Abstract
Conley indices and Morse decompositions of flows can be found by using algorithms which rigorously analyze discrete dynamical systems. This usually involves integrating a time discretization of the flow using interval arithmetic. We compare the old idea of fixing a time step as a parameter to a time step continuously varying in phase space. We present an example where this second strategy necessarily yields better numerical outputs and prove that our outputs yield a valid Morse decomposition of the given flow.
Highlights
While the numerical approximation of ordinary differential equations has a long history, the systematic study of how to compute time invariant structures began in the 1980s
Rigorous computations of these structures is an even more recent phenomenon. These latter efforts can be roughly divided into two approaches: direct computation of invariant sets, e.g., periodic orbits, heteroclinic and homoclinic orbits, invariant manifolds, and a more indirect approach based on identification of isolating neighborhoods
From a theoretical perspective this approach allows us to recover the local and global dynamics associated with isolated invariant sets generated by ordinary differential equations
Summary
While the numerical approximation of ordinary differential equations has a long history, the systematic study of how to compute time invariant structures began in the 1980s. From a theoretical perspective this approach allows us to recover the local and global dynamics associated with isolated invariant sets generated by ordinary differential equations. As indicated above, [14, Theorem 1] guarantees that on the level of isolated invariant sets the dynamics of φh captures the dynamics of φ Since this is not always true for a φτ with varying time steps we prove Lemma 3.18, which provides a criterion under which we obtain the desired isolation. Their proofs require some background in dynamical systems, which can be found in [22] and other textbooks about the field
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
