- Series
- Dynamical Systems Working Seminar
- Time
- Tuesday, October 1, 2019 - 2:00pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Jiaqi Yang – Georgia Tech – jyang373@gatech.edu
- Organizer
- Adrian P. Bustamante
Given an analytic two-dimensional ordinary differential equation which admits a limit cycle, we consider the singular perturbation of it by adding a state-dependent delay. We show that for small enough perturbation, there exist a limit cycle and a two-dimensional family of solutions to the perturbed state-dependent delay equation (SDDE), which resemble the solutions of the original ODE.
More precisely, for the original ODE, there is a parameterization of the limit cycle and its stable manifold. We show that, there is a very similar parameterization that gives a 2-dimensional family of solutions of the SDDE.
In our work, we analyze the parameterization, and find functional equations to be satisfied (invariance equations). We prove a theorem in \emph{``a posteriori''} format, that is, if there are approximate solutions of the invariance equations, then there are true solutions of the invariance equations nearby (with appropriate choices of norms). An algorithm which follows from the constructive proof of above theorem has been implemented.
This is a joint work with Joan Gimeno and Rafael de la Llave.