Spike-adding canard explosion of bursting oscillations

Spike-adding canard explosion of bursting oscillations Paul Carter Mathematical Institute Leiden University Abstract This paper examines a spike-adding bifurcation phenomenon whereby small amplitude canard cycles transition into large amplitude bursting oscillations along a single continuous branch in parameter space. We consider a class of three-dimensional singularly perturbed ODEs with two fast variables and one slow variable and singular perturbation parameter ε 1 under general assumptions which guarantee such a transition occurs. The primary ingredients include a cubic critical manifold and a saddle homoclinic bifurcation within the associated layer problem. The continuous transition from canard cycles to N-spike bursting oscillations up to N O(1/ε) spikes occurs upon varying a single bifurcation parameter on an exponentially thin interval. We construct this transition rigorously using geometric singular perturbation theory; critical to understanding this transition are the existence of canard orbits as well as slow passage through the saddle homoclinic bifurcation, which are analyzed in detail. 1 Introduction The phenomenon of bursting has been widely studied in models of neurons and neuroendocrine cells. These solutions are typically time-periodic and alternate between slow quiescent phases and active bursting phases comprised of a sequence of action potentials or spikes. One of the earliest models is due to Chay and Keizer [3], which formed the basis in [23, 24] for analyzing the bursting phenomenon in the context of fast-slow ordinary differential equations, which has since been the primary mathematical formulation for understanding bursting in numerical and analytical studies. In this context, this paper is concerned with the notion of continuous transition between different solutions upon variation in parameters. Within the study of bursting oscillations, one frequently observes so-called spike-adding, in which small variation in system parameters can result in additional spikes during the bursting phase. This has been observed numerically in a variety of models [5, 6, 11, 19, 21, 22, 3] and has frequently been linked to the phenomenon of canards [8, 9, 17]. Canards have been posed as a mechanism whereby small parameter changes can produce continuous transitions between globally distinct solutions in many contexts, for example in the classical planar canard explosion [17], or in transitions between different traveling pulse solutions in FitzHugh Nagumo system of nerve impulse propagation [2]. In this spirit, this paper aims to rigorously analyze the link between canards and the spike adding phenomenon and identify general techniques which can be used in the analysis of similar global transitions in singular perturbation problems. With regards to spike-adding, the fundamental observation is that bursting solutions with different numbers of spikes can exist in nearby parameter regimes, and numerical analysis of many models suggest that different branches of spiking solutions are connected. In particular it has been observed [6, 28] that a periodic orbit with N spikes can be continuously deformed into one with N + 1 (or more) spikes by adjusting a single system parameter. In this work, our interest lies the spike adding process that arises when adjusting a canard-unfolding parameter [6], though we remark that it has been suggested [18, 28] that spike-adding can also occur when 1

varying the singular perturbation parameter itself. Here we focus on the canard phenomenon as it has been identified as an important component in the onset of spike-adding in the context of various models, including the Hindmarsh-Rose model [12, 13] of neuronal bursting and the Morris Lecar Terman model [2, 25, 28] describing bursting in pancreatic β-cells. Canards have also been cited in more complex bifurcation scenarios in bursting applications and in the study of mixed-mode bursting oscillations [6]. The Morris Lecar system [2] v = y.5 (v +.5) 2w (v +.7) m (v) (v 1) ẇ = 1.15 (w (v) 2w) τ(v) ẏ = ε(k + k r 2v) (1.1) w (v) = 1 + tanh ( ) v.1, m (v) = 1 + tanh.145 ( v +.1.15 ) ( ) v.1, τ(v) = cosh,.29 was originally proposed as a model of electrical activity in barnacle muscle fibers. In that context, v is interpreted as membrane potential, w as fraction of open potassium ion channels, and y a potassium current variable. The quantity k + k r determines the equilibrium potential corresponding to potassium conductance. We identify (1.1) as being among the simplest examples of onset of spike-adding of bursting oscillations through canard explosion. Figure 1 depicts the transition from local canard explosion to large scale bursting oscillations, obtained numerically in (1.1) for ε =.1. The lower panels show the v-profiles of bursting oscillations with 1 and 2 spikes in bursting phase; all solutions along the transition from local canard explosion born at a Hopf bifurcation to large amplitude bursting oscillations were found along the same branch in parameter space and were obtained in the numerical continuation software AUTO upon varying the parameter k for fixed ε =.1. In [28], Terman developed general assumptions in a class of three-dimensional ODEs which ensure that the geometry of the equations is qualitatively similar to that of (1.1). See Figure 2 for a visualization of the singular limit geometry. The primary features are a cubic critical manifold M with three branches: an attracting bottom branch M b, a saddle-type middle branch M m, and upper branch M u (typically repelling). In the fast layer dynamics for a value of y = ȳ h, the system undergoes a saddle-homoclinic bifurcation along the middle branch, from which bifurcates a family P of periodic orbits for y > ȳ h. The cubic critical manifold also admits two fold points: one of classical fold type (F l ) and one of canard type (F r ); in (1.1), the constant k r denotes the v-coordinate of the fold F r, which can be approximated numerically as k r.2449. The fold points and the saddle homoclinic bifurcation are the key pieces (and main challenges) to understanding the spike adding transitions in this general model setting. The spike-adding sequence is then generated as follows (see Figure 1 for sample periodic orbits along the transition in the Morris Lecar system (1.1) obtained using AUTO). At the lower right fold F r are born small amplitude canard orbits; see for example the blue orbit in the left panel of Figure 1. As the parameter k is varied on an exponentially thin interval, the orbits grow into large amplitude canards until reaching the upper left fold F l, though when continuing along the repelling upper branch M u, eventually they begin to interact with the family P of periodic orbits. The number of spikes in a given bursting solution is determined by the number of excursions around the upper branch M u, and the family of periodic orbits P allows for many such excursions. In particular, passing near the saddle homoclinic bifurcation allows for a fast spike which follows the singular homoclinic orbit to be deposited, while the growth continues back along the middle branch M m towards the fold, and back to the saddle homoclinic bifurcation to deposit another spike, and so on. The fact that the slow portion of the bursting orbits passes near the lower fold from an attracting slow manifold to a saddle slow manifold (that is, along a canard segment) is what allows each successive spike adding event to occur, each taking place within an exponentially thin interval of the parameter k. Figure 3 depicts a bursting solution obtained numerically in (1.1) after many such spike-adding events by continuing numerically in parameter space from the local canard explosion at the fold F r. 2

