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

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whereα,β1 andβ2 are: α(t) = C|a|( ˆX+ǫ(1+ ˆZ)) β1(t) = −µ+ C(δ+|a| ˆX) β2(t) = K[(ǫ+|b|+ǫ|a| ˆX) ˆZ+ǫ+|b|] The constantµ>0 carries over from Lemma 2.5.1 andδ>0 defines the neighbourhood size of B as given in (2.22). Proof. We start by proving (2.31). A direct calculation as in the previous lemma gives | ˆZi| ′ =− η11 ˆZ 2 ˆZi i +η1i Z1| ˆZi| Z1| ˆZi| (2.33) The chain and quotient rules used for calculating (2.33) are only valid for|Zi|0, when Zi=0 we only obtain left and right limits for the derivative of opposite sign. If we estimate the right-hand side of (2.33) then | ˆZi| ′ ≤ η11 | ˆZi|+ η1i Z1 which is independent of such left and right limits. Using Lemma 2.5.1 again we find η11 Z1 η1i Z1 ≤ ˜C|a|(1+ˆX)+ǫ ˜K|a|(1+ ˆZ+ˆX) ≤ ˜C|a|(| ˆZi|+ˆX)+ǫ ˜K|a|(1+ ˆZ+ˆX) The desired estimate (2.31) now follows if we choose C again such that C > ˜C+ǫ ˜K and C> ˜K. This completes the first part of the proof. For the second part we again use calculus first: ˆX ′ i = d dt Xi Z1 = Z1X ′ i Z1 ′ − XiZ 1 Z2 1 = Z1(BiiXi+η2i)− Xi(ΛZ1+η11) = (Bii−Λ) ˆXi+ 47 Z2 1 η2i Z1 − η11 Z1 ˆXi
This result can be written more compactly in vector form as: From Lemma 2.5.1 we get that η2 Z1 ˆX ′ = (B−Λ) ˆX+ η2 Z1 − η11 Z1 ˆX ≤ Ĉ|a| ˆX+ ˆK(ǫ+|b|)(1+ ˆZ+ˆX) Since we have already found an estimate on|η11/Z1| above it now follows that where η2 Z1 − η11 Z1 ˆX ≤β3(t) ˆX+β2(t) (2.34) β3(t)=C ∗ |a|(1+ˆX)+ K(ǫ+|b|) and the constants are chosen so that C ∗ > ˜C+ǫ ˜K+ ˜C and K> max( ˜K, ˆK). Basi- cally the estimate (2.34) controls the nonlinear term for the equations of ˆX. We proceed by considering an integrating factor: J(t) := exp − t 0 (B−ΛId)ds The standard integrating factor calculation then reads ˆX ′ J− (B−Λ) ˆXJ=J ⇒ ( ˆXJ) ′ η2 = J − η11 ˆX ⇒ t 0 ( ˆXJ) ′ ds= η2 Z1 − η11 Z1 Z1 Z1 t η2 J − 0 Z1 η11 Z1 t η2 ⇒ ˆX=J(t) −1 ˆX0+J(t) −1 The inverse of the matrix J(t) is 0 J Z1 J(t) −1 t = exp (B−ΛId)ds 0 48 ˆX ˆX ds − η11 Z1 ˆX ds
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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
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: Proof. First, we work near q, then|
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 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
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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
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This explicit solution provides a b
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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,
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eturns directly to q. Figure 4.5(b)
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are complicated [107]: we expect th
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t∈[0, 1]: z ′ = T F(z) H0(z(0))
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CHAPTER 5 PAPER III: “HOMOCLINIC
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s 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 H
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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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