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

More documents

Recommendations

Info

CHAPTER 4 PAPER II: “COMPUTING SLOW MANIFOLDS OF SADDLE-TYPE” 4.1 Abstract Slow manifolds are important geometric structures in the state spaces of dy- namical systems with multiple time scales. This paper introduces an algorithm for computing trajectories on slow manifolds that are normally hyperbolic with both stable and unstable fast manifolds. We present two examples of bifurcation problems where these manifolds play a key role and a third example in which saddle type slow manifolds are part of a traveling wave profile of a partial dif- ferential equation. Initial value solvers are incapable of computing trajectories on saddle-type slow manifolds, so the slow manifold of saddle-type (SMST) algorithm presented here is formulated as a boundary value method. We take an empirical approach here to assessing the accuracy and effectiveness of the algorithm. Remark: Copyright (c)[2009] Society for Industrial and Applied Mathematics. Reprinted with permission. All rights reserved. 4.2 Introduction Slow-fast vector fields have the form ǫ ˙x = f (x, y,ǫ) ˙y = g(x, y,ǫ) (4.1) 95
with x∈R m the vector of fast variables, y∈R n the vector of slow variables andǫ a small parameter that represents the ratio of time scales. The pair (x, y) will be denoted by z and the vector field will be written ˙z=F(z). Invariant slow manifolds on which the motion of the system has speed that is O(1) are a common feature of slow-fast systems. Nevertheless, simulation of these systems with explicit numerical integration algorithms is limited to time steps that are O(ǫ) due to numerical instabilities. Indeed, trajectories often spend most of their time fol- lowing attracting slow manifolds. Implicit “stiff” integration methods [61] compute trajectories along the attracting slow manifolds, taking time steps that are O(1) while avoiding the numerical instabilities of explicit methods. However, no initial value solver will compute forward trajectories that evolve on non- attracting slow manifolds because the geometric instability of these trajectories is such that nearby initial conditions diverge from one another at exponential rates commensurate with the fast time scale. Even an exact initial value solver in the presence of round-off errors of magnitudeδwill amplify this round-off error to unit magnitude in a time that is O(−ǫ log(δ)). Trajectories on slow manifolds that are repelling in all normal directions can be computed by reversing time, but different strategies are needed to compute trajectories that lie on slow manifolds of saddle type. This paper presents an algorithm that directly com- putes accurate trajectories of slow manifolds of saddle-type. The most similar work on computing these manifolds have been AUTO computations that con- tinue families of trajectories to obtain portions of a saddle-type slow manifold. Examples include a slow manifold that lies in the unstable manifold of a three dimensional model of a cardiac pacemaker [79] and segments of homoclinic or- bits in the model of FitzHugh-Nagumo traveling-waves [18] studied further in Section 3.3 of this paper. 96
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 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 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
Page 111: Then (3.24) transforms to a three t
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).
Page 149 and 150: y x2 W s (q) C0 Cl,ǫ x1 W s (Cl,ǫ
Page 151 and 152: 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
Page 169 and 170:
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
Page 175 and 176:
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
show all

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

Create successful ePaper yourself

Delete template?

Save as template?