Canard dynamics: applications to the biosciences and theoretical aspects
|
|
- Cory Cole
- 6 years ago
- Views:
Transcription
1 Canard dynamics: applications to the biosciences and theoretical aspects M. Krupa INRIA Rocquencourt (France) Collaborators: M. Desroches (INRIA), T. Kaper (BU)
2 Outline 1 Canards in two dimensions 2 Mixed-mode oscillations 3 Spike adding in square wave bursters 4 Mixed-mode bursting oscillations 5 Analysis of the canard phenomenon 6 Other directions
3 Take VdP with α large and constant forcing a Second order ODE:... x + α(x 2-1)x + x = a Rewritten as first order system:. εx = y-x 3 /3+x. y = a-x where: 0 < ε=1/α 1 Long-term dynamics of when a is varied: from a=1.2 via a= to a= x 1.25 x 0 x time time time
4 Benoît, Callot, Diener & Diener (1981) Van der Pol oscillator. εx. = y y-x 3 /3+x a-x 2 O(exp( c ε )) O(ε)-away from the Hopf point the branch becomes almost vertical. 1 HB a ε =0.001
5 Benoît, Callot, Diener & Diener (1981) Van der Pol oscillator. εx. = y y-x 3 /3+x a-x O(exp( c ε )) 2. Hopf bifurcation at a=1 O(ε)-away from the Hopf point the branch becomes almost vertical. 1 HB a ε =0.001
6 Time-scale analysis: from ε > 0 to ε = 0. x O(1/ε) x is fast. y O(1) y is slow
7 Time-scale analysis: from ε > 0 to ε = 0. x O(1/ε) x is fast. y O(1) y is slow Limiting system for the slow dynamics: ε > 0 ε = 0: Reduced sys.. εx = y-x 3 /3+x 0 = y-x 3 /3 +x.. y = a-x y = a-x
8 Time-scale analysis: from ε > 0 to ε = 0. x O(1/ε) x is fast. y O(1) y is slow Limiting system for the slow dynamics: ε > 0 ε = 0: Reduced sys.. εx = y-x 3 /3+x 0 = y-x 3 /3 +x.. y = a-x y = a-x Limiting system for the fast dynamics: ε > 0 ε = 0: Layer sys. x = y-x 3 /3+x x = y-x 3 /3 +x y = ε(a-x) y = 0
9 Time-scale analysis: from ε > 0 to ε = 0 ẋ O(1/ε) x is fast. y O(1) y is slow Limiting system for the slow dynamics: ε = 0: Reduced sys. 0 = y-x 3 /3+x. y = a-x slow subsystem ODE defined on the cubic S:={y=x 3 -x} /3 Limiting system for the fast dynamics: ε = 0: Layer sys. x = y-x 3 /3+x y = 0 fast subsystem family of ODEs param. by y S is a set of equilibria
10 Time-scale analysis: dynamics from ε = 0 to ε > S a S r S a away from the slow curve S, the overall dynamics is fast in an ε-neighbourhood of S, the overall dynamics is slow Transition: bifurcation points of the fast dynamics Note S has 2 fold points different stability on each side: a r S is attracting and S is repelling
11 Fenichel theory: dynamics from ε = 0 to ε > For ε>0 there are Fenichel slow manifolds or rivers
12 Back to Benoît et al. The VdP system has limit cycles which follow the attracting part S of the cubic nullcline S... all the way down to the fold point and then... r continue along the repelling part S of S! a
13 Back to Benoît et al. The VdP system has limit cycles which follow the attracting part S of the cubic nullcline S... all the way down to the fold point and then... r continue along the repelling part S of S! a They have been called canards by the French mathematicians who discovered them
14 Back to Benoît et al. The VdP system has limit cycles which follow the attracting part S of the cubic nullcline S... all the way down to the fold point and then... r continue along the repelling part S of S! a They have been called canards by the French mathematicians who discovered them
15 Answer: Why do canard cycles exist in an exponentially small parameter range? S r is repelling so to follow it for a time t = O(1/ε) the solution (cycle) must be exponentially close to it.
16 Applications aerodynamics M. Brøns, Math. Eng. Industry 2(1): 51-63, 1988 chemical reactions M. Brøns & K. Bar-Eli, J. Phys. Chem. 95: , 1991 B. Peng et al., Phil. Trans. R. Soc. Lond. A 337: , 1991 M. Brøns & K. Bar-Eli, Proc. R. Soc. London A 445: , 1994
17 Applications (...) J. Moehlis, J. Math. Biol. 52: , 2006 H.G. Rotstein, N. Kopell, A.M. Zhabotinsky, and I.R. Epstein, Canard phenomenon and localization of oscillations in the Belousov-Zhabotinsky reaction with global feedback. J. Chemical Physics, 2003;119:
18 4D Hodgkin-Huxley Cdv/dt = I I L I Na I K I (applied current) is const I L = g L (v V L ) I Na = g Na m 3 h(v V Na ) I K = g K n 4 (v V K ) x = m, h, n gating variables. dx/dt = 1 τ x (v) (x x (v)) 4D 2D reduction Known time scale separation: v and m are fast, h and m are slow. 4D 3D reduction: m = m (v) 3D 2D reduction (Rinzel): h(t) =0.8 n(t)
