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with F(z)=O(z 2 ) and M= (mi j). Taylor expanding F yields z ′ = Mz+ 1 1 B(z, z)+ C(z, z, z) 2 6 where the multilinear functions B and C are given by: Bi(u, v) = Ci(u, v, w) = N ∂2 Fi(ξ) ∂ξ j∂ξk N ∂3 Fi(ξ) ∂ξ j∂ξk∂ξl j,k=1 j,k,l=1 u jvk ξ=0 u jvkwl ξ=0 The matrix M has eigenvaluesλ1,2=±iω0 forω0> 0. Let q∈C N be the eigenvector ofλ1 and p∈C N the corresponding eigenvector of the transpose M T i.e. Mq=iω0q, M ¯q=−iω0 ¯q, M T p=−iω0p, M ¯p=iω0 ¯p where the the overbar denotes componentwise complex conjugation. We can always normalize p so that the standard complex inner product with q satisfies ¯p T q= N j=1 ¯p jq j= 1. The first Lyapunov coefficient of the Hopf bifurcation can then be defined by (Kuznetsov [88], p.180): l Ku 1 1 T T −1 T = ¯p C(q, q, ¯q)−2 ¯p B(q, L B(q, ¯q))+ ¯p B(¯q, (2iω0IN− M) 2ω0 −1 B(q, q)) (6.9) In the case of a two-dimensional vector field F= (F 1 , F 2 ) the formula (6.9) can be expressed in the simpler form (Kuznetsov [88], p.98): where l Ku 1 = 1 2ω 2 0 Re(ig20g11+ω0g21) (6.10) g20= ¯p T B(q, q), g11= ¯p T B(q, ¯q), g21= ¯p T C(q, q, ¯q) It is important to note that l Ku 1 is not uniquely defined until we choose a normal- ization of the eigenvector q. We adopt the convention using unit norm ¯q T q=1. 153
A slight modification of the formula (6.9) is used to evaluate the Lyapunov coefficient l MC 1 numerically in the bifurcation software MatCont [46]. Using the current MatCont convention 1 we note that ω0l Ku 1 = lMC 1 Other expressions for the first Lyapunov coefficient can be found in the litera- ture. We consider only the planar case using simpler notation (z1, z2)=(x, y): ⎛ x ⎜⎝ ′ y ′ ⎞ ⎟⎠ = ⎛ F ⎜⎝ 1 (x, y) F2 ⎞ ⎛ ⎞ x ⎟⎠ =: M ⎜⎝ ⎟⎠ (x, y) y + ⎛ ⎞ f (x, y) ⎜⎝ ⎟⎠ (6.11) g(x, y) Then another convention for l1 is (Chow, Li and Wang [19], p.211): l CLW 1 = m12 16ω 4 0 [ω 2 0 [( fxxx+ gxxy)+2m22( fxxy+ gxyy)−m21( fxyy+ gyyy)] −m12m22( f 2 xx− fxxgxy− fxygxx− gxxgyy− 2gxy) −m21m22(g 2 yy− gyy fxy− gxy fyy− fxx fyy− 2 f 2 xy) (6.12) +m 2 12 ( fxxgxx+ gxxgxy)−m 2 21 ( fyygyy+ fxy fyy) −(ω 2 0 + 3m222 )( fxx fxy− gxygyy)] where all evaluations in (6.12) are at (x, y)=(0, 0). Next, assume that we have applied a preliminary linear coordinate change ⎛ ⎞ ⎛ ⎞ x u ⎜⎝ ⎟⎠ = N ⎜⎝ ⎟⎠ where N : R y v 2 → R 2 to the system (6.11) to transform M into Jordan normal form. Then we look at: ⎛ u ⎜⎝ ′ v ′ ⎞ ⎟⎠ = ⎛ ⎞⎛ ⎞ ⎛ ⎞ 0 −ω0 u f (u, v) ⎜⎝ ⎟⎠⎜⎝ ⎟⎠ + N−1 ⎜⎝ ⎟⎠ ω0 0 v g(u, v) ⎛ ⎞⎛ ⎞ 0 −ω0 u = ⎜⎝ ⎟⎠⎜⎝ ⎟⎠ ω0 0 v + ⎛ f ⎜⎝ ∗ (u, v) g∗ ⎞ ⎟⎠ (6.13) (u, v) 1 MatCont version 2.5.1 - December 2008 154
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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
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Proof. First, we work near q, then|
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This result can be written more com
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Proof. (of Theorem 2.4.2) The argum
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ˆZ is still exponentially small. P
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in the FitzHugh-Nagumo equation ǫ
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center-unstable manifold W cu (0, 0
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where the second “fast jump” oc
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CHAPTER 3 PAPER I: “HOMOCLINIC OR
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[39, 40, 41, 42]). It states that f
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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
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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
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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.
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
Page 165 and 166: are called fold points. We can use
Page 167 and 168: curve C= Cl∪{(0, 0)}∪ Cr where
Page 169: explicitly. Note that currently no
Page 173 and 174: 6.5 Relating l1 and K Krupa and Szm
Page 175 and 176: nates. The computer algebra system
Page 177 and 178: atλ=λH= 0 forǫ= 0.05 is l MC 1
Page 179 and 180: 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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