Oscillations on one dimensional time dependent center manifolds: algebraic curves approach

Rabiei Motlagh, O. ^[1] ; Molaei Derakhtenjani, M. ^[1] ; Mohammadi Nejad ^[1]
1. [1] University of Birjand
  
  University of Birjand
  
  Irán
Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 73, Fasc. 3, 2022, págs. 433-456
Idioma: inglés
DOI: 10.1007/s13348-021-00328-3
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Resumen
- We consider a one dimensional time dependent ODE of degree n≥2 as the restriction of an arbitrary nonautonomous ODE to the associated one dimensional center manifold. Then, we present an algorithm for computing time dependent algebraic curves of m-th degree with m≤n−1. This computation leads us to a m dimensional time dependent bifurcation equation. We determine oscillatory behaviors of the system with the help of the bifurcation equation. Finally, we complete the method for a general parametric planar system and find periodic solutions. The method can be applied for a wide range of nonautonomous systems and does not have restrictions of classical methods such as the Poincaré map.