19 2D Hodgkin-Huxley (! (a) (! (b) Γ wh V V $%'! $%'! Γ m $%&!!*&( n!*(( H I appl H! "!! #!! Γ m Γ wh (c) Γ h S 0 S ε r V Γ h H $%&! $%'!! '! "! (! $%"( I appl $%'!!!!*( ' (! $%"( $%'!!!!*( ' (! (d) Γ wh Γ m!*+( $%)! $%"! #! V $%"( Γ h $%'!!!!*( ' t/t
20 Mixed mode oscillations Canard explosion with a drift
21 Mixed mode oscillations ' # v "'( " "'( # )*+ v # "'( " "'( # ),+ #'(!"" #""" #$"" #%"" #&"" #!"" t #'(!"" #""" #$"" #%"" #&"" #!"" t v )-+ Γ 6 Γ 7 L r S r ε ξ 5 ξ 6 ξ 7 S a ε h ' s
22 Mixed mode oscillations ' # v "'( " "'( # )*+ v # "'( " "'( # ),+ #'(!"" #""" #$"" #%"" #&"" #!"" t #'(!"" #""" #$"" #%"" #&"" #!"" t v )-+ Γ 6 Γ 7 L r S r ε ξ 5 ξ 6 ξ 7 S a ε h ' s M. Brons, M. Krupa and M. Wechselberger. Mixed mode oscillations due to the generalized canard phenomenon. Fields Inst. Comm. 49, (2006) M. Krupa, N. Popovic and N. Kopell. Mixed-mode oscillations in three timescale systems--a prototypical example. SIAM J. Appl. Dyn. Sys. 7, (2008)
23 Applications, mixed mode oscillations H. Rotstein, T. Oppermann, J. White, and N. Kopell The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells. J. Comput. Neurosci., 2006 J. Rubin and M. Wechselberger Giant squid-hidden canard: the 3D geometry of the Hodgkin Huxley model Biol Cybern (2007) 97:5 32 DOI /s M. Krupa, N. Popovic, N. Kopell and H. G. Rotstein. Mixed-mode oscillations in a three timescale model of a dopamine neuron. Chaos, 18, p (2008). Review: M. Deroches, J. Guckenheimer, B. Krauskopf, C. Kuehn, H. Osinga, M. Wechselberger. Mixed-mode oscillations with multiple time scales. Mixed-mode oscillations wit SIAM Rev. 54 (2012)
24 Spike adding canard explosion and mixed-mode bursting oscillations MMBOs References: D. Terman, Chaotic spikes arising from a model of bursting in excitable membranes, SIAM Journal on Applied Mathematics 51 (5) (1991) J. Guckenheimer, C. Kuehn, Computing slow manifolds of saddle type, SIADS 8, (2009) M. Desroches, T. J. Kaper and M. Krupa, Mixed-mode bursting oscillations: Dynamics created by a slow passage through spike-adding canard explosion in a squarewave burster. Chaos 23(4), pp (2013).
25 Square wave burster Context: Morris-Lecar type system (extra slow variable): v =I 0.5(v +0.5) 2w(v +0.7) 0.5(1 + tanh( v w =1.15(0.5(1 + tanh( v ) w) cosh(v ) I =ε(k v) ))(v 1) Two fast and one slow variable
26 Bursting (square wave burster)
27 The Hindmarsh-Rose burster x = y ax 3 + bx 2 + I z y = c dx 2 y z = ε(s(x x 1 ) z) x (b1) (b2) (b3) (a) 0 Ho HB LP Bifurcation diagram fast system LP Ho z
28 xx x Spike-adding via canards zt tt z zzz 0.75 t x xx x x x t z t t z (g) xxx (g) x z z 0.75z 0.75 t t (h) t z t t z (f) (g) (f) (e) (e) zzz t t x t (e) x x (e) (f) x x x xx x 0.75 t 0.75 z tt z (d) (d) (h) (h) z 0.75 zz t 0.75 t (i)
29 Spike-adding via canards (cont) t x 0 x z z
30 Adding a spike within canard explosion maximal canard 1 spike intermediate canard 2 spikes maximal canard 2 spikes
31 Combination of MMO and bursting (II) + =...? MMOs slow oscillations (+ 1 spike) Bursting fast oscillations (+ quiescent phase)
32 Combination of MMO and bursting (II) + =...? MMOs slow oscillations (+ 1 spike) Bursting fast oscillations (+ quiescent phase) different time scales, fast and slow complex oscillations Minimal system: 2 fast & 2 slow variables
33 MMOs + Bursting = Mixed-Mode Bursting Oscillations (MMBOs) 2 fast variables 2 slow variables
34 MMOs + Bursting = Mixed-Mode Bursting Oscillations (MMBOs) 2 fast variables 2 slow variables 2 fast: ε 1 x 1 = f 1 ε 2 x 2 = f 2
35 MMOs + Bursting = Mixed-Mode Bursting Oscillations (MMBOs) 2 fast variables 2 slow variables 2 fast: 2 slow: ε 1 x 1 = f 1 ε 2 x 2 = f 2 y 1 = g 1 y 2 = g 2
36 MMOs + Bursting = Mixed-Mode Bursting Oscillations (MMBOs) 2 fast variables 2 slow variables 2 fast: 2 slow: ε 1 x 1 = f 1 ε 2 x 2 = f 2 y 1 = g 1 y 2 = g 2 MMOs
37 MMOs + Bursting = Mixed-Mode Bursting Oscillations (MMBOs) 2 fast variables 2 slow variables 2 fast: 2 slow: ε 1 x 1 = f 1 ε 2 x 2 = f 2 y 1 = g 1 y 2 = g 2 MMOs
38 MMOs + Bursting = Mixed-Mode Bursting Oscillations (MMBOs) 2 fast variables 2 slow variables 2 fast: 2 slow: ε 1 x 1 = f 1 ε 2 x 2 = f 2 y 1 = g 1 y 2 = g 2 Bursting
39 MMOs + Bursting = Mixed-Mode Bursting Oscillations (MMBOs) 2 fast variables 2 slow variables 2 fast: 2 slow: ε 1 x 1 = f 1 ε 2 x 2 = f 2 y 1 = g 1 y 2 = g 2 Bursting
40 MMOs + Bursting = Mixed-Mode Bursting Oscillations (MMBOs) 2 fast variables 2 slow variables 2 fast: 2 slow: ε 1 x 1 = f 1 ε 2 x 2 = f 2 y 1 = g 1 y 2 = g 2 Combine theories... and questions: what patterns of oscillations? what organising centres?