.1.1 -.1 -.1 B A v -.2 -.3 3 2 4 1 v -.2 -.3 6 7 5 8 -.4 -.4 -.5 -.5 -.2.2.4.6.8 y -.2.2.4.6.8 y.1 4.1 8 A B v -.1 v -.1 -.2 -.2 -.3 -.3 t t Figure 1: Shown is the continuous spike-adding transition in (1.1) for ε =.1 obtained in AUTO; all solutions exist for values of k.2 on an interval of width O(e 1/ε ). The upper left panel shows the transition sequence labeled 1 4 from small amplitude canard orbits (blue label 1) to a 1-spike bursting solution (green label 4), and the critical manifold M is shown in dashed red. The lower left panel depicts the the v profile for the 1-spike solution. The upper right panel shows the transition sequence labeled 5 8 from a 1-spike bursting solution to a 2-spike bursting solution (green label 8). The solutions labelled 5 8 all traverse the spike labelled A. The second spike is grown from right to left until reaching the upper fold F l, where it turns back (see solution with orange label 6) and continues from left to right to solution 7 before finally being deposited at the spike labelled B, culminating in a 2-spike bursting solution (green label 8); the v profile for the 2-spike solution is shown in the lower right panel. 3

v M u F ` M m P M b F r w y h y Figure 2: Shown is the singular limit geometry for a spike-adding system. The cubic critical manifold admits three branches: M u (repelling), M m (saddle-type), and M b (attracting). There are two folds: F l (classical fold) and F r (canard point). In the fast layer dynamics, there is a saddle homoclinic bifurcation at y = ȳ h which results in a family of periodic orbits P for y > ȳ h. v.1 -.1 -.2 -.3 -.4 -.5 v.2.1 -.1 -.2 -.3 -.4 -.2.2.4.6.8 y -.5 t Figure 3: Spikes are continuously added to the bursting oscillations along the spike-adding transition, achieving an O(1/ε) number of spikes. The left panel depicts a bursting solution of (1.1) along the transition from 8 to 9 spikes, and the right panel depicts the corresponding v-profile. The solution was obtained in AUTO for ε =.1 and k.2. 4

The aim of this work is to outline this transition in detail for < ε 1 and rigorously construct the spike-adding sequence from small amplitude canard cycles to bursting solutions with an O(1/ε) number of spikes. We will show that this transition occurs along a single continuous branch under exponentially small variations in the single bifurcation parameter k for fixed ε. The primary technical challenges relate to analysis near the fold points F l,r as well as tracking solutions near the saddle homoclinic bifurcation. We present a detailed analysis of slow passage near the saddle homoclinic bifurcation in order to understand how solutions behave in this region; this analysis is critical in showing how branches of periodic orbits with different numbers of spikes are connected. The remainder of this paper is outlined as follows. The general setup and assumptions are detailed in 2, as well as the statement of the main result, Theorem 2.2. The proof of Theorem 2.2 is given in 3, followed by a brief discussion in 4. 2 Setup The model system under consideration is a three-dimensional singularly perturbed ordinary differential equation with two fast variables and one slow variable, which we write in the form v = f 1 (v, w, y, k, ε) ẇ = f 2 (v, w, y, k, ε) ẏ = εg(v, w, y, k, ε), (2.1) where = d dt, k is a bifurcation parameter, ε > is a small parameter and f 1, f 2, g are C r+1 -smooth functions of their arguments for some r 3. We refer to (2.1) as the fast system. By rescaling τ = εt, we obtain the corresponding slow system εv = f 1 (v, w, y, k, ε) εw = f 2 (v, w, y, k, ε) y = g(v, w, y, k, ε), (2.2) where = d. These two systems are equivalent for any ε >, though the dynamics are best understood dτ by perturbing from the distinct singular limits obtained by setting ε = in each of (2.1), (2.2). We outline hypotheses with respect to each of these limits in 2.1 and 2.2, respectively, and we state the main result in 2.3. 2.1 Layer problem Setting ε = in (2.1) results in the layer problem v = f 1 (v, w, y, k, ) ẇ = f 2 (v, w, y, k, ) ẏ =, (2.3) which we consider for k [ k, k ] for some k >. The dynamics are restricted to planes y =const, and this system admits a manifold of equilibria ( ) f 1 (v, w, y, k, ε) M := {(v, w, y) : F (v, w, y, k, ) = }, F (v, w, y, k, ε) := (2.4) f 2 (v, w, y, k, ε) which is called the critical manifold. For simplicity we assume M can be written as a graph over the v-coordinate. We also assume the following (see Figure 2). 5

h p u p u p u p u p p m p h p m b b b v w p b v w p b v w p b (a) (b) (c) Figure 4: Shown is the structure of the layer problem (2.3) in the cases: (a) y (ȳ l, ȳ h ), (b) y = ȳ h, and (c) y (ȳ h, ȳ r ). Pictured in each phase portrait for y (ȳ l, ȳ r ) are the heteroclinic orbits φ u (y), φ b (y), φ p (y); note that for y = ȳ h, the orbit φ u (y) coincides with the homoclinic orbit γ h. In (c), also pictured are the periodic orbits γ p ( ; y) which bifurcate from γ h for y > ȳ h. Hypothesis 1. (S-shaped critical manifold) We assume the critical manifold is S-shaped, consisting of three branches; that is, we assume there exists ȳ r, ȳ l such that the layer problem (2.3) admits a single equilibrium p b (y) for y (, ȳ l ), three equilibria p b (y), p m (y), p u (y) for y (ȳ l, ȳ r ) and a single equilibrium p u (y) for y (ȳ r, ), with two equilibria colliding at saddle-node bifurcations at each of y = ȳ l, ȳ r. We denote the fold points by We can therefore decompose M as F l,r = ( v l,r, w l,r, ȳ l,r ). (2.5) M = M b F r M m F l M u, (2.6) where the three branches M b,m,u (bottom, middle, upper) are contained in the regions { < y < ȳ r }, {ȳ l < y < ȳ r }, {ȳ l < y < }, respectively. We will sometimes write M (y 1, y 2 ), for = b, m, u to refer the intersection M {y 1 y y 2 }. Hypothesis 2. The bottom and middle branches of the critical manifold M satisfy the following. (i) The bottom branch M b is normally attracting, that is, D (v,w) F M b part. has two eigenvalues with negative real (ii) The middle branch M m is of saddle type, so that D (v,w) F M m has one positive and one negative eigenvalue. Crucial to the spike-adding process is a saddle-homoclinic bifurcation in the layer problem which occurs along the middle branch M m. Hypothesis 3. (Saddle homoclinic orbit) There exists y = ȳ h (k) (ȳ l, ȳ r ) such that (2.3) admits a homoclinic orbit γ h (t) = (v h (t), w h (t)) bi-asymptotic to the saddle equilibrium p h := p m (ȳ h ); further, the homoclinic orbit γ h (t) surrounds the equilibrium p u (ȳ h ) (see Figures 2 and 4). 6

