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3.2 Introduction 3.2.1 Fast-Slow Systems Fast-slow systems of ordinary differential equations (ODEs) have the general form: ǫ ˙x = ǫ dx = f (x, y,ǫ) dτ (3.1) ˙y = dy = g(x, y,ǫ) dτ where x∈R m , y∈R n and 0≤ǫ≪ 1 represents the ratio of time scales. The functions f and g are assumed to be sufficiently smooth. In the singular limitǫ→ 0 the vector field (3.1) becomes a differential-algebraic equation. The algebraic constraint f= 0 defines the critical manifold C0={(x, y)∈R m ×R n : f (x, y, 0)=0}. Where Dx f (p) is nonsingular, the implicit function theorem implies that there exists a map h(x)=y parameterizing C0 as a graph. This yields the implicitly defined vector field ˙y=g(h(y), y, 0) on C0 called the slow flow. We can change (3.1) to the fast time scale t=τ/ǫ and letǫ→ 0 to obtain the second possible singular limit system x ′ = dx dt y ′ = dy dt = f (x, y, 0) (3.2) We call the vector field (3.2) parametrized by the slow variables y the fast subsystem or the layer equations. The central idea of singular perturbation analysis is to use information about the fast subsystem and the slow flow to under- stand the full system (3.1). One of the main tools is Fenichel’s Theorem (see 61 = 0
[39, 40, 41, 42]). It states that for everyǫ sufficiently small and C0 normally hy- perbolic there exists a family of invariant manifolds Cǫ for the flow (3.1). The manifolds are at a distance O(ǫ) from C0 and the flows on them converge to the slow flow on C0 asǫ→ 0. Points p∈C0 where Dx f (p) is singular are referred to as fold points 1 . Beyond Fenichel’s Theorem many other techniques have been developed. More detailed introductions and results can be found in [1, 71, 49] from a geometric viewpoint. Asymptotic methods are developed in [93, 47] whereas ideas from nonstandard analysis are introduced in [28]. While the theory is well developed for two-dimensional fast-slow systems, higher-dimensional fast-slow systems are an active area of current research. In the following we shall fo- cus on the FitzHugh-Nagumo equation viewed as a three-dimensional fast-slow system. 3.2.2 The FitzHugh-Nagumo Equation The FitzHugh-Nagumo equation is a simplification of the Hodgin-Huxley model for an electric potential of a nerve axon [65]. The first version was developed by FitzHugh [43] and is a two-dimensional system of ODEs: ǫ ˙u = v− u3 + u+ p 3 (3.3) ˙v = − 1 s (v+γu−a) 1 The projection of C0 onto the x coordinates may have more degenerate singularities than fold singularities at some of these points. 62
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MULTIPLE TIME SCALE DYNAMICS WITH T
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MULTIPLE TIME SCALE DYNAMICS WITH T
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To my family. iv
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ing mechanical systems.”, [E.T. P
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TABLE OF CONTENTS Biographical Sket
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LIST OF TABLES 5.1 Euclidean distan
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3.1 Bifurcation diagram of (3.6). H
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4.6 Boundary conditions are blue an
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6.3 For all computations of the ful
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where (x, y)∈R m × R n are varia
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the general definition of normal hy
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fast-slow system and yields: ˙x=(D
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whereΛ,Γare matrix-valued functio
Page 27 and 28: forǫ> 0. Therefore they have no im
Page 29 and 30: The Hopf bifurcation is supercritic
Page 31 and 32: where the algorithm can be used to
Page 33 and 34: slow flow. In particular the manifo
Page 35 and 36: f ast whereφ t is the flow of the
Page 37 and 38: whereµ∈R p are parameters. Usual
Page 39 and 40: We shall not prove this result but
Page 41 and 42: d as given in equation (2.5). Let p
Page 43 and 44: variables x1= u, x2= v and y=w we g
Page 45 and 46: The geometry of the system for one
Page 47 and 48: (0, 0, 0). 2.3 The Exchange Lemma I
Page 49 and 50: withδ>0 sufficiently small. The sa
Page 51 and 52: Theorem 2.3.2. Let ¯q ∈ M∩{|a|
Page 53 and 54: exit point of a trajectory starting
Page 55 and 56: Hence we have k=1=l and n=2 in view
Page 57 and 58: The variational equations describin
Page 59 and 60: (iii) Denote byη1i the i-th row of
Page 61 and 62: Then let H(Z, X, t) := X ′ − BX
Page 63 and 64: Proof. First, we work near q, then|