41 Our strategy to construct and analyse a 4D slow-fast system with MMBOs start from a burster (Hindmarsh-Rose in our case) add a slow variable similarly to the MMO case, we want MMBOs to be the result to a slow passage through a canard explosion we will construct a slow passage through a spike-adding canard explosion 1.5 x 1 (a) 1.5 x 1 (b) t z x = y ax 3 + bx 2 + I z y = c dx 2 y z = ε(s(x x 1 ) z) Hindmarsh-Rose I = ε(k h x (x x fold ) 2 h y (y y fold ) 2 h I (I I fold ))
42 Understanding MMBOs as a slow passage x = y ax 3 + bx 2 + I z y = c dx 2 y z = ε(s(x x 1 ) z) I = ε(k h x (x x fold ) 2 h y (y y fold ) 2 h I (I I fold ))
43 x = y ax 3 + bx 2 + I z y = c dx 2 y z = ε(s(x x 1 ) z) Controlling the number of SAOs using folded node theory I = ε(k h x (x x fold ) 2 h y (y y fold ) 2 h I (I I fold )) (a) 2 ε = 10 4 (b) 2 ε = 10 3 x (c) ! x 10 4 (d) t 2 x I I t 4.5! x t 2 x 10 4
44 x = y ax 3 + bx 2 + I z y = c dx 2 y z = ε(s(x x 1 ) z) I = ε(k h x (x x fold ) 2 h y (y y fold ) 2 h I (I I fold )) Reducing the value of epsilon to match theoretical formulas ε = 10 5 (a) 2 x 1 (b) 2 x t 2 x t 2 x 10 5
45 Analysis: canard phenomenon and MMOs Naive approach: rescaling We first translate the singularity to the origin: ẋ = y x x3 ẏ = λ x λ = a 1 Now rescale: x = ε x, y = εȳ λ = ε λ
46 Rescaled equations (we drop the bars): ẋ = ε(y x εx 3 ) ẏ = ε(λ x) After time rescaling: ẋ = y x ẏ = λ x εx 3
47 What is the problem? x = ε x, y = εȳ λ = ε λ If ( x, ȳ) were assumed uniformly bounded with respect to ε then the corresponding neighborhood in (x, y) is O(ε) in size. Too small for Fenichel theory!
48 Possible approaches: Nonstandard analysis E. Benoît, J.-L. Callot, F. Diener and M. Diener, Chasse au canard, Collectanea Mathematica 32 (1-2): , Classical analysis, using stretch variables W. Eckhaus, Standard chase on French Ducks, Springer LNM Vol. 985: , 1983 Blow-up F. Dumortier and R. Roussarie, Canard cycles and center manifolds, Memoirs of the American Mathematical Society 121(577), M. Krupa and P. Szmolyan, Relaxation oscillations and canard explosion, Journal of Differential Equations 174(2) , 2001.
49 Blow-up Singular coordinate transformation: Φ:R + S 4 R 4 x = r x, y = r 2 ȳ, ε = r 2 ε, λ = r λ -it contains the rescaling -it covers a neighborhood of fixed size (wrt to ε)
50 Charts of the blow-up, i.e. charts of the sphere, correspond to parts of the phase space y K1 K1 K2 x Chart K1 connects to the slow flow Chart K2 is the rescaling Chart K3 connects to the fast flow
51 2D problems lead to 3D problems Case 1: Simple fold εẋ = y + x 2 ẏ = g(x, y), g(0, 0) < 0. S a S r
52 Case II: canard point Canard point is a degenerate fold defined by the condition g(0, 0) = 0 The following equations give an example: εẋ = y + x 2 ẏ = x λ λ 0 S a S r
53 ε =0 Unfoldings of a canard point, ε>0 S a S r S a S r S a S r
54 Case III: folded node (two slow one fast dimensions) εẋ = y + x 2 ẏ = x z ż = µ
55 The slow subsystem 0= y + x 2 ẏ = x z ż = µ reduced equations
56 Explanation: λ unfoldings of a canard point y λ>0! > 0 x y λ =0! > 0 x y λ<0! < 0 x
57 Explanation: λ unfoldings of a canard point!
58 Folded node! εẋ = y + x 2 ẏ = x z ż = µ Folded node is a canard point with a drift
59 Folded node! ε>0 The green trajectory and the magenta trajectory are called The green trajectory and the magenta trajectory are called primary canards primary canards The black trajectory is a secondary canard The black trajectory is a secondary canards
60 Computed slow manifolds and canards Computed slow manifolds and canards z C + 2 z 1 C + 0. C η 1 η 3 η 2 γ s!1 γ s γ w C0!2!3!1 1 3 x 5 C 0 Σ 0 C. y x computed by Mathieu Desroches
61 Secondary canard obtained by continuation. ξ r 6 L r S r Σ fn F ξ a 6 L a. S a computed by Mathieu Desroches
62 Selected references on folded node E. Benoît, Canards et enlacements, Publications Mathématiques de l IHES 72(1): 63 91, P. Szmolyan and M. Wechselberger, Canards in R 3, Journal of Differential Equations 177(2): , M. Wechselberger, Existence and bifurcations of canards in R 3 in the case of the folded node, SIAM Journal on Applied Dynamical Systems 4(1): , M. Brøns, M. Krupa and M. Wechselberger, Mixed-mode oscillations due to the generalized canard phenomenon, in Bifurcation Theory and spatio-temporal pattern formation, Fields Institute Communications vol. 49, pp , 2006.
