Topological-numerical approach to the existence of periodic trajectories in ODE's
Paweł Pilarczyk
Conference Publications 2003, 2003(Special): 701-708 doi: 10.3934/proc.2003.2003.701
We discuss an application of a topological-numerical method for proving the existence of a periodic trajectory in a smooth dynamical system in $\mathbb(R)^n$ where a periodic trajectory is numerically observed. The method is based on the Conley index theory and rigorous numerics for ODEs and it is a generalization of the method introduced in [13]. We apply this method to the Rössler equations.
keywords: algorithm. periodic orbit Conley index RÄossler equations isolating neighbor-hood
Shadowing is generic---a continuous map case
Piotr Kościelniak Marcin Mazur Piotr Oprocha Paweł Pilarczyk
Discrete & Continuous Dynamical Systems - A 2014, 34(9): 3591-3609 doi: 10.3934/dcds.2014.34.3591
We prove that shadowing (the pseudo-orbit tracing property), periodic shadowing (tracing periodic pseudo-orbits with periodic real trajectories), and inverse shadowing with respect to certain families of methods (tracing all real orbits of the system with pseudo-orbits generated by arbitrary methods from these families) are all generic in the class of continuous maps and in the class of continuous onto maps on compact topological manifolds (with or without boundary) that admit a decomposition (including triangulable manifolds and manifolds with handlebody).
keywords: inverse shadowing semidynamical system Shadowing manifold pseudotrajectory periodic shadowing $C^{0}$ topology. continuous surjection genericity continuous map

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