We now consider the dynamics for nearby values of y. We first consider the linearization of the layer problem (2.3) about the equilibrium p h, which by Hypothesis 2 admits one positive and one negative eigenvalue, which we denote by λ ± h, respectively. We next linearize (2.3) about γ h, which results in the system The associated adjoint equation is given by Φ = D (v,w) F (v h (t), w h (t), ȳ h, k, )Φ. (2.7) Ψ = D (v,w) F (v h (t), w h (t), ȳ h, k, ) T Ψ, (2.8) which admits a unique bounded solution Ψ h (t) (up to multiplication by a constant). We assume the following regarding the bifurcation of periodic orbits from the homoclinic orbit γ h ; see Figure 4. Hypothesis 4. (Periodic manifold) The saddle quantity ν h := λ + h λ h associated with the equilibrium p h satisfies ν h < and the Melnikov integral M h = D y F (v h (t), w h (t), ȳ h,, ) Ψ h (t)dt, (2.9) is nonzero so that γ h breaks transversely as y is varied near y ȳ h. Therefore from the homoclinic orbit γ h bifurcates a family of attracting periodic orbits [14] for either y < ȳ h or y > ȳ h ; we assume the latter and denote this family by for some ȳ h < ȳ p < ȳ r. As a result we have the following. P = {γ p ( ; y) : y (ȳ h, ȳ p )} (2.1) (i) The periodic orbits {γ p ( ; y) : y (ȳ h, ȳ p )} have corresponding periods T p (y), y (ȳ h, ȳ p ), where T p (y) is a smooth function of y and T p (y) as y ȳ h. (ii) Each periodic orbit γ p ( ; y), y (ȳ h, ȳ p ) admits a single nontrivial Floquet multiplier e µp(y)tp(y) < 1, where µ p (y) > is a smooth function of y. We note that away from the endpoints y = ȳ h, ȳ p, the family P forms an invariant manifold, which is normally attracting; this manifold is shaped as a cylinder which surrounds the upper branch M u ; see Figure 2. The next hypothesis concerns the existence of heteroclinic orbits connecting the middle branch M m bottom branch M b as well as heteroclinic orbits between M m and P; see Figure 4. to the Hypothesis 5. (Behavior of W u (M m )) For each value of y (ȳ l, ȳ r ), the saddle equilibrium p m (y) has a one dimensional unstable manifold W u (p m (y)) which is composed of two orbits W m, W m +. (i) For each y (ȳ l, ȳ r ), W m is given by a heteroclinic orbit φ b (y) which limits onto the stable equilibrium p b (y) on the bottom branch M b. (ii) The behavior of W+ m varies: For y = ȳ h, W+ m is precisely the homoclinic orbit γ h. For y (ȳ l, ȳ h ), W+ m is given by a second heteroclinic orbit φ u (y) which limits onto the stable equilibrium p b (y) on the bottom branch M b, while for y (ȳ h, ȳ p ), W+ m is a heteroclinic orbit φ p (y) which limits onto the attracting periodic orbit γ p ( ; y). The behavior of W+ m for y ȳ p is not relevant. 7

2.2 Reduced problem Taking ε = in (2.2) results in the associated reduced problem = f 1 (v, w, y, k, ) = f 2 (v, w, y, k, ) y = g(v, w, y, k, ), (2.11) which is a differential-algebraic system in which the flow is restricted to the critical manifold M. Regarding the slow flow, we have the following. Hypothesis 6. (Slow flow). The function g (v, w, y) = g(v, w, y,, ) satisfies g M m <, g P <, g M b > g ( v r, w r, ȳ r ) =, g ( v l, w l, ȳ l ) < (2.12) Finally, regarding the fold points F l,r, we have the following. Hypothesis 7. The fold points F l, F r satisfy the following. (i) (Nondegenerate repelling fold point) The point F l is a normally repelling fold point, in the sense that D (v,w) F ( v l, w l, ȳ l, k, ) (2.13) has one positive eigenvalue for k [ k, k ]. The full system (2.1) therefore admits a two-dimensional local center manifold W c (F l ), on which we assume the point F l is a nondegenerate fold (or jump) point in the sense of [16, 2.1]. (ii) (Nondegenerate attracting canard point) The point F r is an attracting canard point, i.e. D (v,w) F ( v r, w r, ȳ r, k, ) (2.14) has one negative eigenvalue for k [ k, k ]. The full system (2.1) therefore admits a two-dimensional local center manifold W c (F r ), on which we assume the point F r is a nondegenerate canard point with unfolding parameter k in the sense of [16, 3.1]. This two-dimensional system therefore admits a singular Hopf bifurcation for k = ε =, which we assume is nondegenerate, in the sense of [17, 3.4]. Remark 2.1. The nondegeneracy condition for the singular Hopf bifurcation can be determined from the normal form of the reduced equations on W c (F r ); we refer to Theorem 3.2 below. 2.3 Statement of the main result We are now able to state our main result. Theorem 2.2. Consider system (2.1) satisfying Hypotheses 1 7. Then there exist ρ, η, ε > such that for each ε (, ε ), there exists a continuous one-parameter family θ (k sa (θ, ε), B(θ, ε)), θ (, Θ(ε)) (2.15) of periodic orbits B(θ, ε) originating at a Hopf bifurcation near the fold point F r, where k sa, B are C 1 in (θ, ε). For θ (N, N + 1), the periodic orbit B(θ, ε) is an N-spike bursting solution, and the quantity Θ(ε) satisfies lim ε εθ(ε) = θ >. Further, for θ (ρ, Θ(ε)), the parameter k sa (θ, ε) satisfies for a C r function k mc ( ε) = O(ε). k sa (θ, ε) k mc ( ε) = O(e η/ε ) (2.16) 8

