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

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y 0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06 p=0.0598 p=0.0801 0.05 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 x 1 (a) Projection onto (x1, y) C 0 x 2 0.015 0.01 0.005 0 −0.005 −0.01 −0.015 −0.02 p=0.0598 p=0.0801 −0.025 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 x 1 (b) Projection onto (x1, x2) Figure 3.6: Hopf bifurcation at p≈0.083, s=1 andǫ= 0.01. The critical manifold C0 is shown in red and periodic orbits are shown in blue. Only the first and the last critical manifold for the continuation run are shown; not all periodic orbits obtained during the continuation are displayed. The analysis of the slow subsystems (3.17) and (3.18) gives a conjecture about the shape of the periodic orbits in the FitzHugh-Nagumo equation. Consider the parameter regime close to a Hopf bifurcation point. From (3.17) we expect one part of the small periodic orbits generated in the Hopf bifurcation to lie close to the slow manifolds Cl,ǫ and Cm,ǫ. Using the results about equation (3.18) we anticipate the second part to consist of an excursion in the x2 direction whose length is governed by the wave speed s. Figure 3.6 shows a numerical continuation in MatCont [46] of the periodic orbits generated in a Hopf bifurcation and confirms the singular limit analysis for small amplitude orbits. Furthermore we observe from comparison of the x1 and x2 coordinates of the periodic orbits in Figure 3.6(b) that orbits tend to lie close to the plane defined by x2= 0. More precisely, the x2 diameter of the periodic orbits is observed to be O(ǫ) in this case. This indicates that the rescaling of Section 3.3.3 can help to describe the system close to the U-shaped Hopf curve. Note that it is difficult to 81 C 0
check whether this observation of an O(ǫ)-diameter in the x2-coordinate persists for values ofǫ< 0.01 since numerical continuation of canard-type periodic orbits is difficult to use for smallerǫ. s 0.7 0.6 0.5 0.4 0.3 0.2 0.1 ε=0.01 ε=10 −7 0 0.1706 0.1707 0.1708 0.1709 0.171 0.1711 0.1712 (a) GH ǫ 1 p s 4 3.95 3.9 3.85 3.8 ε=10 −7 ε=0.01 3.75 0.051 0.052 0.053 0.054 0.055 0.056 0.057 0.058 0.059 (b) GH ǫ 2 Figure 3.7: Tracking of two generalized Hopf points (GH) in (p, s,ǫ)parameter space. Each point in the figure corresponds to a different value ofǫ. The point GHǫ 1 in 3.7(a) corresponds to the point shown as a square in Figure 3.1 and the point GHǫ 2 in 3.7(b) is further up on the left branch of the U-curve and is not displayed in Figure 3.1. In contrast to this, it is easily possible to compute the U-shaped Hopf curve using numerical continuation for very small values ofǫ. We have used this possibility to track two generalized Hopf bifurcation points in three parameters (p, s,ǫ). The U-shaped Hopf curve has been computed by numerical continuation for a mesh of parameter values forǫ between 10 −2 and 10 −7 using MatCont [46]. The two generalized Hopf points GHǫ 1,2 on the left half of the U-curve were detected as codimension two points during each continuation run. The results of this “three-parameter continuation” are shown in Figure 3.7. The two generalized Hopf points depend onǫ and we find that their singular 82 p
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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
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
Page 83 and 84: 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
Page 93 and 94: Also recall that the y-nullcline pa
Page 95 and 96: is negative for all values h∈(0,
Page 97: 3.4 The Full System 3.4.1 Hopf Bifu
Page 101 and 102: 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).
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y x2 W s (q) C0 Cl,ǫ x1 W s (Cl,ǫ
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periodic orbit and q are restricted
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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
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was carried out with the stiff solv
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6.2 Introduction Our framework in t
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are called fold points. We can use
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curve C= Cl∪{(0, 0)}∪ Cr where
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explicitly. Note that currently no
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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
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6.8 Additions It is important to no
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−7.85 −7.95 −8.05 λ x −7.9
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BIBLIOGRAPHY [1] V.I. Arnold. Encyc
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[26] M. Desroches, B. Krauskopf, an
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icated to Floris Takens. Eds.: Henk
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[74] T.J. Kaper and C.K.R.T. Jones.
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[98] H.G. Rotstein, M. Wechselberge
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