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Loss of normal hyperbolicity of unbounded critical manifolds
 Nonlinearity
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SCALING IN SINGULAR PERTURBATION PROBLEMS: BLOWINGUP A RELAXATION OSCILLATOR
"... Scaling in singular perturbation problems: blowingup a relaxation oscillator by ..."
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Scaling in singular perturbation problems: blowingup a relaxation oscillator by
c ○ 2011 Society for Industrial and Applied Mathematics A Geometric Model for MixedMode Oscillations in a Chemical System ∗
"... Abstract. This paper presents a detailed analysis of mixedmode oscillations in the “autocatalator, ” a threedimensional, two time scale vector field that is a chemical reactor model satisfying the law of mass action. Unlike earlier work, this paper investigates a return map that simultaneously exhi ..."
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Abstract. This paper presents a detailed analysis of mixedmode oscillations in the “autocatalator, ” a threedimensional, two time scale vector field that is a chemical reactor model satisfying the law of mass action. Unlike earlier work, this paper investigates a return map that simultaneously exhibits full rank and rank deficient behavior in different regions of a cross section. Canard trajectories that follow a twodimensional repelling slow manifold separate these regions. Ultimately, onedimensional induced maps are constructed from approximations to the return maps. The bifurcations of these induced maps are used to characterize the bifurcations of the mixedmode oscillations. It is further shown that the mixedmode oscillations are associated with a singular Hopf bifurcation that occurs in the system.
Wavetrain solutions of a reactiondiffusionadvection model of musselalgae interaction
, 2015
"... We consider a system of coupled partial differential equations modeling the interaction of mussels and algae in advective environments. A key parameter is the relative rate of advection of the algae concentration and diffusion of the mussel species. When advection dominates diffusion, one observes l ..."
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We consider a system of coupled partial differential equations modeling the interaction of mussels and algae in advective environments. A key parameter is the relative rate of advection of the algae concentration and diffusion of the mussel species. When advection dominates diffusion, one observes largeamplitude solutions representing bands of mussels propagating slowly in the upstream direction. Here, we prove the existence of a family of such periodic wavetrain solutions. Our proof relies on Geometric Singular Perturbation Theory to construct these solutions as periodic orbits of the associated traveling wave equations in the largeadvection/smalldiffusion limit. The construction encounters a number of mathematical obstacles which necessitate a compactification of phase space, geometric desingularization to deal with a loss of normal hyperbolicity, and the application of a generalized Exchange Lemma to handle lossofstability turning points. In particular, our analysis uncovers logarithmic (switchback) corrections to the leadingorder solution.
A Remark on Geometric Desingularization of a NonHyperbolic Point using Hyperbolic Space
, 2014
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A Function Space Model for Canonical Systems with an Inner Singularity
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