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The existence of normally hyperbolic slow manifolds is established by Fenichel theory [42, 71]. The singular limitǫ= 0 of system (4.1) is a differential algebraic equation with trajectories confined to the critical manifold S= S 0 defined by f= 0. At points of S where Dx f is a regular m×m matrix, the implicit function theorem implies that S is locally the graph of a function x=h(y). This equation yields the vector field ˙y=g(h(y), y, 0) for the slow flow on S . The geometry is more complicated at fold points of S where Dx f is singular. It is often possible to extend the slow flow to the fold points after a rescaling of the vector field [51]. Fenichel proved the existence of invariant slow manifolds Sǫ where all eigenvalues of Dx f have nonzero real parts. Forǫ> 0 small, these normally hyperbolic slow manifolds are within an O(ǫ) distance from the critical manifold S 0 and the flow on Sǫ converges to the slow flow on S 0 asǫ→ 0. Fenichel theory is usually developed in the context of overflowing slow manifolds with boundaries. Trajectories may leave these manifolds through their boundaries. In this setting, slow manifolds are not unique, but the distance between a pair of slow manifolds is “exponentially small,” i.e. of order O(exp(−c/ǫ)) for a suitable positive c, independent ofǫ [71]. The next section of this paper presents the SMST (slow manifold of saddle- type) algorithm. This section gives an estimate of the order of accuracy of the algorithm, augmented by analysis of a linear system for which there are explicit solutions of both the solutions of the differential equations and the boundary value solver. The third section of the paper presents numerical investigations of three ex- amples: 1. A three-dimensional version of the Morris-Lecar model for bursting neu- 97
ons that was used by David Terman in his analysis of the transition between bursts with different numbers of spikes [105, 90], 2. A three-dimensional system whose homoclinic orbits yield traveling-wave profiles for the FitzHugh-Nagumo model [18], 3. A four-dimensional model of two coupled neurons studied by Gucken- heimer, Hoffman and Weckesser [53]. Empirical tests of the precision of the algorithm are given for the Morris-Lecar model. 4.3 The SMST Algorithm This section describes a collocation method called the SMST algorithm for computing slow manifolds of saddle-type in slow-fast systems. Collocation meth- ods [21, 3, 30, 31] are a well established method for solving boundary value problems. The algorithm described in this paper is not a new collocation method [59, 48]: the subtlety lies in the formulation of a boundary value prob- lem that yields discrete systems of equations with well-conditioned Jacobians. The crucial part of the geometry is to specify boundary conditions for trajectory segments on a slow manifold that yield well-conditioned discretizations. Two issues that must be dealt with in formulating the algorithm are that (1) the boundary conditions must determine a unique slow manifold in circumstances where there is an entire “tube” of such manifolds, and (2) any pair of trajectories that lie close to the slow manifold are “exponentially close” along most of their length. 98
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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
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forǫ> 0. Therefore they have no im
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The Hopf bifurcation is supercritic
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where the algorithm can be used to
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slow flow. In particular the manifo
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f ast whereφ t is the flow of the
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whereµ∈R p are parameters. Usual
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We shall not prove this result but
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d as given in equation (2.5). Let p
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variables x1= u, x2= v and y=w we g
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The geometry of the system for one
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(0, 0, 0). 2.3 The Exchange Lemma I
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withδ>0 sufficiently small. The sa
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Theorem 2.3.2. Let ¯q ∈ M∩{|a|
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exit point of a trajectory starting
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Hence we have k=1=l and n=2 in view
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The variational equations describin
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(iii) Denote byη1i the i-th row of
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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 and 78: CHAPTER 3 PAPER I: “HOMOCLINIC OR
Page 79 and 80: [39, 40, 41, 42]). It states that f
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: with x∈R m the vector of fast var
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,
Page 129 and 130: eturns directly to q. Figure 4.5(b)
Page 131 and 132: are complicated [107]: we expect th
Page 133 and 134: t∈[0, 1]: z ′ = T F(z) H0(z(0))
Page 135 and 136: CHAPTER 5 PAPER III: “HOMOCLINIC
Page 137 and 138: s 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 H
Page 139 and 140: For a point p∈C0 we say that C0 i
Page 141 and 142: conjugate pair of eigenvalues for t
Page 143 and 144: Proof. (Sketch) The Lyapunov coeffi
Page 145 and 146: Table 5.1: Euclidean distance in (p
Page 147 and 148: gency of W s (q) with E u (Cl,ǫ).
Page 149 and 150: y x2 W s (q) C0 Cl,ǫ x1 W s (Cl,ǫ
Page 151 and 152: periodic orbit and q are restricted
Page 153 and 154: yΣ1=ψ(Σ1) andΣ2=ψ(Σ2); see Fi
Page 155 and 156: eigenvalues, Shilnikov [102] proved
Page 157 and 158: s 1.39 1.385 1.38 1.375 1.37 1.365
Page 159 and 160: Figure 5.9(a)-(b). It was obtained
Page 161 and 162: was carried out with the stiff solv
Page 163 and 164: 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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