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

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vious situation. Clearly we also want to include the case when a heteroclinic or homoclinic orbit involves fast and slow dynamics. The simplest case for this situation is to assume the existence of two n-dimensional normally hyperbolic invariant manifolds C 1 and C 2 that are contained in the critical manifold C of (2.1). Suppose C 1 and C 2 contain hyperbolic invariant sets P 1 resp. P 2 for the slow flow (2.2); to simplify the abstract setting one can simply think of P 1 and P 2 as equilibrium points. The fast subsystem will be used to find a heteroclinic orbit between P 1 and P 2 . Remark: All the results that follow are going to be applicable in the case C 1 = C 2 (and P 1 = P 2 ) which yields homoclinic orbits. Therefore we restrict the discussion, for now, to the case of heteroclinic orbits. The stable and unstable manifolds of P 1 and P 2 with respect to the slow flow have to be defined. Without loss of generality let us assume we are interested in the unstable manifold of P 1 and the stable manifold of P 2 defined by: W u (P 1 ) = {p∈C 1 :φ slow τ (y)→P 1 asτ→−∞} W s (P 2 ) = {p∈C 2 :φ slow τ (y)→P 1 asτ→∞} whereφ slow τ is the slow flow for equation (2.2). Furthermore certain submani- folds of the stable and unstable manifolds of the critical manifold C have to be defined N 1 = N 2 = p∈W u (P 1 ) p∈W s (P 2 ) {q∈R n+m :φ f ast t (q)→ p as t→−∞} {q∈R n+m f ast :φ t (q)→ p as t→∞} (2.3) 17
f ast whereφ t is the flow of the fast subsystem given by x ′ = f (x, y, 0) y ′ = 0 and the y-coordinates to be used for a particular flow in (2.3) is determined by the y-coordinates of p. Instead of requiring a transversal intersection of manifolds defined entirely by the slow flow we require a transversal intersection of N 1 and N 2 . The situation is shown in Figure 2.2. W u (P 1 ) C 1 P 1 N 1 N 2 P 2 C 2 W s (P 2 ) Figure 2.2: Transversal intersection of N 1 and N 2 yields a heteroclinic connection between P 1 and P 2 consisting of two slow segments and one fast segment. The heteroclinic orbit consists of a trajectory segment of the slow flow on C 1 starting at P 1 . This segment connects to a trajectory of a fast subsystem which lies in the transversal intersection of N 1 and N 2 . Then the last segment lies on C 2 connecting to P 2 . The orbit we obtain in this way is also called a singular orbit as it consists of trajectory segments obtained in the singular limitǫ= 0. 18
Page 1 and 2: 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: slow flow. In particular the manifo
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 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
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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
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waves” (see e.g. [63, 15, 69]). 3
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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
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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
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
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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)
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
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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
Page 163 and 164:
6.2 Introduction Our framework in t
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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
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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
Page 181 and 182:
6.8 Additions It is important to no
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−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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