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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 8 (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
Canards at folded node
 Mosc. Math. J
"... Folded singularities occur generically in singularly perturbed systems of differential equations with two slow variables and one fast variable. The folded singularities can be saddles, nodes or foci. Canards are trajectories that flow from the stable sheet of the slow manifold of these systems to th ..."
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Cited by 6 (3 self)
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Folded singularities occur generically in singularly perturbed systems of differential equations with two slow variables and one fast variable. The folded singularities can be saddles, nodes or foci. Canards are trajectories that flow from the stable sheet of the slow manifold of these systems to the unstable sheet of their slow manifold. Benoît has given a comprehensive description of the flow near a folded saddle, but the phase portraits near folded nodes have been only partially described. This paper examines these phase portraits, presenting a picture of the flows in the case of a model system with a folded node. We prove that the number of canard solutions in these systems is unbounded. 1
Bursting induced by excitatory synaptic coupling in nonidentical conditional relaxation oscillators or squarewave bursters
, 2006
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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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Cited by 2 (1 self)
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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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Cited by 1 (0 self)
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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
Lecture Notes Summer Semester 2001
"... This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Summer term 2001, to undergraduate Mathematics and Physics students. It covers a few selected topics from perturbation theory at an introductory level. Only certain results are proved, and for some of the ..."
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This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Summer term 2001, to undergraduate Mathematics and Physics students. It covers a few selected topics from perturbation theory at an introductory level. Only certain results are proved, and for some of the most important theorems, sketches of the proofs are provided. Chapter 2 presents a topological approach to perturbations of planar vector fields. It is based on the idea that the qualitative dynamics of most vector fields does not change under small perturbations, and indeed, the set of all these structurally stable systems can be identified. The most common exceptional vector fields can be classified as well. In Chapter 3, we use the problem of stability of elliptic periodic orbits to develop perturbation theory for a class of dynamical systems of dimension 3 and larger, including (but not limited to) integrable Hamiltonian systems. This will bring us, via averaging and LieDeprit series, all the way to KAMtheory. Finally, Chapter 4 contains an introduction to singular perturbation theory, which is concerned with systems that do not admit a welldefined limit when the perturbation parameter goes to zero. After stating a fundamental result on existence of invariant
Control of oscillation periods and phase durations
"... in halfcenter central pattern generators: a comparative mechanistic analysis ..."
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in halfcenter central pattern generators: a comparative mechanistic analysis
MAPPINGS FOR BIOLOGICAL OSCILLATIONS by
"... Becauseofintrinsiccellularproperties, previousconditionsandnetworkconnections, neurons have many distinct oscillatory behaviors. The qualitative theory of differential equations offers tools that can be used to describe solutions for the differential equations that model neurons. Primarily, Poincaré ..."
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Becauseofintrinsiccellularproperties, previousconditionsandnetworkconnections, neurons have many distinct oscillatory behaviors. The qualitative theory of differential equations offers tools that can be used to describe solutions for the differential equations that model neurons. Primarily, Poincaré Return Maps are used for investigating global stability. The maps capitalize on the recurrent nature of oscillations and are able to analyze changes in dynamics, even at bifurcation points where most other methods fail. Elliptic bursting models are found in numerous biological systems, including the external Globus Pallidus (GPe) section of the brain; the focus for studies of epileptic seizures and Parkinsons disease. However, the bifurcation structure for changes in dynamics remains incomplete. This dissertation develops computerassisted Poincaré maps for mathematical and biologically relevant elliptic bursting neuron models and central pattern generators (CPGs). Thefirst method, used forindividual neurons, offerstheadvantageofanentire family of computationally smooth and complete mappings, which can explain all of the systems dynamical transitions. A complete bifurcation analysis was performed detailing the mechanisms for the transitions from tonic spiking to quiescence in elliptic bursters. A previously
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"... Local network parameters can affect internetwork phase lags in central pattern generators ..."
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Local network parameters can affect internetwork phase lags in central pattern generators
Loss of stability scenario in the Ziegler system
"... This paper is devoted to the analysis of the Ziegler system by using the methods of the singular perturbation theory ..."
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This paper is devoted to the analysis of the Ziegler system by using the methods of the singular perturbation theory