram in all details. However, using the expansion (3.4), we obtain the asymptotic behavior of (x(t, ε, λ),ỹ(t, ε, λ)), far from the λ = 0 axis. When dλ + i dε (0) = dλ − i dε (0) for either i = 1, 4, we cannot exactly determine the behavior of (x(t, ε, λ),ỹ(t, ε, λ)) for (ε, λ) close to the curves λ = λ ± i (ε).  , we cannot say whether the curves λ ± i are above or below the line λ = 0. We think it is worth observing that in the previous case a new scenario may arise. In fact a priori we could have uncountably many intersections between λ + i and λ − i . These intersections would correspond to heteroclinic trajectories with fast convergence and following the unusual direction: when ε and λ are positive the trajectory tends to S in the past and to U in the future. So (x(t, ε, λ),ỹ(t, ε, λ)), together with the heteroclinic connection between U and S contained in M c (ε, λ), form a heteroclinic cycle.