Martin Wechselberger
University of Sydney
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Publication
Featured researches published by Martin Wechselberger.
Siam Review | 2012
Mathieu Desroches; John Guckenheimer; Bernd Krauskopf; Christian Kuehn; Hinke M. Osinga; Martin Wechselberger
Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.
Siam Journal on Applied Dynamical Systems | 2005
Martin Wechselberger
We give a geometric analysis of canards of folded node type in singularly perturbed systems with two-dimensional (2D) folded critical manifold using the blow-up technique. The existence of two primary canards is known provided a nonresonance condition
Siam Journal on Applied Dynamical Systems | 2008
Horacio G. Rotstein; Martin Wechselberger; Nancy Kopell
\mu\notin\mathbb{N}
Chaos | 2008
Jonathan E. Rubin; Martin Wechselberger
is satisfied, where
Nonlinearity | 2006
John Guckenheimer; Martin Wechselberger; Lai Sang Young
\mu=\lambda_1/\lambda_2
Siam Journal on Applied Dynamical Systems | 2009
Bard Ermentrout; Martin Wechselberger
denotes the ratio of the eigenvalues of the associated folded singularity of the reduced flow. We show that, due to resonances, bifurcation of secondary canards occurs. We give a detailed geometric explanation of this phenomenon using an extension of Melnikov theory to prove a transcritical bifurcation of canards for odd
Siam Journal on Applied Dynamical Systems | 2005
Janet Best; Alla Borisyuk; Jonathan E. Rubin; David Terman; Martin Wechselberger
\mu
Journal of Computational Neuroscience | 2010
Theodore Vo; Richard Bertram; Joël Tabak; Martin Wechselberger
. Furthermore, we show numerically the existence of a pitchfork bifurcation for even
Journal of Mathematical Neuroscience | 2011
Wondimu Teka; Joël Tabak; Theodore Vo; Martin Wechselberger; Richard Bertram
\mu
Journal of Mathematical Neuroscience | 2013
John Mitry; Michelle M. McCarthy; Nancy Kopell; Martin Wechselberger
and a novel turning point bifurcation close to