Theorem 2.2 guarantees the existence of a single connected branch of bursting solutions which encompasses the transition from canard explosion (i.e. small amplitude Hopf cycles local to the fold point F r ) to large amplitude bursting oscillations with an O(1/ε) number of spikes. The number of spikes is determined by the number of excursions around the upper branch M u. Each spike is added sequentially throughout the spike-adding process as the single bifurcation parameter k varies on an interval of size O(e η/ε ). The remainder of this paper is concerned with the proof of Theorem 2.2. 3 Construction of spike-adding sequence In this section, we present the proof of Theorem 2.2 by constructing the entire spike-adding sequence of bursting solutions for small ε >. We begin in 3.1 by collecting facts regarding the perturbation of normally hyperbolic portions of the critical manifold M and their (un)stable manifolds, which follow from standard results of geometric singular perturbation theory [1]. In 3.2 3.3, we analyze the fold point F r and the canard explosion which occurs in a local two-dimensional center manifold W c (F r ) containing the fold. We then proceed by constructing bursting solutions which complete large excursions in phase space, that is, periodic orbits which do not remain in a small neighborhood of the fold point F r. We describe in 3.4 the general strategy for constructing such solutions, and in 3.5 3.7 we construct the transition from -spike solutions to 1-spike solutions. To understand how additional spikes are generated, a detailed understanding of the flow near the saddle-homoclinic bifurcation is needed, which we present in 3.8, and the proof of the key technical result is given in 3.9. In 3.1-3.11, we construct N-spike solutions for any N and show how the branches of N-spike solutions and (N + 1)-spike solutions are connected. Finally, the proof of Theorem 2.2 is briefly concluded in 3.12. 3.1 Persistence of invariant manifolds We collect several preliminary results which follow from standard geometric singular perturbation theory and center manifold theory. For sufficiently small ε, k, we have the following: 1. Away from the fold points F l,r, the three branches M b,m,u are normally hyperbolic and persist for (k, ε) ( k, k ) (, ε ) as locally invariant slow manifolds M b,m,u ε. The middle branch M m has two-dimensional stable and unstable manifolds W s (M m ), W u (M m ) which persist as locally invariant manifolds W s (M m ε ), W u (M m ε ) for (k, ε) ( k, k ) (, ε ). Similarly the bottom branch M b has a three-dimensional stable manifold W s (M b ) which persists as a locally invariant manifold W s (M b ε) for (k, ε) ( k, k ) (, ε ). 2. Near the fold point F r there is a local two-dimensional attracting C r -smooth center manifold W c (F r ) which persists for (k, ε) ( k, k ) (, ε ). The slow manifolds M b ε and M m ε extend into a neighborhood of F r, where they shadow corresponding basepoint solution orbits M b,r ε and M m,r ε which lie on W c (F r ). 3. Near the fold point F l there is a local two-dimensional repelling C r -smooth center manifold W c (F l ) which persists for (k, ε) ( k, k ) (, ε ). 4. Away from the saddle homoclinic bifurcation at y = ȳ h, the periodic manifold P persists as a twodimensional normally attracting locally invariant manifold P ε for (k, ε) ( k, k ) (, ε ). 9

z r M m r y r x r M b Figure 5: Pictured is a schematic of the singular flow in a neighborhood of the right fold point F r. The normally attracting center manifold W c (F r ) corresponds to the plane {x r = }. 3.2 Local coordinates near F r and maximal canards By Hypothesis 7, in a neighborhood of the fold F r, after a change of coordinates we obtain the system ẋ r = x r ( c r (k) + O(x r, y r, z r, ε)) ż r = y r h 1 (y r, z r, k, ε) + zrh 2 2 (y r, z r, k, ε) + εh 3 (y r, z r, k, ε) ẏ r = ε ( z r h 4 (y r, z r, k, ε) + kh 5 (y r, z r, k, ε) + y r h 6 (y r, z r, k, ε)) k = (3.1) ε =, where c r (k) >, and the functions h j, j = 1,..., 6 are C r and satisfy h 3 (y r, z r, k, ε) = O(y r, z r, k, ε) h j (y r, z r, k, ε) = 1 + O(y r, z r, k, ε), j = 1, 2, 4, 5. (3.2) At the linear level, the slow variable y r in these local coordinates corresponds to a rescaling of the original slow variable (y ȳ r ). See Figure 5 for a schematic of the singular ε = flow near F r. The manifold W c (F r ) is given by x r =, where the strong stable fibers have been straightened; we recall that by construction W c (F r ) contains the one-dimensional (shadowed) slow manifolds M b,r ε and M m,r ε. We note that the (z r, y r ) coordinates are in the canonical form for a canard point (compare [16]). Canard points are characterized by canard trajectories which follow a strongly attracting manifold (in this case M b,r ε ), pass near the equilibrium and continue along a strongly repelling manifold (in this case M m,r ε ) for some time. To understand the flow near this point, we use blowup methods as in [16]. Restricting to the center manifold x r =, the blow up transformation is given by y r = r 2 ȳ, z r = r z, k = r k, ε = r 2 ε, (3.3) defined on the manifold B = S 2 [, r ] [ k, k ] for sufficiently small r, k with (ȳ, z, ε) S 2. There is one relevant coordinate chart which will be needed for the matching analysis; in the literature, this is frequently referred to as the family rescaling chart, which corresponds to an ε-rescaling of the variables and parameters. 1

