We present theoretical settings of three approaches to computer-assisted proving the existence of periodic orbits of flows generated by ordinary differential equations. The first and the second methods are based on the notion a brick, i.e. an isolating block having the invariant part empty. If a (possibly) large isolating block is filled-in with bricks, some clusters of bricks are properly ordered, and suitable relations among the bricks hold (distinct for each one of these methods), then the flow has a periodic orbit. The reference to the first approach is [1], while the second one is motivated by the paper [4] and it will be given in [2]. In the third approach, based on the papers [3] and [5], the existence of a periodic orbit in a rotating flow is guaranteed, provided some conditions on singular homology cycles of a section of an index pair with respect to a time-discretization of the flow hold.

References:

[1] M.Mrozek, R.Srzednicki, J.Thorpe, T.Wanner, Combinatorial vs. classical dynamics: Recurrence, Comm. Nonlin. Sc. Numerical Simulation 108 (2022) 106226, 1-30.

[2] M.Mrozek, R.Srzednicki, J.Thorpe, T.Wanner, Combinatorial vs. classical dynamics: Recurrence: A Lefschetz fixed point theorem for periodic orbits, in preparation.

[3] M.Mrozek, R.Srzednicki, F.Weilandt, A topological approach to the algorithmic computation of the Conley index for Poincare Maps, SIAM J. Appl. Dynamical Systems 14 (2015), 1348-1386.

[4] R.Srzednicki, On periodic solutions inside isolating chains, J. Differential Equations 165 (2000), 42-60.

[5] R.Srzednicki, On determining the homological Conley index of Poincare maps in autonomous systems, Topol. Methods Nonlinear Anal. 60 (2022), 5-32.