Discretized Fast–Slow Systems with Canards in Two Dimensions
| Autor: | Maximilian Engel, Christian Kuehn, Matteo Petrera, Yuri Suris |
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| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: |
Mathematics::Dynamical Systems
Applied Mathematics General Engineering 500 Naturwissenschaften und Mathematik::510 Mathematik::510 Mathematik Blow-up method Dynamical Systems (math.DS) Slow manifolds Loss of normal hyperbolicity Canards Invariant manifolds Modeling and Simulation Maps FOS: Mathematics Mathematics - Dynamical Systems 34E15 34E20 37M99 37G10 34C45 39A99 Discretization |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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