63 Multiple secondary canards. Σ fn L r S r ε v S a ε ξ 12 ξ 11 ξ 10 ξ 9 ξ 8 ξ 7 ξ 6 ξ 5 ξ 4 n. h Computed by M. Desroches
64 References on secondary canards M. Wechselberger, Existence and bifurcations of canards in Rin 3 the case of the folded node, SIAM Journal on Applied Dynamical Systems 4(1): , M. Krupa, N. Popovic and N. Kopell, Mixed-mode oscillations in the three time-scale systems: A prototypical example, SIAM Journal on Applied Dynamical Systems 7(2): , M. Krupa and M. Wechselberger, Local analysis near a folded saddle-node singularity, Journal of Differential Equations 248(12): , M. Krupa, A. Vidal, M. Desroches and F. Clément, Multiscale analysis of mixed-mode oscillations in a phantom bursting model, SIAM Journal on Applied Dynamical Systems (2012)
65 Torus canards: transition from spiking to bursting (related to discrete canards?) Early work by Izhikievich, SIAM Review, 43, , Canonical system: ż =(u + iω)z +2z z 2 z z u = ε(a z 2 )+... Canard explosion for amplitude equations
66 Transition from bursting to spiking bursting: relaxation oscillation modulated spiking: canard with head modulated spiking: small canard fast spiking
67 Activity of a Purkinje cell The big picture Burke, Barry, Kramer, Kaper, Desroches Increasing excitation (J) Bursting Bursting [fold fold-cycle] [canard] AM Spiking [canard] Bif Diag (Fast system) V (Full system) Fixed point Slow nullcline & attracting FP of fast system intersect J = (AM spiking) Torus canard explosion Torus bifurcation (TR) in full system
68 Noise driven canards Other directions Extensions to infinite dimensions, delay eqs, PDES Krupa, Touboul Fine aspects of the dynamics, e.g. chaotic MMOs Krupa, Popovic, Kopell, SIADS 2008 Systems with more than two timescales Krupa, Vidal, Desroches, Clémént, SIADS 2012 Networks of canard oscillators, synchronization Canards in boundary value problems Torus canards? Touboul, Krupa, Desroches, submitted 2013
MIXED-MODE OSCILLATIONS WITH MULTIPLE TIME SCALES
MIXED-MODE OSCILLATIONS WITH MULTIPLE TIME SCALES MATHIEU DESROCHES, JOHN GUCKENHEIMER, BERND KRAUSKOPF, CHRISTIAN KUEHN, HINKE M OSINGA, MARTIN WECHSELBERGER Abstract Mixed-mode oscillations (MMOs) are
More informationNew Developments in Dynamical Systems Arising from the Biosciences. The Effect of Noise on Mixed-Mode Oscillations
Barbara Gent gent@math.uni-bielefeld.de http://www.math.uni-bielefeld.de/ gent New Developments in Dynamical Systems Arising from the Biosciences Mathematical Biosciences Institute, 22 26 March 2011 The
More informationJournal of Differential Equations
J. Differential Equations 248 (2010 2841 2888 Contents lists available at ScienceDirect Journal of Differential Equations www.elsevier.com/locate/jde Local analysis near a folded saddle-node singularity
More informationA showcase of torus canards in neuronal bursters
Journal of Mathematical Neuroscience (2012) 2:3 DOI 10.1186/2190-8567-2-3 RESEARCH OpenAccess A showcase of torus canards in neuronal bursters John Burke Mathieu Desroches Anna M Barry Tasso J Kaper Mark
More informationarxiv: v2 [math.ds] 8 Feb 2016
Canard explosion and relaxation oscillation in planar, piecewise-smooth, continuous systems Andrew Roberts Department of Mathematics, Cornell University (Dated: February 9, 2016) arxiv:1412.0263v2 [math.ds]
More informationTHREE SCENARIOS FOR CHANGING OF STABILITY IN THE DYNAMIC MODEL OF NERVE CONDUCTION
THREE SCENARIOS FOR CHANGING OF STABILITY IN THE DYNAMIC MODEL OF NERVE CONDUCTION E.A. Shchepakina Samara National Research University, Samara, Russia Abstract. The paper deals with the specific cases
More informationPiecewise-linear (PWL) canard dynamics : Simplifying singular perturbation theory in the canard regime using piecewise-linear systems
Piecewise-linear (PWL) canard dynamics : Simplifying singular perturbation theory in the canard regime using piecewise-linear systems Mathieu Desroches, Soledad Fernández-García, Martin Krupa, Rafel Prohens,
More informationFeedback control of canards
Feedback control of canards CHAOS 18, 015110 2008 Joseph Durham a and Jeff Moehlis b Department of Mechanical Engineering, University of California, Santa Barbara, California 93106, USA Received 9 January
More informationA Study of the Van der Pol Equation
A Study of the Van der Pol Equation Kai Zhe Tan, s1465711 September 16, 2016 Abstract The Van der Pol equation is famous for modelling biological systems as well as being a good model to study its multiple
More informationThe selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales
CHAOS 18, 015105 2008 The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales Jonathan Rubin Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania
More informationStochastic differential equations in neuroscience
Stochastic differential equations in neuroscience Nils Berglund MAPMO, Orléans (CNRS, UMR 6628) http://www.univ-orleans.fr/mapmo/membres/berglund/ Barbara Gentz, Universität Bielefeld Damien Landon, MAPMO-Orléans
More informationSpike-adding canard explosion of bursting oscillations
Spike-adding canard explosion of bursting oscillations Paul Carter Mathematical Institute Leiden University Abstract This paper examines a spike-adding bifurcation phenomenon whereby small amplitude canard
More informationCANARDS AND HORSESHOES IN THE FORCED VAN DER POL EQUATION
CANARDS AND HORSESHOES IN THE FORCED VAN DER POL EQUATION WARREN WECKESSER Department of Mathematics Colgate University Hamilton, NY 3346 E-mail: wweckesser@mail.colgate.edu Cartwright and Littlewood discovered
More informationNumerical Computation of Canards
