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where the functions hi are given by: h3(x, y,λ,ǫ) = O(x, y,λ,ǫ) h j(x, y,λ,ǫ) = 1+O(x, y,λ,ǫ), j=1, 2, 4, 5 We define several computable constants, abbreviating (0, 0, 0, 0)=0 in the defi- nitions: a1= (h3)x(0), a2= (h1)x(0), a3= (h2)x(0), a4= (h4)x(0), a5= (h6)x(0) Note that all ai for i=1, 2, 3, 4, 5 only depend on partial derivatives with respect to x. Next we define another constant: A=−a2+ 3a3− 2a4− 2a5 Theorem 6.3.1. (Krupa and Szmolyan [86]) For 0
explicitly. Note that currently no standard bifurcation software such as AUTO [34] or MatCont [46] computes the constants ai and A automatically. Nevertheless bifurcation software can detect Hopf bifurcations so that given a fixedǫ we can approximateλH numerically. Hence the numerical problem that remains is to compute A since λH−λc= A 8 ǫ+ O(ǫ3/2 ) To simplify the notation we define K=A/8. If we know K we can easily approximate the location of the maximal canard byλc=λH− Kǫ+ O(ǫ 3/2 ). The maximal canard organizes the canard explosion [86] and indicates where the rapid amplitude growth of the small orbits generated in the Hopf bifurcation occurs. Our goal is to avoid any additional normal form transformations and center manifold reductions to compute K. The key point to achieve this is to observe that K is just a rescaled version of “the” first Lyapunov coefficient of the Hopf bifurcation atλH. 6.4 The First Lyapunov Coefficient We review and clarify the interpretation, computation and conventions asso- ciated with the first Lyapunov coefficient of a Hopf bifurcation. Consider a general N-dimensional ODE at a non-degenerate Hopf bifurcation point. We assume that the equilibrium has been translated to the origin so that z ′ = Mz+ F(z), for z∈R N 152 (6.8)
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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
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
Page 165 and 166: are called fold points. We can use
Page 167: curve C= Cl∪{(0, 0)}∪ Cr where
Page 171 and 172: A slight modification of the formul
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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