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Published on: July 19, 2016
Homoclinic snaking near a codimension-two Turing-Hopf bifurcation point in the Brusselator model
1Department of Engineering Sciences and Applied Mathematics, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208, USA. tzou.justin@gmail.com
Summary
This study investigates spatiotemporal Turing-Hopf pinning solutions in the Brusselator model, revealing coexistence of stable stationary stripes and time-periodic oscillations. These solutions exhibit complex "snaking" structures organized by saddle-node bifurcations.
Area of Science:
- Nonlinear Dynamics
- Chemical Kinetics
- Pattern Formation
Background:
- The Brusselator model is a classic example used to study pattern formation in chemical reactions.
- Turing and Hopf bifurcations are fundamental concepts in understanding the emergence of spatial and temporal patterns, respectively.
- Pinning solutions represent stable localized patterns within a homogeneous background.
Purpose of the Study:
- To analyze spatiotemporal Turing-Hopf pinning solutions near the codimension-two point in the 1D Brusselator model.
- To characterize the coexistence of stationary stripes and time-periodic oscillations in these solutions.
- To compare these nonvariational solutions with stationary pinning solutions in variational systems.
Main Methods:
- Numerical continuation was employed to solve time-periodic boundary value problems for Fourier amplitudes.
- Analysis of solution branches organized by saddle-node bifurcations.
- Investigation of intertwined solution branches, including those with and without defects and exhibiting collapsed snaking behavior.
Main Results:
- Identified spatiotemporal Turing-Hopf pinning solutions with coexisting stationary stripes and time-periodic oscillations.
- Observed solution branches organized by saddle-node bifurcations, forming snaking structures similar to stationary solutions.
- Found two intertwined pairs of solution branches, differing by a phase shift and connected to collapsed snaking branches.
Conclusions:
- Spatiotemporal Turing-Hopf pinning solutions in the 1D Brusselator model exhibit complex organization and dynamics.
- The study contrasts these nonvariational solutions with those found in variational systems like the Swift-Hohenberg equations.
- Detailed analysis of depinning dynamics, wavelength variation, and comparison with amplitude equation predictions provide insights into pattern stability and evolution.
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