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The Forced van der Pol Equation II: Canards in the Reduced System
 SIAM J. APPLIED DYNAMICAL SYSTEMS
, 2003
"... This is the second in a series of papers about the dynamics of the forced van der Pol oscillator [J. Guckenheimer, K. Hoffman, W. Weckesser ..."
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Cited by 11 (4 self)
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This is the second in a series of papers about the dynamics of the forced van der Pol oscillator [J. Guckenheimer, K. Hoffman, W. Weckesser
A COMPUTATIONAL TOOL FOR THE REDUCTION OF NONLINEAR ODE SYSTEMS POSSESSING MULTIPLE SCALES ∗
"... Abstract. Near an orbit of interest in a dynamical system, it is typical to ask which variables dominate its structure at what times. What are its principal local degrees of freedom? What local bifurcation structure is most appropriate? We introduce a combined numerical and analytical technique that ..."
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Abstract. Near an orbit of interest in a dynamical system, it is typical to ask which variables dominate its structure at what times. What are its principal local degrees of freedom? What local bifurcation structure is most appropriate? We introduce a combined numerical and analytical technique that aids the identification of structure in a class of systems of nonlinear ordinary differential equations (ODEs) that are commonly applied in dynamical models of physical processes. This “dominant scale ” technique prioritizes consideration of the influence that distinguished “inputs ” to an ODE have on its dynamics. On this basis a sequence of reduced models is derived, where each model is valid for a duration that is determined selfconsistently as the system’s state variables evolve. The characteristic time scales of all sufficiently dominant variables are also taken into account to further reduce the model. The result is a hybrid dynamical system of reduced differentialalgebraic models that are switched at discrete event times. The technique does not rely on explicit small parameters in the ODEs and automatically detects changing scale separation both in time and in “dominance strength ” (a quantity we derive to measure an input’s influence). Reduced regimes describing the full system have quantified domains of validity in time and with respect to variation in state variables. This enables the qualitative analysis of the system near known orbits (e.g., to study bifurcations) without sole reliance on numerical shooting methods. These methods have been incorporated into a new software tool named Dssrt, which we demonstrate on a limit cycle of a synaptically driven Hodgkin–Huxley neuron model.
ninety plus thirty years of nonlinear dynamics: Less is more and more is different
 International Journal of Bifurcation and Chaos
, 2005
"... I review the early (1885–1975) and more recent history of dynamical systems theory, identifying key principles and themes, including those of dimension reduction, normal form transformation and unfolding of degenerate cases. I end by briefly noting recent extensions and applications in nonlinear flu ..."
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Cited by 6 (0 self)
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I review the early (1885–1975) and more recent history of dynamical systems theory, identifying key principles and themes, including those of dimension reduction, normal form transformation and unfolding of degenerate cases. I end by briefly noting recent extensions and applications in nonlinear fluid and solid mechanics, with a nod toward mathematical biology. I argue throughout that this essentially mathematical theory was largely motivated by nonlinear scientific problems, and that after a long gestation it is propagating throughout the sciences and technology.
Scaling Effects and SpatioTemporal Multilevel Dynamics in Epileptic Seizures
, 2012
"... Epileptic seizures are one of the most wellknown dysfunctions of the nervous system. During a seizure, a highly synchronized behavior of neural activity is observed that can cause symptoms ranging from mild sensual malfunctions to the complete loss of body control. In this paper, we aim to contribu ..."
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Epileptic seizures are one of the most wellknown dysfunctions of the nervous system. During a seizure, a highly synchronized behavior of neural activity is observed that can cause symptoms ranging from mild sensual malfunctions to the complete loss of body control. In this paper, we aim to contribute towards a better understanding of the dynamical systems phenomena that cause seizures. Based on data analysis and modelling, seizure dynamics can be identified to possess multiple spatial scales and on each spatial scale also multiple time scales. At each scale, we reach several novel insights. On the smallest spatial scale we consider single model neurons and investigate earlywarning signs of spiking. This introduces the theory of critical transitions to excitable systems. For clusters of neurons (or neuronal regions) we use patient data and find oscillatory behavior and new scaling laws near the seizure onset. These scalings lead to substantiate the conjecture obtained from meanfield models that a Hopf bifurcation could be involved near seizure onset. On the largest spatial scale we introduce a measure based on phaselocking intervals and wavelets into seizure modelling. It is used to resolve synchronization between different regions in the brain and identifies timeshifted scaling laws at different wavelet scales. We also compare our waveletbased multiscale approach with maximum linear crosscorrelation and meanphase coherence measures.
1 Periodic Orbit Continuation in Multiple Time Scale Systems
"... Continuation methods utilizing boundary value solvers are an effective tool for computing unstable periodic orbits of dynamical systems. AUTO [1] is the standard implementation of these procedures. However, the collocation methods used in AUTO often require very fine meshes for convergence on ..."
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Continuation methods utilizing boundary value solvers are an effective tool for computing unstable periodic orbits of dynamical systems. AUTO [1] is the standard implementation of these procedures. However, the collocation methods used in AUTO often require very fine meshes for convergence on
Bifurcations of Relaxation Oscillations near Folded Saddles ∗
, 2004
"... Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slowfast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions ..."
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Relaxation oscillations are periodic orbits of multiple time scale dynamical systems that contain both slow and fast segments. The slowfast decomposition of these orbits is defined in the singular limit. Geometric methods in singular perturbation theory classify degeneracies of these decompositions that occur in generic one parameter families of relaxation oscillations. This paper investigates the bifurcations that are associated with one type of degeneracy that occurs in systems with two slow variables, namely orbits that become homoclinic to a folded saddle. 1
The Forced van der Pol Equation II: Canards in the Reduced System∗
"... Abstract. This is the second in a series of papers about the dynamics of the forced van der Pol oscillator ..."
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Abstract. This is the second in a series of papers about the dynamics of the forced van der Pol oscillator
MULTIPLE TIME SCALE DYNAMICS WITH TWO FAST VARIABLES AND ONE SLOW VARIABLE
, 2010
"... This thesis considers dynamical systems that have multiple time scales. The focus lies on systems with two fast variables and one slow variable. The twoparameter bifurcation structure of the FitzHughNagumo (FHN) equation is analyzed in detail. A singular bifurcation diagram is constructed and inva ..."
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This thesis considers dynamical systems that have multiple time scales. The focus lies on systems with two fast variables and one slow variable. The twoparameter bifurcation structure of the FitzHughNagumo (FHN) equation is analyzed in detail. A singular bifurcation diagram is constructed and invariant manifolds of the problem are computed. A boundaryvalue approach to compute slow manifolds of saddletype is developed. Interactions of classical invariant manifolds and slow manifolds explain the exponentially small turning of a homoclinic bifurcation curve in parameter space. Mixedmode oscillations and maximal canards are detected in the FHN equation. An asymptotic formula to find maximal canards is proved which is based on the first Lyapunov coefficient at a singular Hopf bifurcation.