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Abstract
A criterion for the existence of canards in singularly perturbed dynamical systems is presented. Canards are counterintuitive solutions that evolve along both attracting and repelling branches of invariant manifolds. In two dimensions, canards result in periodic oscillations whose amplitude and period grow in a highly nonlinear way: they are slowly varying with respect to a control parameter, except for an exponentially small range of values where they grow extremely rapidly. This sudden growth, called a canard explosion, has been encountered in many applications ranging from chemistry to neuronal dynamics, aerospace engineering and ecology. Here, we give quantitative meaning to the frequently encountered statement that the singular perturbation parameter epsilon, which represents a ratio between fast and slow time scales, is ‘small enough’ for canards to exist. If limit cycles exist, then the criterion expresses the condition that epsilon must be small enough for there to exist a set of zerocurvature in the neighbourhood of a repelling slow manifold, where orbits can develop inflection points, and thus form the nonconvex cycles observed in a canard explosion. We apply the criterion to examples in two and three dimensions, namely to supercritical and subcritical forms of the van der Pol oscillator, and a prototypical three timescale system with slow passage through a canard explosion.
Original language  English 

Pages (fromto)  24042421 
Number of pages  18 
Journal  Proceedings of the Royal Society of London Series A  Mathematical Physical and Engineering Sciences 
Volume  467 
Issue number  2132 
DOIs  
Publication status  Published  2011 
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Dive into the research topics of 'Canards and curvature: the 'smallness of ' in slowfast dynamics'. Together they form a unique fingerprint.Projects
 1 Finished

CA Fellowship for Mike Jeffrey  When Worlds Collide: The Asymptotics of Interacting Systems
Jeffrey, M.
Engineering and Physical Sciences Research Council
1/08/11 → 31/07/12
Project: Research council