Computation of centre manifolds and some codimension-one bifurcations for impulsive delay differential equations
| Autor: | Kevin E. M. Church, Xinzhi Liu |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Hopf bifurcation
Differential equation Euclidean space Applied Mathematics 010102 general mathematics Mathematical analysis Saddle-node bifurcation Codimension Delay differential equation 01 natural sciences 010101 applied mathematics symbols.namesake Method of characteristics symbols Boundary value problem 0101 mathematics Analysis Mathematics |
| Zdroj: | Journal of Differential Equations. 267:3852-3921 |
| ISSN: | 0022-0396 |
| DOI: | 10.1016/j.jde.2019.04.022 |
| Popis: | Based on the centre manifold theorem for impulsive delay differential equations, we derive impulsive evolution equations and boundary conditions associated to a concrete representation of the centre manifold in Euclidean space, as well as finite-dimensional impulsive differential equations associated to the evolution on these manifolds. Though the centre manifolds are not unique, their Taylor expansions agree up to prescribed order, and we present an implicit formula for the quadratic term using a variation of the method of characteristics. We use our centre manifold reduction to derive analogues of the saddle-node and Hopf bifurcation for impulsive delay differential equations, and the latter leads to a novel bifurcation pattern to an invariant cylinder. Examples are provided to illustrate the correctness of the bifurcation theorems and to visualize the geometry of centre manifolds in the presence of impulse effects. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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