Computation of minimal periods for ordinary differential equations

A framework is presented for lower-bounding periods among periodic solutions to an autonomous dynamical system governed by ordinary differential equations.

For a chosen dynamical system, lower bounds can be proved by constructing auxiliary functions that, similarly to Lyapunov functions, satisfy a certain inequality pointwise on state space.

Different formulations can give bounds applying either to all periodic solutions or to only periodic solutions with chosen symmetry.

In the case of differential equations that are polynomial in the state variables, we present computational methods that use semidefinite programming to construct auxiliary functions.

Furthermore, we give an algorithm to rigorously validate the numerically computed bounds via rational arithmetic.

To illustrate these methods, computations are carried out for two chaotic systems that each have an infinite number of periodic solutions: the Lorenz system, which is dissipative, and the Hénon-Heiles system, which is Hamiltonian.

All computed bounds are validated with rational arithmetic.

Separate bounds are computed that apply to all periodic solutions, and to only periodic solutions with certain symmetries.

In all cases, our best validated bounds agree with periods of known periodic solutions to at least 5 digits, which strongly suggests exact sharpness of our framework for these examples.

The question of how broadly our framework is sharp is discussed, but it remains open.