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Computation of minimal periods for ordinary differential equations

Jeremy P. Parker

TL;DR

The paper addresses the problem of determining the shortest possible period $T$ for periodic solutions of autonomous ODEs by introducing a Lyapunov-like framework based on state-space observables and a Lie-derivative inequality. For polynomial vector fields, the bound is computed via sum-of-squares (SOS) optimization implemented as a semidefinite program (SDP), enabling simultaneous optimization over the observable $oldsymbol{ m phi}$ and auxiliary function $V$. The authors derive an analytical bound for symmetric Lorenz orbits and provide rigorously validated SDP bounds that closely match the known Lorenz period (to four decimals), while also discovering a shorter periodic orbit in the Hénon–Heiles system at a fixed energy level $H=1/7$, with $T \\approx 6.0521$. The work demonstrates a versatile approach applicable to polynomial systems (and transformable nonpolynomial systems) and outlines open questions about bound sharpness and extensions to general, non-symmetric orbits and PDE contexts, supported by publicly available code for replication and validation.

Abstract

We consider the problem of finding the shortest possible period for an exactly periodic solution to some given autonomous ordinary differential equation. We show that, given a pair of Lyapunov-like observable functions defined over the state space of the corresponding dynamical system and satisfying a certain pointwise inequality, we can obtain a global lower bound for such periods. We give a method valid for the case of bounding the period of only those solutions which are invariant under a symmetry transformation, as well as bounds for general periodic orbits. If the governing equations are polynomial in the state variables, we can use semidefinite programming to find such auxiliary functions computationally, and thus compute lower bounds which can be rigorously validated using rational arithmetic. We apply our method to the Lorenz and Henon-Heiles systems. For both systems we are able to give validated bounds which are sharp to several decimal places.

Computation of minimal periods for ordinary differential equations

TL;DR

The paper addresses the problem of determining the shortest possible period for periodic solutions of autonomous ODEs by introducing a Lyapunov-like framework based on state-space observables and a Lie-derivative inequality. For polynomial vector fields, the bound is computed via sum-of-squares (SOS) optimization implemented as a semidefinite program (SDP), enabling simultaneous optimization over the observable and auxiliary function . The authors derive an analytical bound for symmetric Lorenz orbits and provide rigorously validated SDP bounds that closely match the known Lorenz period (to four decimals), while also discovering a shorter periodic orbit in the Hénon–Heiles system at a fixed energy level , with . The work demonstrates a versatile approach applicable to polynomial systems (and transformable nonpolynomial systems) and outlines open questions about bound sharpness and extensions to general, non-symmetric orbits and PDE contexts, supported by publicly available code for replication and validation.

Abstract

We consider the problem of finding the shortest possible period for an exactly periodic solution to some given autonomous ordinary differential equation. We show that, given a pair of Lyapunov-like observable functions defined over the state space of the corresponding dynamical system and satisfying a certain pointwise inequality, we can obtain a global lower bound for such periods. We give a method valid for the case of bounding the period of only those solutions which are invariant under a symmetry transformation, as well as bounds for general periodic orbits. If the governing equations are polynomial in the state variables, we can use semidefinite programming to find such auxiliary functions computationally, and thus compute lower bounds which can be rigorously validated using rational arithmetic. We apply our method to the Lorenz and Henon-Heiles systems. For both systems we are able to give validated bounds which are sharp to several decimal places.
Paper Structure (7 sections, 3 theorems, 25 equations, 2 figures, 2 tables)

This paper contains 7 sections, 3 theorems, 25 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. A Lyapunov-like method for lower bounds on periods
  3. Minimal periods for symmetric trajectories in the Lorenz system
  4. Analytical result
  5. Validated numerical results using SDP
  6. Minimal periods in the Hénon-Heiles system at a fixed energy level
  7. Discussion and conclusion

Key Result

Lemma 2.1

For $T>0$ and $f\in C^1([0,T],\mathbb{R}^m)$, if $\int_0^T f(t)\mathrm{d}t = 0$ and $f(T)=f(0)$ then

Figures (2)

  • Figure 1: The LR orbit, the shortest known periodic solution in the Lorenz system at the standard parameters, with period $T\approx1.5587$.
  • Figure 2: Two symmetry-related periodic orbits with period $T\approx6.0521$ in the Hénon-Heiles system \ref{['eq:HH']} at energy level $E=1/7$.

Theorems & Definitions (4)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3