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CHAPTER 2 TRACKING INVARIANT MANIFOLDS In the following chapter we shall restrict to a fast-slow system ǫ ˙x = f (x, y,ǫ) ˙y = g(x, y,ǫ) (2.1) where (x, y)∈R m+n . We make the standing assumption that the critical manifold C0= C is normally hyperbolic, i.e. for all (x, y)∈C it follows that Dx f (x, y, 0) has rank m. By Fenichel’s Theorem this implies that the associated slow manifold Cǫ is also normally hyperbolic forǫ sufficiently small. This assumption shall sometimes be repeated in important theorems but will assumed to hold in any scenario we describe in this chapter if we do not explicitly state any assumptions on C. The main goal of this chapter is to show how the geometric theory of fast- slow systems can be used to prove the existence of homoclinic and heteroclinic orbits. 2.1 Simple Jumps and Transversality Since the critical manifold C0= C is normally hyperbolic we can locally write it as a graph C={(h(y), y)∈R m+n } and consider the slow flow ˙y=g(h(y), y, 0) (2.2) If the slow flow (2.2) has two hyperbolic equilibrium points p= p0 and q=q0 there might be a heteroclinic connection between p and q. The situation is shown in Figure 2.1. In Figure 2.1 we assume a transversal intersection of the stable manifold W s (p) and the unstable manifold W u (q) for the equilibria of the 15
slow flow. In particular the manifolds are stable and unstable manifolds with respect to equation (2.2). C= C0 W s (p) q W p u (q) Figure 2.1: The phase space is R 4 with three slow variables and one fast variable. The critical manifold C is a solid ball D 3 ⊂ R 4 . The equilibria p, q ∈ D 3 are shown together with the transversal intersection of their stable and unstable manifolds W s (p) and W u (q) for the slow flow (2.2). The transversal intersection gives a heteroclinic connection. The fast slow is indicated by double arrows and is directed toward C. Perturbing the system for 0
Page 1 and 2: MULTIPLE TIME SCALE DYNAMICS WITH T
Page 3 and 4: MULTIPLE TIME SCALE DYNAMICS WITH T
Page 5 and 6: To my family. iv
Page 7 and 8: ing mechanical systems.”, [E.T. P
Page 9 and 10: TABLE OF CONTENTS Biographical Sket
Page 11 and 12: LIST OF TABLES 5.1 Euclidean distan
Page 13 and 14: 3.1 Bifurcation diagram of (3.6). H
Page 15 and 16: 4.6 Boundary conditions are blue an
Page 17 and 18: 6.3 For all computations of the ful
Page 19 and 20: where (x, y)∈R m × R n are varia
Page 21 and 22: the general definition of normal hy
Page 23 and 24: fast-slow system and yields: ˙x=(D
Page 25 and 26: 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: where the algorithm can be used to
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 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
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s 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0
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One can view this as a projection o
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for x1. We find that there are thre
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We compute the functionsγl andγr
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3.3.3 Two Slow Variables, One Fast
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Also recall that the y-nullcline pa
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is negative for all values h∈(0,
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3.4 The Full System 3.4.1 Hopf Bifu
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check whether this observation of a
Page 101 and 102:
indistinguishable numerically from
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we should view this curve as an app
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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
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the singular limits but it cannot b
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Then (3.24) transforms to a three t
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with x∈R m the vector of fast var
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ons that was used by David Terman i
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0.15 0.1 0.05 0 −0.05 0.15 0.1 0.
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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
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computing the homoclinic orbit, eve
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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)
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
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conjugate pair of eigenvalues for t
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Proof. (Sketch) The Lyapunov coeffi
Page 145 and 146:
Table 5.1: Euclidean distance in (p
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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
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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
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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
Page 165 and 166:
are called fold points. We can use
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curve C= Cl∪{(0, 0)}∪ Cr where
Page 169 and 170:
explicitly. Note that currently no
Page 171 and 172:
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
Page 181 and 182:
6.8 Additions It is important to no
Page 183 and 184:
−7.85 −7.95 −8.05 λ x −7.9
Page 185 and 186:
BIBLIOGRAPHY [1] V.I. Arnold. Encyc
Page 187 and 188:
[26] M. Desroches, B. Krauskopf, an
Page 189 and 190:
icated to Floris Takens. Eds.: Henk
Page 191 and 192:
[74] T.J. Kaper and C.K.R.T. Jones.
Page 193:
[98] H.G. Rotstein, M. Wechselberge
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