[Paper Review] Globally Resonant Homoclinic Tangencies

Sishu Shankar Muni|arXiv (Cornell University)|Jun 17, 2022

Advanced Differential Equations and Dynamical Systems4 citations

TL;DR

This paper investigates globally resonant homoclinic tangencies in two-dimensional maps, demonstrating that infinitely many stable single-round periodic solutions can coexist due to a codimension-three (orientation-reversing) or codimension-four (orientation-preserving) bifurcation. The key result is that bifurcation sequences scale differently depending on parameter direction, with generic scaling ∼|λ|²ᵏ and degenerate cases showing slower ∼|λ|ᵏ or 1/k scaling, leading to highly complex stability regions.

ABSTRACT

The attractors of a dynamical system govern its typical long-term behaviour. The presence of many attractors is significant as it means the behaviour is heavily dependent on the initial conditions. To understand how large numbers of attractors can coexist, in this thesis we study the occurrence of infinitely many stable single-round periodic solutions associated with homoclinic connections in two-dimensional maps. We show this phenomenon has a relatively high codimension requiring a homoclinic tangency and `global resonance', as has been described previously in the area-preserving setting. However, unlike in that setting, local resonant terms also play an important role. To determine how the phenomenon may manifest in bifurcation diagrams, we also study perturbations of a globally resonant homoclinic tangency. We find there exist sequences of saddle-node and period-doubling bifurcations. Interestingly, in different directions of parameter space, the bifurcation values scale differently resulting in a complicated shape for the stability region for each periodic solution. In degenerate directions, the bifurcation values scale substantially slower as illustrated in an abstract piecewise-smooth $C^{1}$ map.

Motivation & Objective

• To understand the mechanism enabling infinitely many stable single-round periodic solutions in two-dimensional maps.
• To determine the codimension of globally resonant homoclinic tangencies in orientation-preserving and orientation-reversing maps.
• To analyze the bifurcation structure unfolding from such tangencies, particularly saddle-node and period-doubling sequences.
• To characterize the scaling laws of bifurcation points in different parameter directions.
• To explore the implications for multistability in nonlinear dynamical systems, especially in applications like neuron models and impacting systems.

Proposed method

• Used normal form analysis and stability triangle conditions to bound trace and determinant for higher iterates of the map.
• Studied perturbations of the homoclinic tangency in one-parameter families to identify sequences of saddle-node and period-doubling bifurcations.
• Derived scaling laws for bifurcation values by analyzing the stable eigenvalue λ of the fixed point, distinguishing between generic and degenerate parameter directions.
• Employed numerical bifurcation software to locate saddle-node and period-doubling points near the unfolding scenario.
• Analyzed piecewise-smooth C¹ maps to illustrate slower scaling laws in degenerate cases.
• Investigated the role of local resonant terms and global resonance in shaping the stability region.

Experimental results

Research questions

• RQ1 What is the codimension of globally resonant homoclinic tangencies in orientation-preserving and orientation-reversing two-dimensional maps?
• RQ2 How do the scaling laws of bifurcation points vary across different directions in parameter space near the tangency?
• RQ3 What is the structure of the stability region for single-round periodic solutions in the unfolding of such tangencies?
• RQ4 How do local resonant terms influence the emergence of infinitely many stable periodic orbits?
• RQ5 Can the mechanism of infinite coexistence be generalized to higher-order tangencies or higher-dimensional maps?

Key findings

• Globally resonant homoclinic tangencies occur at codimension three for orientation-reversing maps and codimension four for orientation-preserving maps.
• Near the tangency, sequences of saddle-node and period-doubling bifurcations emerge, with bifurcation values scaling generically as |λ|²ᵏ for −1 < λ < 1.
• In degenerate parameter directions, bifurcation values scale more slowly, with examples showing ∼|λ|ᵏ and ∼1/k scaling laws.
• The stability region for each periodic solution exhibits a highly complicated shape, dependent on the parameter direction and scaling behavior.
• The generic scaling law |λ|²ᵏ differs from that in piecewise-linear maps, which instead exhibit |λ|ᵏ scaling.
• Double-round periodic solutions were observed in the orientation-preserving case, suggesting further bifurcations remain to be characterized.