Numerical Computation of Canards John Guckenheimer Kathleen Hoffman Warren Weckesser January 9, 23 Shortened title: Computing Canards Abstract Singularly perturbed systems of ordinary differential equations
More informationarxiv: v1 [math.ds] 30 Jan 2012
Homoclinic Orbits of the FitzHugh-Nagumo Equation: Bifurcations in the Full System John Guckenheimer Christian Kuehn arxiv:1201.6352v1 [math.ds] 30 Jan 2012 Abstract This paper investigates travelling
More informationNeuroscience applications: isochrons and isostables. Alexandre Mauroy (joint work with I. Mezic)
Neuroscience applications: isochrons and isostables Alexandre Mauroy (joint work with I. Mezic) Outline Isochrons and phase reduction of neurons Koopman operator and isochrons Isostables of excitable systems
More informationTHE GEOMETRY OF SLOW MANIFOLDS NEAR A FOLDED NODE
THE GEOMETRY OF SLOW MANIFOLDS NEAR A FOLDED NODE M DESROCHES, B KRAUSKOPF, HM OSINGA Abstract This paper is concerned with the geometry of slow manifolds of a dynamical system with two slow and one fast
More informationCanards at Folded Nodes
Canards at Folded Nodes John Guckenheimer and Radu Haiduc Mathematics Department, Ithaca, NY 14853 For Yulij Il yashenko with admiration and affection on the occasion of his 60th birthday March 18, 2003
More informationSome results on interspike interval statistics in conductance-based models for neuron action potentials
Some results on interspike interval statistics in conductance-based models for neuron action potentials Nils Berglund MAPMO, Université d Orléans CNRS, UMR 7349 et Fédération Denis Poisson www.univ-orleans.fr/mapmo/membres/berglund
More informationUNIVERSITY OF CALIFORNIA SANTA BARBARA. Controlling Canards Using Ideas From The Theory of Mixed-Mode Oscillations. Joseph William Durham
i UNIVERSITY OF CALIFORNIA SANTA BARBARA Controlling Canards Using Ideas From The Theory of Mixed-Mode Oscillations by Joseph William Durham A thesis submitted in partial satisfaction of the requirements
More informationInteraction of Canard and Singular Hopf Mechanisms in a Neural Model
SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 0, No. 4, pp. 443 479 c 0 Society for Industrial and Applied Mathematics Interaction of Canard and Singular Hopf Mechanisms in a Neural Model R. Curtu and J. Rubin
More informationMixed mode oscillations as a mechanism for pseudo-plateau bursting
J Comput Neurosci (2010) 28:443 458 DOI 10.1007/s10827-010-0226-7 Mixed mode oscillations as a mechanism for pseudo-plateau bursting Theodore Vo Richard Bertram Joel Tabak Martin Wechselberger Received:
More informationRelaxation Oscillation and Canard Explosion 1
Journal of Differential Equations 74, 32368 (200) doi:0.006jdeq.2000.3929, available online at http:www.idealibrary.com on Relaxation Oscillation and Canard Explosion M. Krupa and P. Szmolyan Institut
More informationAnalysis of stochastic torus-type bursting in 3D neuron model
Analysis of stochastic torus-type bursting in 3D neuron model Lev B. Ryashko lev.ryashko@urfu.ru Evdokia S. Slepukhina evdokia.slepukhina@urfu.ru Ural Federal University (Yekaterinburg, Russia) Abstract
More information1 1
jmg16@cornell.edu 1 1 Periodic Orbit Continuation in Multiple Time Scale Systems John Guckenheimer and M. Drew LaMar Mathematics Department, Cornell University, Ithaca, NY 14853 1 Introduction Continuation
More informationReduced System Computing and MMOs. Warren Weckesser. Department of Mathematics Colgate University
Reduced System Computing and MMOs Warren Weckesser Department of Mathematics Colgate University Acknowledgments NSF Grant DMS-54468: RUI: Reduced System Computing for Singularly Perturbed Differential
More informationSingular perturbation methods for slow fast dynamics
Nonlinear Dyn (2007) 50:747 753 DOI 10.1007/s11071-007-9236-z ORIGINAL ARTICLE Singular perturbation methods for slow fast dynamics Ferdinand Verhulst Received: 2 February 2006 / Accepted: 4 April 2006
More information1. Introduction. We consider singularly perturbed ordinary differential equations (ODEs) in the standard form
SIAM J. MATH. ANAL. Vol. 33, No. 2, pp. 286 314 c 2001 Society for Industrial and Applied Mathematics EXTENDING GEOMETRIC SINGULAR PERTURBATION THEORY TO NONHYPERBOLIC POINTS FOLD AND CANARD POINTS IN
More informationMixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling
Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling Soledad Fernández-García, Maciej Krupa, Frédérique Clément To cite this version: Soledad Fernández-García, Maciej
More informationarxiv: v1 [math.ds] 5 Apr 2008
A novel canard-based mechanism for mixed-mode oscillations in a neuronal model Jozsi Jalics Department of Mathematics and Statistics, Youngstown State University, Youngstown, OH, 44555 arxiv:0804.0829v1
More informationBIFURCATIONS OF SELF - EXCITATION REGIMES IN OSCILLATORY SYSTEMS WITH NONLINEAR ENERGY SINK
BIFURCATIONS OF SELF - EXCITATION REGIMES IN OSCILLATORY SYSTEMS WITH NONLINEAR ENERGY SINK O.V.Gendelman,, T. Bar 1 Faculty of Mechanical Engineering, Technion Israel Institute of Technology Abstract:.
More informationCanard cycles in global dynamics
Canard cycles in global dynamics Alexandre Vidal, Jean-Pierre Francoise To cite this version: Alexandre Vidal, Jean-Pierre Francoise. Canard cycles in global dynamics. International Journal of Bifurcation
More informationDynamical Systems in Neuroscience: Elementary Bifurcations
Dynamical Systems in Neuroscience: Elementary Bifurcations Foris Kuang May 2017 1 Contents 1 Introduction 3 2 Definitions 3 3 Hodgkin-Huxley Model 3 4 Morris-Lecar Model 4 5 Stability 5 5.1 Linear ODE..............................................