Keeping the same notation as in [16] and [17], the family rescaling chart K 2 uses the coordinates y r = r 2 2y 2, z r = r 2 z 2, k = r 2 k 2, ε = r 2 2. (3.4) Using these blow-up charts, the authors of [16] studied the behavior of the manifolds M b,r ε and M m,r ε and determined conditions under which these manifolds coincide along a canard trajectory. We place a section Σ r = {z r =, x r δ x, y r < ρ} for small fixed δ x, ρ which will serve as a Poincaré section for constructing the periodic orbits. In the chart K 2, the section Σ r is given by Σ r = { z 2 =, x r δ x, r2y 2 2 < ρ }. It was shown in [16] that for all sufficiently small r 2, k 2, the manifolds M b,r ε and M m,r ε reach Σ r at y 2 = y2(k b 2, r 2 ) and y 2 = y2 m (k 2, r 2 ), respectively. We have the following result which describes the distance between M b,r ε Proposition 3.1. [16, Proposition 3.5] The distance between the slow manifolds M b,r ε by and M m,r ε in Σ r. and M m,r ε in Σ r is given y b 2 y m 2 = D (k 2, r 2 ) = d k2 k 2 + d r2 r 2 + O(r 2 2 + k 2 2), (3.5) where the coefficients d k2, d r2 are constants, bounded away from zero independently of k 2, r 2. Hence we can solve for the existence of a maximal canard trajectory within W c (F r ), corresponding to a zero of the distance function D (k 2, r 2 ), which occurs when where µ = d r 2 d k2. This proposition describes the splitting of the manifolds M b,r ε k 2 = k mc 2 = µr 2 + O(r 2 2), (3.6) and M m,r ε as a function of k 2, r 2, and in particular ensures that this splitting occurs in a transverse fashion as the parameter k = k 2 r 2 is varied near k k mc ( ε), where the function denotes the location of the maximal canard solution. k mc ( ε) = k2 mc ε = µε + O(ε 3/2 ) (3.7) Further, it was shown in [16] that the system (3.1) undergoes a singular Hopf bifurcation, which also occurs near the location of the maximal canard. The sub/super criticality of the Hopf bifurcation is determined via the quantity where A H = a 1 + 3a 2 2a 4 2a 5 (3.8) We have the following. a 1 = h 1 z r (,,, ), a 2 = h 2 z r (,,, ), a 3 = h 3 z r (,,, ) a 4 = h 4 z r (,,, ), a 5 = h 6 (,,, ) Theorem 3.2. [16, Theorem 3.1] There exist ε, k > such that for (k, ε) ( k, k ) (, ε ) the system (3.1) admits a single equilibrium. The equilibrium is stable for k < k H ( ε), where k H ( ε) = a 3 + a 6 ε + O(ε 3/2 ) (3.9) 2 and loses stability through a Hopf bifurcation as k passes through k H ( ε). The Hopf bifurcation is nondegenerate if the quantity A H defined in (3.8) is nonzero. It is supercritical if A H < and subcritical if A H >. 11

3.3 Local canard explosion Within the center manifold W c (F r ), we refer to [17] for the bifurcation of local canard orbits from the singular Hopf bifurcation at the equilibrium at the origin. Upon varying the parameter k k mc ( ε), these orbits grow to small, but O(1), size within W c (F r ). We quote the following from [17]. Theorem 3.3. [17, Theorems 4.1, 4.2, Proposition 4.3] Assume that A H and that ρ > is sufficiently small. Then there exists ε > such that for ε (, ε ), the system (3.1) undergoes a Hopf bifurcation at k = k H ( ε), from which bifurcates a continuous family of periodic orbits s (k sc (s, ε), Γ sc (s, ε)), s (, ρ] (3.1) where k sc (s, ε) is C r in (s, ε) with k sc (s, ε) k H ( ε) as s, and k sc (ρ, ε) k mc ( ε) = O(e q/ε ). (3.11) For each s (, ρ], the orbit Γ sc (s, ε)) W c (F r ) passes through the point (x r, y r, z r ) = (, s, ). This theorem guarantees the existence of a singular Hopf bifurcation and the bifurcation of a continuous family of periodic orbits within the center manifold W c (F r ) which grow to O(1) amplitude for all sufficiently small ε (determined via the small parameter ρ which, in general, satisfies < ε ρ). 3.4 General strategy of constructing O(1)-amplitude periodic orbits In this section, we describe a general strategy for constructing a periodic orbit which completes an O(1) excursion in phase space before returning to a neighborhood of the fold F r. The idea is to determine an appropriate onedimensional curve I of initial conditions which can be evolved both forward and backward in time, spanning a two-dimensional manifold I of candidate solution orbits. This manifold is tracked forward and backward until it first intersects the section Σ r ; this intersection is therefore given by two curves I +, I, resulting from the forward and backward evolution, respectively. The geometric setup for the construction strategy is shown in Figure 6. We then consider the Poincaré map Π r : Σ r Σ r, and search for solutions with initial conditions on I which return to Σ r, meeting the curve I +. The desired periodic orbit is then given by a fixed point of this map, corresponding to an intersection of the curves I +, I which occurs along a single solution orbit, lying entirely within the manifold I. We now describe this procedure in more detail, and determine conditions on the initial curve I which guarantee that this strategy results in a unique solution. We assume the following. (i) The curve I lies outside a small -neighborhood of F r. (ii) Under the forward evolution of (2.1), the manifold I is contained in W s (M b ε). (iii) Under the backward evolution of (2.1), the manifold I transversely intersects W u (M m ε ). Under these conditions, we can construct a periodic orbits as follows. When evolving forwards, since the manifold I is contained in W s (M b ε), we can track I as it is exponentially contracted to M b ε until reaching a small neighborhood of F r, whereby I intersects Σ r in a curve I + which is O(e η/ε )-close to M b ε Σ r. On the other hand, since I transversely intersects W u (M m ε ) under the backward evolution of (2.1), by the exchange lemma [27], in backwards time I intersects Σ r in a curve I which is aligned C 1 -O(e η/ε )-close to the strong stable fibers of the manifold W c (F r ). In particular, I intersects W c (F r ) Σ r at a base-point which is O(e η/ε )-close to M m ε Σ r, and I is thus aligned C 1 -O(e η/ε )-close to the strong stable fiber of that base-point. 12

v M m " W u (M m " ) I I + I r M b " w y Figure 6: Shown is a schematic of the strategy for constructing large amplitude bursting oscillations outlined in 3.4. A one-dimensional manifold I of candidate solutions is evolved forwards and backwards under the flow of (2.1) until intersecting the section Σ r. This intersection consists of two curves: I ± (corresponding to the forward and backward evolution, respectively); matching conditions are then determined within the section Σ r which guarantee the existence of a periodic orbit. The Poincaré map Π r : Σ r Σ r by construction satisfies Π r (I ) I +. We use the blow-up coordinates (3.4) to set up fixed point matching conditions in the section Σ r. Within Σ r, it is most natural to parametrize solutions using the coordinates (x r, y 2 ). 3.5 (Lower) -spike orbits In this section, we construct -spike orbits, which encompass the transition from the local canard explosion occurring within the center manifold W c (F r ) to large canard orbits which complete a global excursion. This excursion is characterized by a long canard trajectory, which consists of first following M b ε, then M m ε, and then finally returning to M b ε via one of the heteroclinic orbits φ b (y). We refer to these orbits as lower -spike orbits as they traverse one of the heteroclinic orbits φ b (y), as opposed to one of the upper heteroclinics φ u (y). These orbits are most naturally parameterized by which heteroclinic connection φ b (y) is followed, or equivalently, the minimum y-value achieved along the orbit. Hence for s [ȳ l +, ȳ r ], we search for a -spike periodic orbit which achieves a minimum y-value of y = s, and is obtained as a perturbation from the singular orbit Γ (s) := M b (s, ȳ r ) M m (s, ȳ r ) φ b (s) (3.12) Following the strategy of the previous section, we choose an appropriate one-dimensional curve of candidate initial conditions. For this, we denote by w b the w-coordinate at which the orbit φ b (s) intersects the set {v = v r }. We then set I b (s) := {( v r, w, s) : w w b < δ} for some sufficiently small δ >, and we assume that φ b (s) crosses I b (s) in a transverse fashion within the plane {y = s}. We now determine the behavior of I b (s) under the forwards and backwards evolution of (2.1). Since φ b (s) lies in the intersection W s (M b ) W u (M m ) for ε =, we see that for all sufficiently small ε >, the forward evolution of I b (s) must also lie in W s (M b ε). On the other hand, the backwards evolution of I b (s) 13

