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

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λ3 is the v coordinate of F21(λ1, 0) andρ −1 ,σare O(e −K/ǫ ) for suitable K> 0. We assume further that the domain of F21 is [λ1,λ1+ρ −1 ]×[−1, 1]. Figure 5.7 depicts F21. With these choices, we observe two properties of the C-curve of homoclinic orbits in the geometric model: 1. Ifσv+λ2−ρ 2 (u−λ1) 2 is negative on the domain of F21, then the image of F21 is disjoint from the domain of F12 and there are no recurrent orbits passing near the saddle point. Thus, recurrence implies thatλ2>−σ. 2. Ifλ2> 0, then there are two values ofλ1 for which the saddle-point has a single pulse homoclinic orbit. These points occur for values ofλ1 for which the w-component of F21(0, 0) vanishes:λ1=±ρ −1 |λ2| 1/2 . The magnitude of these values ofλ2 is exponentially small. v Σ2 Σ1 W u (q) λ1 u F21 v W s (q) (λ2,λ3) Figure 5.7: Sketch of the map F21 : Σ2 → Σ1. The (u, v) coordinates are centered at W u (q) and the domain of F21 is in the thin rectangle at distanceλ1 from the origin. The image of this rectangle is the parabolic strip inΣ2. When a vector field has a single pulse homoclinic orbit to a saddle-focus whose real eigenvalue has larger magnitude than the real part of the complex 137 w
eigenvalues, Shilnikov [102] proved that a neighborhood of this homoclinic orbit contains chaotic invariant sets. This conclusion applies to our geometric model when it has a single pulse homoclinic orbit. Consequently, there will be a plethora of bifurcations that occur in the parameter intervalλ2∈ [0,σ], creating the invariant sets asλ2 decreases fromσto 0. The numerical results in the previous section suggest that in the FitzHugh- Nagumo system, some of the periodic orbits in the invariant sets near the homoclinic orbit can be continued to the Hopf bifurcation of the equilibrium point. Note that saddle-node bifurcations that create periodic orbits in the invariant sets of the geometric model lie exponentially close to the curveλ2= 0 that mod- els tangency of W s (q) and W u (Cl,ǫ) in the FitzHugh-Nagumo model. This observation explains why the right most curve of saddle-node bifurcations in Figure 7 of Champneys et al. [18] lies close to the sharp turn of the C-curve. There will also be curves of heteroclinic orbits between the equilibrium point and periodic orbits close to the C-curve. At least some of these form codimen- sion two EP1t bifurcations near the turn of the C-curve as discussed by Champ- neys et al. [18]. Thus, the tangency between W s (q) and W u (Cl,ǫ) implies that there are several types of bifurcation curves that pass exponentially close to the sharp turn of the C-curve in the FitzHugh-Nagumo model. Numerically, any of these can be used to approximately locate the sharp turn of the C-curve. 138
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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
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 and 114: with x∈R m the vector of fast var
Page 115 and 116: 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: yΣ1=ψ(Σ1) andΣ2=ψ(Σ2); see Fi
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 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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