More informationAbrupt changes and intervening oscillations in conceptual climate models
Abrupt changes and intervening oscillations in conceptual climate models Andrew Roberts A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment
More informationCANARDS AT FOLDED NODES
MOSCOW MATHEMATICAL JOURNAL Volume 5, Number 1, January March 2005, Pages 91 103 CANARDS AT FOLDED NODES JOHN GUCKENHEIMER AND RADU HAIDUC For Yulij Ilyashenko with admiration and affection on the occasion
More informationSlow Manifold of a Neuronal Bursting Model
Slow Manifold of a Neuronal Bursting Model Jean-Marc Ginoux 1 and Bruno Rossetto 2 1 PROTEE Laboratory, Université du Sud, B.P. 2132, 83957, La Garde Cedex, France, ginoux@univ-tln.fr 2 PROTEE Laboratory,
More informationarxiv: v1 [math.ds] 24 Aug 2018
Canards Existence in Memristor s Circuits arxiv:1808.09022v1 [math.ds] 24 Aug 2018 The aim of this work is to propose an alternative method for determining the condition of existence of canard solutions
More informationTowards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University
Towards a Global Theory of Singularly Perturbed Dynamical Systems John Guckenheimer Cornell University Dynamical systems with multiple time scales arise naturally in many domains. Models of neural systems
More informationDynamical Systems in Neuroscience: The Geometry of Excitability and Bursting
Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting Eugene M. Izhikevich The MIT Press Cambridge, Massachusetts London, England Contents Preface xv 1 Introduction 1 1.1 Neurons
More informationPERIODIC SOLUTIONS AND SLOW MANIFOLDS
Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 7, No. 8 (27) 2533 254 c World Scientific Publishing Company PERIODIC SOLUTIONS AND SLOW MANIFOLDS FERDINAND VERHULST Mathematisch
More informationSynchronization of Elliptic Bursters
SIAM REVIEW Vol. 43,No. 2,pp. 315 344 c 2001 Society for Industrial and Applied Mathematics Synchronization of Elliptic Bursters Eugene M. Izhikevich Abstract. Periodic bursting behavior in neurons is
More informationWhen Transitions Between Bursting Modes Induce Neural Synchrony
International Journal of Bifurcation and Chaos, Vol. 24, No. 8 (2014) 1440013 (9 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127414400136 When Transitions Between Bursting Modes Induce
More informationCanards and Mixed-Mode Oscillations in a Singularly Perturbed Two Predators-One Prey Model Susmita Sadhu
Proceedings of Dynamic Systems and Applications 7 206 2 29 Canards and Mied-Mode Oscillations in a Singularly Perturbed Two Predators-One Prey Model Susmita Sadhu Department of Mathematics Georgia College
More informationAnalysis of burst dynamics bound by potential with active areas
NOLTA, IEICE Paper Analysis of burst dynamics bound by potential with active areas Koji Kurose 1a), Yoshihiro Hayakawa 2, Shigeo Sato 1, and Koji Nakajima 1 1 Laboratory for Brainware/Laboratory for Nanoelectronics
More informationGeometric Desingularization of a Cusp Singularity in Slow Fast Systems with Applications to Zeeman s Examples
J Dyn Diff Equat (013) 5:95 958 DOI 10.1007/s10884-013-93-5 Geometric Desingularization of a Cusp Singularity in Slow Fast Systems with Applications to Zeeman s Examples Henk W. Broer Tasso J. Kaper Martin
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,
More informationExample of a Blue Sky Catastrophe
PUB:[SXG.TEMP]TRANS2913EL.PS 16-OCT-2001 11:08:53.21 SXG Page: 99 (1) Amer. Math. Soc. Transl. (2) Vol. 200, 2000 Example of a Blue Sky Catastrophe Nikolaĭ Gavrilov and Andrey Shilnikov To the memory of
More informationComputational Neuroscience. Session 4-2
Computational Neuroscience. Session 4-2 Dr. Marco A Roque Sol 06/21/2018 Two-Dimensional Two-Dimensional System In this section we will introduce methods of phase plane analysis of two-dimensional systems.
More informationMixed-mode oscillations in a three time-scale model for the dopaminergic neuron
CHAOS 18, 015106 008 Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron Martin Krupa a Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Department
More informationTHREE TIME-SCALES IN AN EXTENDED BONHOEFFER-VAN DER POL OSCILLATOR
THREE TIME-SCALES IN AN EXTENDED BONHOEFFER-VAN DER POL OSCILLATOR P. DE MAESSCHALCK, E. KUTAFINA, AND N. POPOVIĆ Abstract. We consider an extended three-dimensional Bonhoeffer-van der Pol oscillator which
More informationDelayed phenomenon of loss of stability of solutions in a second-order quasi-linear singularly perturbed boundary value problem with a turning point
RESEARCH Open Access Delayed phenomenon of loss of stability of solutions in a second-order quasi-linear singularly perturbed boundary value problem with a turning point Zheyan Zhou * Jianhe Shen * Correspondence:
More informationDynamical systems in neuroscience. Pacific Northwest Computational Neuroscience Connection October 1-2, 2010
Dynamical systems in neuroscience Pacific Northwest Computational Neuroscience Connection October 1-2, 2010 What do I mean by a dynamical system? Set of state variables Law that governs evolution of state
More informationThe Forced van der Pol Equation II: Canards in the Reduced System
SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 2, No. 4, pp. 57 68 c 23 Society for Industrial and Applied Mathematics The Forced van der Pol Equation II: Canards in the Reduced System Katherine Bold, Chantal
More informationBifurcation Analysis of Neuronal Bursters Models
Bifurcation Analysis of Neuronal Bursters Models B. Knowlton, W. McClure, N. Vu REU Final Presentation August 3, 2017 Overview Introduction Stability Bifurcations The Hindmarsh-Rose Model Our Contributions
More informationFrom neuronal oscillations to complexity
1/39 The Fourth International Workshop on Advanced Computation for Engineering Applications (ACEA 2008) MACIS 2 Al-Balqa Applied University, Salt, Jordan Corson Nathalie, Aziz Alaoui M.A. University of