Page 65 and 66: This result can be written more com
Page 67 and 68: Proof. (of Theorem 2.4.2) The argum
Page 69 and 70: ˆZ is still exponentially small. P
Page 71 and 72: in the FitzHugh-Nagumo equation ǫ
Page 73 and 74: center-unstable manifold W cu (0, 0
Page 75 and 76: where the second “fast jump” oc
Page 77: CHAPTER 3 PAPER I: “HOMOCLINIC OR
Page 81 and 82: papers. Geometric singular perturba
Page 83 and 84: s 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0
Page 85 and 86: One can view this as a projection o
Page 87 and 88: for x1. We find that there are thre
Page 89 and 90: We compute the functionsγl andγr
Page 91 and 92: 3.3.3 Two Slow Variables, One Fast
Page 93 and 94: Also recall that the y-nullcline pa
Page 95 and 96: is negative for all values h∈(0,
Page 97 and 98: 3.4 The Full System 3.4.1 Hopf Bifu
Page 99 and 100: check whether this observation of a
Page 101 and 102: indistinguishable numerically from
Page 103 and 104: we should view this curve as an app
Page 105 and 106: s 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 sin
Page 107 and 108: waves” (see e.g. [63, 15, 69]). 3
Page 109 and 110: the singular limits but it cannot b
Page 111 and 112: Then (3.24) transforms to a three t
Page 113 and 114: with x∈R m the vector of fast var
Page 115 and 116: ons that was used by David Terman i
Page 117 and 118: 0.15 0.1 0.05 0 −0.05 0.15 0.1 0.
Page 119 and 120: from the input data. (If b−a is a
Page 121 and 122: This explicit solution provides a b
Page 123 and 124: 4.4.2 Traveling Waves of the FitzHu
Page 125 and 126: computing the homoclinic orbit, eve
Page 127 and 128: y 0.15 0.1 0.05 0 −0.05 ε=0.001,
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eturns directly to q. Figure 4.5(b)
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are complicated [107]: we expect th
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t∈[0, 1]: z ′ = T F(z) H0(z(0))
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CHAPTER 5 PAPER III: “HOMOCLINIC
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s 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 H
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For a point p∈C0 we say that C0 i
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conjugate pair of eigenvalues for t
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Proof. (Sketch) The Lyapunov coeffi
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Table 5.1: Euclidean distance in (p
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gency of W s (q) with E u (Cl,ǫ).
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y x2 W s (q) C0 Cl,ǫ x1 W s (Cl,ǫ
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periodic orbit and q are restricted
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yΣ1=ψ(Σ1) andΣ2=ψ(Σ2); see Fi
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eigenvalues, Shilnikov [102] proved
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s 1.39 1.385 1.38 1.375 1.37 1.365
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Figure 5.9(a)-(b). It was obtained
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was carried out with the stiff solv
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6.2 Introduction Our framework in t
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are called fold points. We can use
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curve C= Cl∪{(0, 0)}∪ Cr where
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explicitly. Note that currently no
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A slight modification of the formul
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6.5 Relating l1 and K Krupa and Szm
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nates. The computer algebra system
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atλ=λH= 0 forǫ= 0.05 is l MC 1
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imum and maximum of the cubic. The
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6.8 Additions It is important to no
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−7.85 −7.95 −8.05 λ x −7.9
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BIBLIOGRAPHY [1] V.I. Arnold. Encyc
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[26] M. Desroches, B. Krauskopf, an
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icated to Floris Takens. Eds.: Henk
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[74] T.J. Kaper and C.K.R.T. Jones.
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[98] H.G. Rotstein, M. Wechselberge
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