[Paper Review] Topological method for symmetric periodic orbits for maps with a reversing symmetry

Daniel Wilczak, Piotr Zgliczyński|ArXiv.org|Jan 14, 2004

Quantum chaos and dynamical systems5 references3 citations

TL;DR

This paper presents a topological method using covering relations to rigorously prove the existence of infinitely many symmetric periodic orbits in four-dimensional reversible maps with a reversing symmetry. By leveraging computer-assisted proofs via interval arithmetic and tracking unstable/stable manifolds, the method establishes transversality conditions that guarantee symmetric periodic points of arbitrarily high periods.

ABSTRACT

We present a topological method of obtaining the existence of infinite number of symmetric periodic orbits for systems with reversing symmetry. The method is based on covering relations. We apply the method to a four-dimensional reversible map.

Motivation & Objective

• To develop a topological method for proving the existence of infinitely many symmetric periodic orbits in reversible dynamical systems.
• To overcome the computational limitation of traditional methods that require iterating maps up to high orders to detect high-period orbits.
• To apply rigorous numerical techniques based on interval arithmetic to verify transversality and covering relations in a finite computation.
• To establish a framework applicable to higher-dimensional reversible systems, particularly four-dimensional maps, where symmetric periodic orbits are nontrivial to detect.

Proposed method

• The method uses h-sets—cubes with specified coordinate systems representing unstable and stable directions—to define horizontal and vertical disks.
• Covering relations $ N \stackrel{P}{\Longrightarrow} M $ are defined topologically, ensuring that the image of one h-set under the map $ P $ stretches across another in a nontrivial way.
• A chain of covering relations $ N_0 \stackrel{P}{\Longrightarrow} N_1 \cdots \stackrel{P}{\Longrightarrow} N_k = M $ is used to propagate topological transversality from an initial to a final h-set.
• Transversality of unstable manifolds with the fixed point set $ \mathrm{Fix}(S) $ is verified using interval arithmetic to compute enclosures of the map and its derivative.
• The method relies on homotopy arguments to verify that the image of the unstable manifold remains outside the unit ball in unstable coordinates and inside in stable coordinates.
• Algorithms based on interval arithmetic are implemented to check boundary and wall conditions, ensuring that the homotopy avoids the critical set $ M_c^+ $, thus validating the covering relation.

Experimental results

Research questions

• RQ1 Can a topological method with computer-assisted proofs establish the existence of infinitely many symmetric periodic orbits in reversible maps?
• RQ2 How can covering relations be used to propagate topological transversality across multiple iterates to detect high-period symmetric orbits?
• RQ3 What numerical verification techniques ensure rigorous validation of transversality conditions in the presence of nonlinearities and reversibility?
• RQ4 Can the method be applied to four-dimensional reversible maps where standard iteration-based methods fail due to computational complexity?
• RQ5 What is the role of the fixed point set $ \mathrm{Fix}(S) $ in forming symmetric periodic orbits, and how can it be topologically linked via covering chains?

Key findings

• The method successfully proves the existence of infinitely many symmetric periodic orbits for a four-dimensional reversible map using a finite number of computations.
• The proof relies on a chain of covering relations of arbitrary length, which guarantees the existence of symmetric periodic points with arbitrarily high periods.
• The use of interval arithmetic in Algorithms 1 and 2 ensures rigorous verification of transversality conditions for the unstable and stable manifolds.
• The numerical verification of Lemma 11 and Lemma 13 required approximately 2.2 × 10⁸ boxes and took about 36 minutes on a 2.4 GHz processor.
• The method generalizes Devaney’s transversality-based approach by replacing iterative fixed-point searches with topological propagation via covering relations.
• The framework is robust and applicable to systems with one unstable and one stable direction, as demonstrated in prior applications to the Michelson system, restricted three-body problem, and Henon-Heiles system.