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

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To understand how the rescaling (6.17) affects the Lyapunov coefficient we consider the Hopf normal form case. We start with a planar vector field with linear part in Jordan form (6.13). Assume that the equilibrium is at the origin (x, y)=0 and Hopf bifurcation occurs forλ=0. Applying the rescaling (6.17) we get: ⎛ ⎞ dx2/dt2 ⎜⎝ ⎟⎠ dy2/dt2 = ⎛ 0 −ω0/ ⎜⎝ √ ǫ ω0/ √ ⎞⎛ ⎟⎠⎜⎝ ǫ 0 x2 y2 ⎞ ⎛ ⎟⎠ + ⎜⎝ 1 ǫ f∗ ( √ ǫx2,ǫy2) 1 ǫ3/2 g∗ ( √ ǫx2,ǫy2) ⎞ ⎟⎠ (6.19) In a fast-slow system with singular Hoof bifurcation we know that g ∗ (.,.)=ǫ(...) and thatω0= O( √ ǫ). Setting kω=ω0/ √ ǫ the Lyapunov coefficient can be computed to leading order by (6.14): ¯l GH 1 1 ∗ = fx2x2 kω (0, 0)[g∗x2x2 (0, 0)+ f∗ x2y2 (0, 0)]+kω fx2x2x2(0, 0) √ ǫ+ O(ǫ) (6.20) Equation (6.20) explains the leading-order behavior more clearly and shows that due to the rescaling certain derivative terms in the Lyapunov coefficient for a singular Hopf bifurcation are non-leading terms with respect toǫ → 0. The point is that the rescaling modifies the order with respect toǫ of the linear and nonlinear terms. Also, applying the chain rule to the nonlinear terms to calcu- late the necessary derivatives can affect which terms contribute. To make Proposition (6.5.1) more useful in an applied framework we have computed all the different versions of the Lyapunov coefficient defined in Sec- tion (6.4) up to leading order for equation (6.5) in original non-rescaled coordi- 157
nates. The computer algebra system Maple [66] was used in this case: l Ku 1 l MC 1 = 4K √ ǫ + O( √ ǫ) = 4Kω0 √ √ + O(ω0 ǫ) ǫ l GH 1 = K+ O(ǫ) (6.21) l CLW 1 l Pe 1 = K+ O(ǫ) 3πK = + O(ǫ/ω0) 64ω0 Using the results (6.21) we now have a direct strategy how to analyze a canard explosion generated in a singular Hopf bifurcation. 1. Compute the location of the Hopf bifurcation. This givesλH. 2. Find the first Lyapunov coefficient at the Hopf bifurcation, e.g. we get l MC 1 ≈ 4Kω0/ √ ǫ. 3. Compute the location of the maximal canard, and hence the canard explosion, byλc≈λH− Kǫ. For example, using MatCont we would get λc≈λH− lMC 1 ǫ 4ω0 3/2 (6.22) Observe that the previous calculation may require calculating the eigenvalues at the Hopf bifurcation to determineω0 but does not require any center manifold calculations nor additional normal form transformations; these have basically been encoded in the calculation of the Lyapunov coefficient. 158
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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
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We shall not prove this result but
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d as given in equation (2.5). Let p
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variables x1= u, x2= v and y=w we g
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The geometry of the system for one
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(0, 0, 0). 2.3 The Exchange Lemma I
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withδ>0 sufficiently small. The sa
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Theorem 2.3.2. Let ¯q ∈ M∩{|a|
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exit point of a trajectory starting
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Hence we have k=1=l and n=2 in view
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The variational equations describin
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(iii) Denote byη1i the i-th row of
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Then let H(Z, X, t) := X ′ − BX
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Proof. First, we work near q, then|
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This result can be written more com
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Proof. (of Theorem 2.4.2) The argum
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ˆZ is still exponentially small. P
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in the FitzHugh-Nagumo equation ǫ
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center-unstable manifold W cu (0, 0
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where the second “fast jump” oc
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CHAPTER 3 PAPER I: “HOMOCLINIC OR
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[39, 40, 41, 42]). It states that f
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papers. Geometric singular perturba
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s 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0
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One can view this as a projection o
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for x1. We find that there are thre
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We compute the functionsγl andγr
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3.3.3 Two Slow Variables, One Fast
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Also recall that the y-nullcline pa
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is negative for all values h∈(0,
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3.4 The Full System 3.4.1 Hopf Bifu
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check whether this observation of a
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indistinguishable numerically from
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we should view this curve as an app
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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
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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: 6.5 Relating l1 and K Krupa and Szm
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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