More informationc 2000 Society for Industrial and Applied Mathematics
SIAM J. APPL. MATH. Vol. 60, No. 2, pp. 503 535 c 2000 Society for Industrial and Applied Mathematics SUBCRITICAL ELLIPTIC BURSTING OF BAUTIN TYPE EUGENE M. IZHIKEVICH Abstract. Bursting behavior in neurons
More informationDynamics and complexity of Hindmarsh-Rose neuronal systems
Dynamics and complexity of Hindmarsh-Rose neuronal systems Nathalie Corson and M.A. Aziz-Alaoui Laboratoire de Mathématiques Appliquées du Havre, 5 rue Philippe Lebon, 766 Le Havre, FRANCE nathalie.corson@univ-lehavre.fr
More informationNeural Excitability in a Subcritical Hopf Oscillator with a Nonlinear Feedback
Neural Excitability in a Subcritical Hopf Oscillator with a Nonlinear Feedback Gautam C Sethia and Abhijit Sen Institute for Plasma Research, Bhat, Gandhinagar 382 428, INDIA Motivation Neural Excitability
More information1 Introduction and neurophysiology
Dynamics of Continuous, Discrete and Impulsive Systems Series B: Algorithms and Applications 16 (2009) 535-549 Copyright c 2009 Watam Press http://www.watam.org ASYMPTOTIC DYNAMICS OF THE SLOW-FAST HINDMARSH-ROSE
More informationBLUE SKY CATASTROPHE IN SINGULARLY-PERTURBED SYSTEMS
BLUE SKY CATASTROPHE IN SINGULARLY-PERTURBED SYSTEMS ANDREY SHILNIKOV, LEONID SHILNIKOV, AND DMITRY TURAEV Abstract. We show that the blue sky catastrophe, which creates a stable periodic orbit whose length
More informationPeriodic Orbits of Vector Fields: Computational Challenges
Periodic Orbits of Vector Fields: Computational Challenges John Guckenheimer Mathematics Department, Ithaca, NY 14853 Abstract Simulation of vector fields is ubiquitous: examples occur in every discipline
More informationA Mathematical Study of Electrical Bursting in Pituitary Cells
A Mathematical Study of Electrical Bursting in Pituitary Cells Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Collaborators on
More informationBIFURCATIONS OF MULTIPLE RELAXATION OSCILLATIONS IN POLYNOMIAL LIÉNARD EQUATIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 139, Number 6, June 2011, Pages 2073 2085 S 0002-9939(201010610-X Article electronically published on November 3, 2010 BIFURCATIONS OF MULTIPLE RELAXATION
More informationSingle neuron models. L. Pezard Aix-Marseille University
Single neuron models L. Pezard Aix-Marseille University Biophysics Biological neuron Biophysics Ionic currents Passive properties Active properties Typology of models Compartmental models Differential
More informationChimera states in networks of biological neurons and coupled damped pendulums
in neural models in networks of pendulum-like elements in networks of biological neurons and coupled damped pendulums J. Hizanidis 1, V. Kanas 2, A. Bezerianos 3, and T. Bountis 4 1 National Center for
More informationarxiv: v3 [math.ds] 27 Nov 2013
Modeling the modulation of neuronal bursting: a singularity theory approach arxiv:1305.7364v3 [math.ds] 27 Nov 2013 Alessio Franci 1,, Guillaume Drion 2,3, & Rodolphe Sepulchre 2,4 1 INRIA Lille-Nord Europe,
More informationTWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY. P. Yu 1,2 and M. Han 1
COMMUNICATIONS ON Website: http://aimsciences.org PURE AND APPLIED ANALYSIS Volume 3, Number 3, September 2004 pp. 515 526 TWELVE LIMIT CYCLES IN A CUBIC ORDER PLANAR SYSTEM WITH Z 2 -SYMMETRY P. Yu 1,2
More informationChapter 24 BIFURCATIONS
Chapter 24 BIFURCATIONS Abstract Keywords: Phase Portrait Fixed Point Saddle-Node Bifurcation Diagram Codimension-1 Hysteresis Hopf Bifurcation SNIC Page 1 24.1 Introduction In linear systems, responses
More informationDynamical modelling of systems of coupled oscillators
Dynamical modelling of systems of coupled oscillators Mathematical Neuroscience Network Training Workshop Edinburgh Peter Ashwin University of Exeter 22nd March 2009 Peter Ashwin (University of Exeter)
More informationAnalysis of the Takens-Bogdanov bifurcation on m parameterized vector fields
Analysis of the Takens-Bogdanov bifurcation on m parameterized vector fields Francisco Armando Carrillo Navarro, Fernando Verduzco G., Joaquín Delgado F. Programa de Doctorado en Ciencias (Matemáticas),
More informationDynamic Characteristics of Neuron Models and Active Areas in Potential Functions K. Nakajima a *, K. Kurose a, S. Sato a, Y.
Available online at www.sciencedirect.com Procedia IUTAM 5 (2012 ) 49 53 IUTAM Symposium on 50 Years of Chaos: Applied and Theoretical Dynamic Characteristics of Neuron Models and Active Areas in Potential
More informationDiscretized Fast-Slow Systems near Pitchfork Singularities
Discretized Fast-Slow Systems near Pitchfork Singularities Luca Arcidiacono, Maximilian Engel, Christian Kuehn arxiv:90.065v [math.ds] 8 Feb 09 February 9, 09 Abstract Motivated by the normal form of a
More informationNonlinear dynamics vs Neuronal Dynamics
Nonlinear dynamics vs Neuronal Dynamics Cours de Methodes Mathematiques en Neurosciences Brain and neuronal biological features Neuro-Computationnal properties of neurons Neuron Models and Dynamical Systems
More informationarxiv: v3 [math.ds] 4 Jan 2017
WILD OSCILLATIONS IN A NONLINEAR NEURON MODEL WITH RESETS: (II) MIXED-MODE OSCILLATIONS JONATHAN E. RUBIN, JUSTYNA SIGNERSKA-RYNKOWSKA, JONATHAN D. TOUBOUL, AND ALEXANDRE VIDAL arxiv:1509.08282v3 [math.ds]
More informationInternal and external synchronization of self-oscillating oscillators with non-identical control parameters
Internal and external synchronization of self-oscillating oscillators with non-identical control parameters Emelianova Yu.P., Kuznetsov A.P. Department of Nonlinear Processes, Saratov State University,
More informationDynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited
Dynamics of Two Coupled van der Pol Oscillators with Delay Coupling Revisited arxiv:1705.03100v1 [math.ds] 8 May 017 Mark Gluzman Center for Applied Mathematics Cornell University and Richard Rand Dept.
