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

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the components of tangent planes of M at points of trajectories in B will always have non-vanishing components in the ai and the y1 directions. In fact, the hy- perplane H spanned by the ai and the y1 coordinates can be defined by requiring that Pσ1···σk+1 (v)=0 for all v∈Hand (σ1,...,σk+1)(a1,...,ak, y1) Therefore we can easily restate the Exchange Lemma in an equivalent form us- ing differential forms. Theorem 2.4.2 (Exchange Lemma - differential form conclusion). Under the same assumptions as in Theorem 2.4.1 we conclude that at ¯q: for all (σ1,...,σk+1)(a1,...,ak, y1). ˆPσ1···σk+1 = O(ǫ) In the next section we give a proof of the Exchange Lemma for the simplest non-trivial case in phase space dimension four for a system with two fast and two slow variables. 2.5 A Proof in R 4 Here we prove Theorem 2.4.2 for a (2, 2)-fast-slow system in Fenichel Normal Form near the slow manifold Sǫ with z := (a, b, y1, y2) and a, b, y1, y2∈ R a ′ = Λ(z,ǫ)a b ′ = Γ(z,ǫ)b (2.21) y ′ 1 = ǫg1(z,ǫ)=ǫ(1+ H1(a, b, y,ǫ)ab) y ′ 2 = ǫg2(z,ǫ)=ǫ(H2(a, b, y,ǫ)ab) 37
Hence we have k=1=l and n=2 in view of the notation in the previous two sections. Also we still work in a suitable neighbourhood/box B={(a, b, y) :|a|
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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 and 10: TABLE OF CONTENTS Biographical Sket
Page 11 and 12: 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: exit point of a trajectory starting
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 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
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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
Page 115 and 116:
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
Page 127 and 128:
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)
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
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For a point p∈C0 we say that C0 i
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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
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periodic orbit and q are restricted
Page 153 and 154:
yΣ1=ψ(Σ1) andΣ2=ψ(Σ2); see Fi
Page 155 and 156:
eigenvalues, Shilnikov [102] proved
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
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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
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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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