v v M u M u M m M b b P F r M m M b u P F r w s y w s y (a) (b) Figure 7: Shown are singular periodic orbits in the case of (a) lower -spike orbits, and (b) upper 1-spike orbits. The lower/upper descriptor refers to which of the heteroclinic orbits φ b, φ u is followed. A lower -spike orbit follows M b, then M m, then the heteroclinic orbit φ b. An upper 1-spike orbit follows M b, then M m, then the heteroclinic orbit φ u ; the fast increase then decrease in the v-variable along the orbit φ u constitutes the spike. transversely intersects the manifold W u (M m ε ). By the exchange lemma, the backwards evolution of I b (s) traces out a two-dimensional manifold I b (s) which intersects Σ r in a curve I (s) which is aligned C 1 -O(e η/ε )-close to the stable fiber of a base-point on W c (F r ) which itself is O(e η/ε )-close to M m,r ε Σ r. We sum this up in the following Lemma 3.4. Within Σ r, the curve I (s) is given as a graph where for ν = x r, s, k. I (s) = {(x r, y 2 ) : y 2 = y m 2 + I (x r, s, k, ε), x r δ x } (3.13) I (, s, k, ε) = O(e η/ε ), ν I (, s, k, ε) = O(e η/ε ) (3.14) We now consider the forward evolution of I (s), which is contained in the two-dimensional manifold I b (s). By construction and by the above discussion, we have that I b (s) W s (M b ε), and therefore I b (s) will be C 1 - exponentially contracted to M b ε; in particular, the forward evolution of I (s) thus meets the section Σ r in a curve I + (s) which is C 1 -O(e η/ε )-close to M b,r ε Σ r. We have the following Lemma 3.5. Consider the Poincaré map Π r : Σ r Σ r. We have that Π r (I (s)) = I + (s); parameterizing points on I (s) by their initial x r coordinate given by x r = x for x δ x, we have that the curve I + (s) is given by where for ν = x, s, k. I + (s) = {( x r y 2 ) = ( x + (x, s, k, ε) y b 2 + I + (x, s, k, ε) ), x δ x } I + (x, s, k, ε) = O(e η/ε ), ν I + (x, s, k, ε) = O(e η/ε ) x + (x, s, k, ε) = O(e η/ε ), ν x + (x, s, k, ε) = O(e η/ε ) (3.15) (3.16) 14

r y I M b " I + O(e /" ) M m " x Figure 8: Shown are the matching conditions within the Poincaré section Σ r. Note that under the Poincaré map Π r : Σ r Σ r, we have that Π(I ) I +. A periodic orbit can be therefore be found when the curves I ± intersect along a single solution orbit. It remains to solve for a fixed point of Π r which lies on the intersection of the curves I ± (s); this corresponds to a periodic orbit which is a perturbation of the singular orbit Γ (s). An intersection of I ± (s) occurs along a single solution orbit if x = x + (x, s, k, ε) y m 2 + I (x, s, k, ε) = y b 2 + I + (x, s, k, ε) (3.17) for some value of x δ x. Using the estimates (3.16), the first equation can be solved for x = x (s, k, ε) = O(e η/ε ), k x (s, k, ε) = O(e η/ε ). (3.18) Plugging this into the second equation and rearranging results in the equation which by Proposition 3.1 can be rewritten as y b 2 y m 2 + I + (x (s, k, ε), s, k, ε) I (x (s, k, ε), s, k, ε) =, (3.19) D (k 2, r 2 ) + I + (x (s, k, ε), s, k, ε) I (x (s, k, ε), s, k, ε) =. (3.2) Using the estimates (3.14), (3.16), (3.18), and the implicit function theorem, this equation can be solved for a unique solution when k = k sa (s, ε) = k mc ( ε) + O(e η/ε ) (3.21) 3.6 Upper 1-spike orbits In this section, we construct 1-spike orbits which complete an excursion around the upper branch M u ε, corresponding to a single spike. We first consider the simpler case of orbits which stay away from the upper left fold F l and the saddle-homoclinic bifurcation occurring along M m ε, as these orbits can be constructed in a very similar manner to the -spike orbits from 3.5. These solutions are are again characterized by a long canard trajectory, which consists of first following M b ε, then M m ε, and then finally returning to M b ε; however, in contrast to the solutions constructed in 3.5, the fast jump down to M b ε instead follows one of the heteroclinic orbits φ u (y). Similarly, these orbits are most naturally parameterized by which heteroclinic connection φ u (y) is followed, or equivalently, the minimum y-value achieved 15