More informationMathematical Analysis of Bursting Electrical Activity in Nerve and Endocrine Cells
Mathematical Analysis of Bursting Electrical Activity in Nerve and Endocrine Cells Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University
More informationBIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs
BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits: equilibria cycles connecting orbits compact invariant manifolds strange
More informationThe relationship between two fast/slow analysis techniques for bursting oscillations
The relationship between two fast/slow analysis techniques for bursting oscillations Wondimu Teka, Joël Tabak, and Richard Bertram Citation: Chaos 22, 043117 (2012); doi: 10.1063/1.4766943 View online:
More informationarxiv: v2 [math.ds] 21 Feb 2014
Mied mode oscillations in a conceptual climate model Andrew Roberts a,, Esther Widiasih b, Martin Wechselberger c, Chris K. R. T. Jones a arxiv:1311.518v [math.ds] 1 Feb 014 Abstract a Department of Mathematics,
More informationBursting and Chaotic Activities in the Nonlinear Dynamics of FitzHugh-Rinzel Neuron Model
Bursting and Chaotic Activities in the Nonlinear Dynamics of FitzHugh-Rinzel Neuron Model Abhishek Yadav *#, Anurag Kumar Swami *, Ajay Srivastava * * Department of Electrical Engineering, College of Technology,
More information8 Example 1: The van der Pol oscillator (Strogatz Chapter 7)
8 Example 1: The van der Pol oscillator (Strogatz Chapter 7) So far we have seen some different possibilities of what can happen in two-dimensional systems (local and global attractors and bifurcations)
More informationWIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY
WIDELY SEPARATED FREQUENCIES IN COUPLED OSCILLATORS WITH ENERGY-PRESERVING QUADRATIC NONLINEARITY J.M. TUWANKOTTA Abstract. In this paper we present an analysis of a system of coupled oscillators suggested
More informationBifurcation structure of a model of bursting pancreatic cells
BioSystems 63 (2001) 3 13 www.elsevier.com/locate/biosystems Bifurcation structure of a model of bursting pancreatic cells Erik Mosekilde a, *, Brian Lading a, Sergiy Yanchuk b, Yuri Maistrenko b a Department
More informationThe Rela(onship Between Two Fast/ Slow Analysis Techniques for Burs(ng Oscilla(ons
The Rela(onship Between Two Fast/ Slow Analysis Techniques for Burs(ng Oscilla(ons Richard Bertram Department of Mathema(cs Florida State University Tallahassee, Florida Collaborators and Support Joël
More informationNoise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle
CHAOS 18, 015111 2008 Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle Cyrill B. Muratov Department of Mathematical Sciences, New Jersey Institute of Technology,
More informationHopf Bifurcations in Problems with O(2) Symmetry: Canonical Coordinates Transformation
Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part 1, 65 71 Hopf Bifurcations in Problems with O(2) Symmetry: Canonical Coordinates Transformation Faridon AMDJADI Department
More informationCyclicity of common slow fast cycles
Available online at www.sciencedirect.com Indagationes Mathematicae 22 (2011) 165 206 www.elsevier.com/locate/indag Cyclicity of common slow fast cycles P. De Maesschalck a, F. Dumortier a,, R. Roussarie
More informationMANY scientists believe that pulse-coupled neural networks
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 10, NO. 3, MAY 1999 499 Class 1 Neural Excitability, Conventional Synapses, Weakly Connected Networks, and Mathematical Foundations of Pulse-Coupled Models Eugene
More informationCHAPTER 1. Bifurcations in the Fast Dynamics of Neurons: Implications for Bursting
CHAPTER 1 Bifurcations in the Fast Dynamics of Neurons: Implications for Bursting John Guckenheimer Mathematics Department, Cornell University Ithaca, NY 14853 Joseph H. Tien Center for Applied Mathematics,
More informationRESEARCH ARTICLE. Delay Induced Canards in High Speed Machining
Dynamical Systems Vol., No., February 29, 1 15 RESEARCH ARTICLE Delay Induced Canards in High Speed Machining Sue Ann Campbell Department of Applied Mathematics University of Waterloo Waterloo, Ontario,
More informationMATH 723 Mathematical Neuroscience Spring 2008 Instructor: Georgi Medvedev
MATH 723 Mathematical Neuroscience Spring 28 Instructor: Georgi Medvedev 2 Lecture 2. Approximate systems. 2. Reduction of the HH model to a 2D system. The original HH system consists of 4 differential
More informationChaos in the Hodgkin Huxley Model
SIAM J. APPLIED DYNAMICAL SYSTEMS Vol. 1, No. 1, pp. 105 114 c 2002 Society for Industrial and Applied Mathematics Chaos in the Hodgkin Huxley Model John Guckenheimer and Ricardo A. Oliva Abstract. The
More informationPhase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 265 269 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
More informationThe hunt on canards in population dynamics: a predator-prey system
The hunt on canards in population dynamics: a predator-prey system Ferdinand Verhulst Mathematisch Instituut, University of Utrecht PO Box 8., 358 TA Utrecht, The Netherlands December 7, 4 Running title:
More informationCanards, canard cascades and black swans
Canards, canard cascades and black swans Vladimir Sobolev * Elena Shchepakina ** Department of Technical Cybernetics Samara State Aerospace University (SSAU) Samara 443086 Russia * hsablem@gmail.com **
More informationSelf-Replication, Self-Destruction, and Spatio-Temporal Chaos in the Gray-Scott Model
Letter Forma, 15, 281 289, 2000 Self-Replication, Self-Destruction, and Spatio-Temporal Chaos in the Gray-Scott Model Yasumasa NISHIURA 1 * and Daishin UEYAMA 2 1 Laboratory of Nonlinear Studies and Computations,
More informationCanards, Folded Nodes, and Mixed-Mode Oscillations in Piecewise-Linear Slow-Fast Systems
SIAM REVIEW Vol. 58, No. 4, pp. 653 691 c 2016 Society for Industrial and Applied Mathematics Canards, Folded Nodes, and Mied-Mode Oscillations in Piecewise-Linear Slow-Fast Systems Mathieu Desroches Antoni
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: January 08, 2018, at 08 30 12 30 Johanneberg Kristian
More informationarxiv:nlin/ v1 [nlin.cd] 13 May 2006
On transition to bursting via deterministic chaos arxiv:nlin/0605030v1 [nlin.cd] 13 May 2006 Georgi S. Medvedev September 23, 2018 Abstract We study statistical properties of the irregular bursting arising
More information