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3.12 Sketch of the singular bifurcation diagram for the FitzHugh- Nagumo equation (3.6). The points A, B and C are defined in (3.23). The part of the diagram obtained from equations (3.17),(3.18) corresponds to the case “ǫ 2 = 0 andǫ 0 and small”. In this scenario the canards to the right of p= p− are stable (see Proposition 3.3.4). The phase portrait in the upper right for equation (3.17) shows the geometry of a small periodic orbit generated in the Hopf bifurcation of (3.17). The two phase portraits below it show the geometry of these periodic orbits further away from the Hopf bifurcation for (3.17),(3.18). . . . . . . . . . . . . . 91 4.1 Boundary conditions for the SMST algorithm are illustrated with a three dimensional example with one slow and two fast variables. The slow manifold of saddle type is drawn black and labeled S . A trajectory that approaches the slow manifold along a strong stable direction and departs along a strong unstable manifold is drawn blue. The initial point of this trajectory lies in a two dimensional manifold Bl transverse to the stable manifold of S , and the final point lies in a one dimensional manifold Br transverse to the unstable manifold of S . . . . . . . . . . . . . . . 100 4.2 Illustration of transversal intersection of stable and unstable manifolds of the slow manifolds W s (S l,ǫ) (green) and W u (S r,ǫ) (magenta). The manifolds are truncated at the yellow section Σ and the trajectoryγ f w∪γbw started onΣat the transversal intersection point xsu is shown in red. . . . . . . . . . . . . . . . . . . . 109 4.3 Illustration of the algorithm for computing homoclinic orbits in the FitzHugh-Nagumo equation. (a)Slow manifolds S l,ǫ and S r,ǫ are shown in black and the unstable manifold of the equilibrium W u (q) is displayed in red. (b) Pieces of the homoclinic orbit; slow segments in black, fast segments in red and S shown in blue. . . 110 4.4 Homoclinic orbit (green) of the FitzHugh-Nagumo equation representing a fast wave. The equilibrium point q is shown in red. . 110 4.5 Homoclinic orbits (green) representing slow waves in the FitzHugh-Nagumo equation. The slow manifold S is shown in blue and the equilibrium q in red. (a) “Single pulse” homoclinic orbit. (b) “Double pulse” homoclinic orbit. This trajectory returns to S l,ǫ before approaching S r,ǫ, then leaves S l,ǫ along its repelling manifold, approaches S r,ǫ briefly and then returns to S l,ǫ a second time, finally flowing along S l,ǫ back to q. . . . . . . . . . 111 xiii
4.6 Boundary conditions are blue and red, the critical manifold C0 is black and the trajectory that lies (disregarding transients) in the slow manifold is green. (a) The initialization step is shown. The solution is identically constant for all t∈[0, 1]. (b) The primary continuation parameterαhas been moved, T will increase and a slow manifold piece is computed. . . . . . . . . . . . . . . . . . . 116 5.1 Bifurcation diagram for the FitzHugh-Nagumo equation (3.6). Shil’nikov homoclinic bifurcations (solid red) and Hopf bifurcations (solid blue) are shown forǫ = 0.01. The dashed curves show the singular limit (ǫ= 0) bifurcation curves for the homoclinic and Hopf bifurcations; see [56] and Section 5.3 for details on the singular limit part of the diagram. . . . . . . . . . . . . . . 120 5.2 Sketch of a homoclinic orbit to the unique equilibrium q. Fast (red) and slow (green) segments decompose the orbit into segments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.3 Figures (a)-(c) show the movement of the stable manifold W s (q) (cyan) with respect to E u (Cl,ǫ) (red) and E s (Cl,ǫ) (green) in phase space on the section y=0.09 forǫ= 0.01. The parameter space diagram (d) shows the homoclinic C-curve (solid red), an extension of the C-curve of parameters where W u (q)∩W s (Cr,ǫ) is nonempty, a curve that marks the tangency of W s (q) to E u (Cl,ǫ) (blue) and a curve that marks a distance between Cl,ǫ and W s (q) (dashed blue) of 0.01 where the arrows indicate the direction in which the distance is bigger than 0.01. The solid black squares in (d) show the parameter values for (a)-(c). . . . . . . . . . . . . . . 129 5.4 Phase space along the C-curve near its sharp turn: the parameter valuesǫ= 0.01, p=0.05 and s≈1.3254 lie on the C-curve. The homoclinic orbit (red), two periodic orbits born in the subcritical Hopf (blue), C0 (thin black), Cl,ǫ and Cr,ǫ (thick black) are shown. The manifold W s (q) (cyan) has been truncated at a fixed coordinate of y. Furthermore W s (Cl,ǫ) (green) is separated by Cl,ǫ into two components shown here by dark green trajectories interacting with Cm,ǫ and by light green trajectories that flow left from Cl,ǫ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 xiv
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: 3.1 Bifurcation diagram of (3.6). H
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
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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
Page 77 and 78:
CHAPTER 3 PAPER I: “HOMOCLINIC OR
Page 79 and 80:
[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
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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
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
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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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