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

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homoclinic orbits to the unique saddle equilibrium (0, 0, 0) (cf. [69]). Note that the existence of the heteroclinics in the fast subsystem was proved in a special case analytically [2] but Figure 3.3 is - to the best of our knowledge - the first explicit computation of where fast subsystem heteroclinics are located. The paper [78] develops a method for finding heteroclinic connections by the same basic approach we used, i.e. defining a codimension one hyperplane H that separates equilibrium points. Figure 3.3 suggests that there exists a double heteroclinic connection for s= 0. Observe that the Hamiltonian in our case is H(x1, x2)= function V(x1) is: V(x1)= px1 5 − (x1) 2 100 + 11(x1) 3 150 − (x1) 4 20 (x2) 2 2 + V(x1) where the The solution curves of (3.12) are given by x2=± √ 2(const. − V(x1)). The struc- ture of the solution curves entails symmetry under reflection about the x1-axis. Suppose ¯p∈[ ¯pl, ¯pr] and recall that we denoted the two saddle points of (3.10) by xl and xr and that their location depends on ¯p. Therefore, we conclude that the two saddles xl and xr must have a heteroclinic connection if they lie on the same energy level, i.e. they satisfy V(xl)−V(xr)=0. This equation can be solved numerically to very high accuracy. Proposition 3.3.1. The fast subsystem of the FitzHugh-Nagumo equation for s=0 has a double heteroclinic connection for ¯p= ¯p ∗ ≈−0.0619259. Given a particular value y=y0 there exists a double heteroclinic connection for p= ¯p ∗ + y0 in the fast subsystem lying in the plane y=y0. 73
3.3.3 Two Slow Variables, One Fast Variable From continuation of periodic orbits in the full system - to be described in Sec- tion 3.4.1 - we observe that near the U-shaped curve of Hopf bifurcations the x2-coordinate is a faster variable than x1. In particular, the small periodic orbits generated in the Hopf bifurcation lie almost in the plane x2= 0. Hence to ana- lyze this region we set ¯x2=x2/ǫ to transform the FitzHugh-Nagumo equation (3.6) into a system with 2 slow and 1 fast variable: ˙x1 = ¯x2 ǫ 2 ˙¯x2 = 1 5 (sǫ ¯x2−x1(x1− 1)( 1 10 − x1)+y− p) (3.14) ˙y = 1 s (x1− y) Note that (3.14) corresponds to the FitzHugh-Nagumo equation in the form (cf. (3.4)): ⎧ ⎪⎨ ⎪⎩ uτ= 5ǫ 2 uxx+ f (u)−w+ p wτ=ǫ(u−w) (3.15) Therefore the transformation ¯x2=x2/ǫ can be viewed as a rescaling of the diffusion strength byǫ 2 . We introduce a new independent small parameter ¯δ=ǫ 2 and then let ¯δ=ǫ 2 → 0. This assumes that O(ǫ) terms do not vanish in this limit, yielding the diffusion free system. Then the slow manifold S 0 of (3.14) is: S 0= (x1, ¯x2, y)∈R 3 : ¯x2= 1 sǫ ( f (x1)−y+ p) (3.16) Proposition 3.3.2. Following time rescaling by s, the slow flow of system (3.14) on S 0 in the variables (x1, y) is given by ǫ ˙x1 = f (x1)−y+ p ˙y = x1− y (3.17) 74
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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
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
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: We compute the functionsγl andγr
Page 93 and 94: Also recall that the y-nullcline pa
Page 95 and 96: 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
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
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conjugate pair of eigenvalues for t
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Proof. (Sketch) The Lyapunov coeffi
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Table 5.1: Euclidean distance in (p
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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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