1. Discretized Fast–Slow Systems with Canards in Two Dimensions.
- Author
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Engel, Maximilian, Kuehn, Christian, Petrera, Matteo, and Suris, Yuri
- Subjects
- *
QUADRATIC fields , *INVARIANT manifolds - Abstract
We study the problem of preservation of maximal canards for time discretized fast–slow systems with canard fold points. In order to ensure such preservation, certain favorable structure-preserving properties of the discretization scheme are required. Conventional schemes do not possess such properties. We perform a detailed analysis for an unconventional discretization scheme due to Kahan. The analysis uses the blow-up method to deal with the loss of normal hyperbolicity at the canard point. We show that the structure-preserving properties of the Kahan discretization for quadratic vector fields imply a similar result as in continuous time, guaranteeing the occurrence of maximal canards between attracting and repelling slow manifolds upon variation of a bifurcation parameter. The proof is based on a Melnikov computation along an invariant separating curve, which organizes the dynamics of the map similarly to the ODE problem. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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