multiple time scale dynamics with two fast variables and one slow ...

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4 Paper II: “Computing slow manifolds of saddle-type” 95 4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3 The SMST Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3.1 Slow manifolds of a linear system . . . . . . . . . . . . . . 103 4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.1 Bursting Neurons . . . . . . . . . . . . . . . . . . . . . . . . 105 4.4.2 Traveling Waves of the FitzHugh-Nagumo Model . . . . . 106 4.4.3 A Model of Reciprocal Inhibition . . . . . . . . . . . . . . . 112 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6 Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5 Paper III: “Homoclinic orbits of the FitzHugh-Nagumo equation: Bifurcations in the full system” 118 5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3 Fast-Slow Decomposition of Homoclinic Orbits . . . . . . . . . . . 123 5.4 Interaction of Invariant Manifolds . . . . . . . . . . . . . . . . . . 127 5.5 Homoclinic Bifurcations in Fast-Slow Systems . . . . . . . . . . . 134 5.6 Canards and Mixed Mode Oscillations . . . . . . . . . . . . . . . . 139 5.6.1 Canard Explosion . . . . . . . . . . . . . . . . . . . . . . . . 139 5.6.2 Mixed-Mode Oscillations . . . . . . . . . . . . . . . . . . . 141 5.7 Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6 Paper IV: “From first Lyapunov coefficients to maximal canards” 145 6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.3 Canard Explosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.4 The First Lyapunov Coefficient . . . . . . . . . . . . . . . . . . . . 152 6.5 Relating l1 and K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.8 Additions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Bibliography 168 ix
LIST OF TABLES 5.1 Euclidean distance in (p,s)-parameter space between the Hopf curve and the location of the tangency point between W s (q) and W u (Cl,ǫ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.1 Comparison between the actual location of the maximal canard (canard explosion) Ic and the first-order approximation Ic(Lyapunov) computed using the first Lyapunov coefficient at the Hopf bifurcation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 x
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: TABLE OF CONTENTS Biographical Sket
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 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 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
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
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Also recall that the y-nullcline pa
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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
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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,
Page 129 and 130:
eturns directly to q. Figure 4.5(b)
Page 131 and 132:
are complicated [107]: we expect th
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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
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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
Page 151 and 152:
periodic orbit and q are restricted
Page 153 and 154:
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
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 and 170:
explicitly. Note that currently no
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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