[Paper Review] Exotic Bifurcations Inspired by Walking Droplet Dynamics

Aminur Rahman, Denis Blackmore|arXiv (Cornell University)|Aug 25, 2017

Nonlinear Dynamics and Pattern Formation11 references3 citations

TL;DR

This paper introduces two novel types of dynamical bifurcations—homoclinic and heteroclinic-homoclinic—arising from interactions between an attracting invariant Jordan curve and stable/unstable manifolds of saddle points in discrete dynamical systems inspired by walking droplet (pilot-wave) models. The key contribution is the identification and analysis of exotic bifurcations driven by parameter variation, particularly the emergence of robust chaotic strange attractors following sequences of tangent and transverse homoclinic orbits, with extensions to higher-dimensional systems and applications to Gilet's map model.

ABSTRACT

We identify two rather novel types of (compound) dynamical bifurcations generated primarily by interactions of an invariant attracting submanifold with stable and unstable manifolds of hyperbolic fixed points. These bifurcation types - inspired by recent investigations of mathematical models for walking droplet (pilot-wave) phenomena - are introduced and illustrated. Some of the one-parameter bifurcation types are analyzed in detail and extended from the plane to higher-dimensional spaces. A few applications to walking droplet dynamics are analyzed.

Motivation & Objective

• To identify and analyze previously unreported bifurcation types in discrete dynamical systems arising from walking droplet dynamics.
• To investigate the interaction between attracting invariant Jordan curves and stable/unstable manifolds of saddle points as a parameter is varied.
• To extend the analysis of exotic bifurcations from planar systems to higher-dimensional spaces.
• To demonstrate the relevance of these bifurcations to the Gilet map model and its variants, particularly in generating robust chaotic strange attractors.
• To lay the groundwork for future study of diffeomorphic models and invariant measures in pilot-wave systems.

Proposed method

• The study employs a one-parameter family of $ C^1 $ maps $ F: ℝ^2 \times \mathbb{R} \to \mathbb{R}^2 $, specifically the Gilet map with $ \mu $ fixed and $ C $ (denoted $ \sigma $) as the varying parameter.
• The analysis focuses on the interaction between a positively invariant Jordan curve (attracting limit cycle) and the stable and unstable manifolds of hyperbolic fixed points (saddles).
• Key mechanisms include the formation of tangent homoclinic orbits, followed by sequences of transverse and tangent homoclinic orbits, leading to dynamical crises and chaotic attractor development.
• The existence of orientation-reversing, non-injective slice regions $ Z $ and $ \breve{Z} $ is established via analysis of the Jacobian determinant's sign change across stable manifolds.
• A modified Gilet map $ \tilde{F} $ is introduced to model heteroclinic-homoclinic bifurcations, with piecewise definitions ensuring continuity while preserving bifurcation structure.
• Higher-dimensional extensions are discussed by generalizing the geometric and topological conditions of the planar bifurcations to $ \mathbb{R}^m $.

Experimental results

Research questions

• RQ1 What novel bifurcation types emerge from the interaction between an attracting invariant Jordan curve and the stable/unstable manifolds of saddle points in walking droplet models?
• RQ2 How do sequences of tangent and transverse homoclinic orbits contribute to the onset of robust chaotic strange attractors?
• RQ3 In what ways do parameter variations—particularly $ \sigma = C $—induce dynamical crises and shape transitions in the attractor?
• RQ4 How can the bifurcation phenomena observed in non-injective maps like Gilet’s be generalized to higher-dimensional dynamical systems?
• RQ5 What role do orientation-reversing, non-injective slice regions play in enabling exotic bifurcations and chaotic dynamics?

Key findings

• The Gilet map exhibits a sequence of bifurcations as $ \sigma = C $ increases, including multiple tangent homoclinic orbits interspersed with transverse homoclinic orbits, culminating in a robust chaotic strange attractor.
• For $ \sigma \in (0.45, 0.528) $, a chaotic strange attractor emerges and persists in shape up to $ \sigma \approx 0.7 $, after which it undergoes a continuous shape change before breaking up into a chaotic splatter.
• The bifurcation mechanism is driven by the interaction of the attracting Jordan curve with the stable manifold of a saddle point, where orientation reversal in the Jacobian determinant creates non-injective slice regions $ Z $ and $ \breve{Z} $.
• The existence of these slice regions is confirmed by the sign change of $ \det F' $ across the stable manifold, leading to cusp-like boundaries in the slice geometry.
• A modified Gilet map $ \tilde{F} $, which is continuous but not $ C^\infty $, successfully reproduces the bifurcation sequence, validating the robustness of the mechanism to piecewise smoothness.
• The results are extendable to higher-dimensional systems, with the geometric and topological conditions of the planar bifurcations serving as a foundation for $ \mathbb{R}^m $ generalizations.