along the orbit. Hence for each s [ȳ l +, ȳ h ] we search for a 1-spike periodic orbit which achieves a minimum y-value of y = s, and is obtained as a perturbation from the singular orbit M b (s, ȳ r ) M m (s, ȳ r ) φ u (s) (3.22) Following the strategy of 3.5, we choose an appropriate one-dimensional curve of candidate initial conditions. For this, we denote by w u the w-coordinate at which the orbit φ u (s) intersects the set {v = v l }. We then set I u (s) := {( v l, w, s) : w w u < δ} for some sufficiently small δ >, and we assume that φ u (s) intersects I u (s) in a transverse fashion within the plane {y = s}. Since φ u (s) lies in the intersection W s (M b ) W u (M m ) for ε =, the remainder of the analysis follows identically to that in 3.5, with the periodic orbit occurring for and we omit the details. k = k sa,upper 1 (s, ε) = k mc ( ε) + O(e η/ε ), (3.23) 3.7 Overlap of -spike orbits and upper 1-spike orbits: analysis of upper left fold point F l We consider the upper left fold F l. We note that the geometry near the fold is similar to that considered in [1, 4], and hence we draw on the local analysis as presented in [1]. We first move to a local coordinate system in a neighborhood of F l, in which the equations take the form ẋ l = x l (c l (k) + O(x l, y l, z l, ε)), ż l = y l + z 2 l + h l (y l, z l, k, ε) ẏ l = εg l (y l, z l, k, ε) (3.24) where c l (k) >, and h l, g l are C r -functions satisfying h l (y l, z l, k, ε) = O(ε, y l z l, y 2 l, z 3 l ), g l (y l, z l, k, ε) = 1 + O(y l, z l, ε), uniformly in k ( k, k ). The geometry of (3.24) for ε = is depicted in Figure 9. In the transformed system (3.24), the (z l, y l )-dynamics on a local two-dimensional invariant center manifold W c (F l ) is decoupled from the dynamics in the hyperbolic x l -direction along the straightened strong unstable fibers. At the linear level, the slow variable y l in these local coordinates corresponds to a rescaling of the original slow variable (y ȳ l ). We consider the flow of (3.24) on the invariant manifold x l =. We append an equation for ε, arriving at the system ż l = y l + z 2 l + h l (y l, z l, k, ε) ẏ l = εg l (y l, z l, k, ε) ε =. (3.25) For ε =, this system possesses a critical manifold given by {(y l, z l ) : y l + zl 2 + h l (y l, z l, k, ) = }, which in a sufficiently small neighborhood of the origin is shaped as a parabola opening to the right; see Figure 9. The branch of this parabola for z l < is attracting and corresponds to the manifold M m. We define M m,+ to be the singular trajectory obtained by appending the fast trajectory given by the line segment {(y l, z l ) : y l =, z l δ z } to the attracting branch M m of the critical manifold. We have the following Proposition 3.6. ([1, Proposition 4.1]) For all sufficiently small ε >, we have the following. 16

z` M u y` x` M m Figure 9: Shown is a schematic of the singular ε = flow near the upper left fold point F l. The locally invariant attracting center manifold W c (F l ) is given by the set {x l = }. (i) Within the center manifold {x l = }, the singular trajectory M m,+ perturbs to a solution M m,+ ε, which is C -O(ε 2/3 )-close and C 1 -O(ε 1/3 )-close to M m,+, uniformly in k < k. This solution can be represented as a graph M m,+ ε = {(, y l, z l ) : y l = s m,+ (z l, k, ε), z l δ z }. (3.26) (ii) The manifold W u (M m,+ ) composed of the strong unstable fibers of the singular trajectory M m,+ also perturbs to a two-dimensional locally invariant manifold W u (M m,+ ε ) which is C -O(ε 2/3 )-close and C 1 - O(ε 1/3 )-close to W u (M m,+ ), uniformly in k < k. The results of Proposition 3.6 are depicted in Figure 1. We proceed by constructing solutions which pass near the fold. These solutions form a bridge between orbits which depart M m ε along the heteroclinics φ b (y) and those which depart along the heteroclinics φ u (y), which were constructed in 3.5 and 3.6, respectively. The geometric intuition is that the fold acts as a means of continuously transitioning from one side of M m ε to the other. The challenge lies in parameterizing these orbits, as the exact orbit φ b (y) or φ u (y) which is followed when leaving a neighborhood of F l is not naturally determined. We choose < δ δ z and define the section Σ l in (see Figure 1) by We have the following. Σ l in = {(x l, y l, z l ) : y l = s m,+ ( δ z, k, ), x l δ x, z l + δ z δ}. (3.27) Lemma 3.7. For all sufficiently small ε >, we have that Σ l in W s (M b ε). Proof. We first define a collection of potential exit sections for solutions with initial conditions in Σ l in. The first is given by Σ l out,1 = {(x l, y l, δ z ) : x l δ x, y l δ}. (3.28) 17

{z } z` M m,+ " M u y` y x` M m z W u (M m,+ " ) x {z } ìn Figure 1: Depicted are the results of Proposition 3.6. For sufficiently small ε >, the singular trajectory M m,+ perturbs to a solution M m,+ ε within the center manifold W c (F l ) = {x l = }. Furthermore, the two-dimensional manifold W u (M m,+ ) composed of the strong unstable fibers of M m,+ invariant manifold W u (M m,+ ε center manifold W c (F l ). also perturbs to a two-dimensional locally ) depicted by the purple surface. Also shown is the section Σ l in, transverse to the 18

For the other sections, we first define U l to be a planar δ-neighborhood of M m,+ within the center manifold {x l = }. This neighborhood U l is bounded by four curves, given by Σ l in {x l = }, Σ l out,1 {x l = }, as well as two other curves U l upper and U l lower, chosen to lie an O(δ) distance on either side of M m,+, so that the union of these four curves bounds a well-defined planar region U l within {x l = } containing M m,+, with O(δ) thickness. We now define four additional exit sections Σ l out,2 = {(δ x, y l, z l ) : (y l, z l ) U l } Σ l out,3 = {( δ x, y l, z l ) : (y l, z l ) U l } Σ l upper = {(x l, y l, z l ) : x l δ x, (y l, z l ) Uupper} l Σ l lower = {(x l, y l, z l ) : x l δ x, (y l, z l ) Ulower}. l (3.29) Previous blow-up analyses [1, 16] of non-degenerate fold points have studied the behavior of base-point solutions with initial conditions in Σ l in {x l = } for < ε 1. In particular, these analyses show that initially such solutions are quickly contracted O(e η/ε )-close to M m,+ ε and remain O(e η/ε )-close to M m,+ ε until reaching the set Σ l out,1 {x l = }. Hence, when considering the full dynamics of (3.25), i.e. with the x l -dynamics included, since solutions on the strong unstable fibers shadow their respective base-point trajectories, any solution with initial condition in Σ l in must pass through one of the three sections Σ l out,j, j = 1, 2, 3. In particular, such solutions do not pass through Σ l upper or Σ l lower. It remains to show that Σ l out,j, j = 1, 2, 3 are contained in W s (M b ε). To see this, we first consider solutions within Σ l out,2 {y l δ} and Σ l out,3 {y l δ}. Provided δ δ x, the fact that such solutions are contained in W s (M b ) is clear due to their proximity with the heteroclinic orbits φ u (y), φ b (y) which lie in W s (M b ). For sufficiently small ε >, by standard geometric singular perturbation theory, these solutions are contained in W s (M b ε). For the remaining solutions, i.e. those within { y l δ}, we first note that due to Hypothesis 7 as well as Hypothesis 5 regarding the layer problem (2.3) for ε =, any solution within the plane {y l = } lying a small fixed distance δ x from the fold F l must lie in W s (M b ). For sufficiently small δ δ x, by the smooth dependence of the layer problem on y l, this also holds for solutions lying distance δ x from F l, which are contained in the region { y l δ}. Again, the fact that these solutions are contained in W s (M b ε) for small ε > follows from standard geometric singular perturbation theory. Hence by appropriately choosing δ δ x, δ z, we obtain the result. We note that for ε =, we have that W u (M m ) Σ l in = {(x l, y l, δ z ) : y l = s m,+ (δ z, k, ), x l δ x }. (3.3) Therefore, for each x δ x, we can define the interval I l ( x) = Σ l in {x l = x}, (3.31) which clearly intersects W u (M m ) transversely within Σ l in. This transversality persists for sufficiently small ε >. Combining this with Lemma 3.7, we see that I l ( x) satisfies the conditions outlined in the strategy from 3.4, and the construction of periodic orbits which pass through I l ( x) follows as in 3.5. 3.8 The flow near the saddle-homoclinic point Before proceeding to construct N-spike solutions for N > 1, it is necessary to understand the passage of solutions the near the saddle homoclinic point p h. This analysis is also critical in determining how the different branches of bursting solutions are connected. The main result of this section is Proposition 3.8, which concerns the behavior of solutions which spend long times hear the saddle homoclinic point. The proof of Proposition 3.8 is given in 3.9. 19

P " W s (M m " ) M m " v y w W u (M m " ) Figure 11: Shown is the flow near the saddle homoclinic bifurcation for sufficiently small ε >. The stable and unstable manifolds W s,u (M m ) of the critical manifold M m, which intersect transversely for ε =, perturb to two-dimensional locally invariant manifolds W s,u (M m ε ) which again intersect transversely near y ȳ h for < ε 1. Away from the saddle homoclinic bifurcation, the periodic manifold P persists as a locally invariant manifold P ε. We continue by considering the flow in a neighborhood of the saddle homoclinic point p h. The existence of a saddle homoclinic orbit γ h at p h when ε = implies that the manifolds W u (M m ) and W s (M m ) intersect transversely along γ h in the plane y = ȳ h. This transverse intersection therefore persists for the manifolds W u (M m ε ) and W s (M m ε ) for ε > sufficiently small; see Figure 11. In a neighborhood of M m ε, there exists a smooth change of coordinates such that the equations can be written in the Fenichel normal form A = F 1 (A, B, Y, k, ε)a Ḃ = F 2 (A, B, Y, k, ε)b (3.32) Ẏ = ε(g 1 (Y, k, ε) + G 2 (A, B, Y, k, ε)), where F 1 (A, B, Y, k, ε) = α(k) + O(A, B, Y, ε) F 2 (A, B, Y, k, ε) = β(k) + O(A, B, Y, ε) G 1 (Y, k, ε) = γ(k) + O(Y, ε) (3.33) G 2 (A, B, Y, k, ε) = O(AB), where α(k), β(k), γ(k) > uniformly in k < k, and α(k) > β(k) due to Hypothesis 4. In the following we will suppress the dependence on k in the notation. In these coordinates, the set A = corresponds to W u (M m ε ), the set B = coincides with W s (M m ε ), and the slow manifold M m ε is given by A = B = ; see Figure 12. We fix the two-dimensional sections Σ h A = {A =, B, Y δ Y } Σ h B = {B =, A, Y δ Y } (3.34) for small > to be chosen later; see Figure 12. 2

M m " W u (M m " ) W s (M m " ) B Y A I loc (I ) gl loc (I ) h B h A W u (M m " ) Figure 12: Shown is the local geometry associated with the flow of the Fenichel normal form (3.32) near the saddle homoclinic bifurcation for sufficiently small ε >. The manifolds W u (M m ε ) and W s (M m ε ) coincide with the sets A = and B =, respectively, and the slow manifold M m ε is given by A = B =. The sections Σ h A, Σ h B defined in (3.34) are placed at A = and B =, respectively for small fixed >. Due to Hypothesis 4 manifold W u (M m ε ) transversely intersects W s (M m ε ) in the section Σ h A for all sufficiently small ε >. Also depicted are the results of Lemma 3.12, concerning the return map Π gl Π loc : Σ h B Σ h B induced by the backwards flow of (3.32), applied to a curve I which transversely intersects W u (M m ε ) in Σ h B. By the above discussion, we can track W u (M m ε ) along γ h and deduce that this manifold transversely intersects W s (M m ε ) in the section Σ h A for all sufficiently small ε > (see Figure 12). Thus the intersection of W u (M m ε ) with the section Σ h A is given by a curve which can be represented as a graph over the B-coordinate, that is W u (M m ε ) Σ h A = {(, B, Y h (B, k, ε)) : B δ}, (3.35) for some < δ, where we can assume without loss of generality (by shifting coordinates) that Y h (, k, ε)) =, B Y h (, k, ε)) = K(k, ε, ), (3.36) where K 1 < K(k, ε, ) < K 2 uniformly in k < k and < ε 1 for some K j = K j ( ) > for j = 1, 2. In the following, it will also be useful to invert this relation, i.e. represent W u (M m ε ) as a graph B = B h (Y, k, ε) for Y δ Y, where B h (, k, ε)) =, Y B h (, k, ε)) = 1/K(k, ε, ). (3.37) The primary result of this section is the following proposition, the proof of which is given in 3.9. Proposition 3.8. Consider the backwards flow of (3.32). For each sufficiently small >, there exists C, δ Y, k, ε > such the the following holds. For each (k, ε) ( k, k ) (, ε ), consider a two-dimensional manifold I which transversely intersects W u (M m ε ) in the section Σ h B at some Y ( δ Y, Cε log ε ). Then there exists N(ε) = O(1/ε) such that under the backwards flow of (3.32), the manifold I returns to the section Σ h B a total of N times, each time transversely intersecting the manifold W u (M m ε ). Furthermore the transversality is uniform in ε > sufficiently small. Remark 3.9. The uniformity of the transversality with respect to ε means that this intersection does not approach tangency as ε. This is important as the manifold I is tracked over N = O(1/ε) excursions. 21