Numerical results for snaking of patterns over patterns in some 2D Selkov-Schnakenberg Reaction-Diffusion systems (1304.1723v3)

Abstract: For a Selkov--Schnakenberg model as a prototype reaction-diffusion system on two dimensional domains we use the continuation and bifurcation software pde2path to numerically calculate branches of patterns embedded in patterns, for instance hexagons embedded in stripes and vice versa, with a planar interface between the two patterns. We use the Ginzburg-Landau reduction to approximate the locations of these branches by Maxwell points for the associated Ginzburg-Landau system. For our basic model, some but not all of these branches show a snaking behaviour in parameter space, over the given computational domains. The (numerical) non-snaking behaviour appears to be related to too narrow bistable ranges with rather small Ginzburg-Landau energy differences. This claim is illustrated by a suitable generalized model. Besides the localized patterns with planar interfaces we also give a number of examples of fully localized atterns over patterns, for instance hexagon patches embedded in radial stripes, and fully localized hexagon patches over straight stripes.

• The paper demonstrates the emergence of localized spot and stripe patterns exhibiting homoclinic snaking in bistable 2D reaction-diffusion systems.
• The paper utilizes numerical continuation with pde2path to precisely track bifurcation branches and transitions between complex pattern configurations.
• The paper applies Ginzburg-Landau reduction to identify Maxwell points, elucidating how parameter variations influence stability and pattern formation.

Analysis of Bistable Reaction-Diffusion Patterns: Numerical and Theoretical Perspectives

The paper "Localized patterns, stationary fronts, and snaking in bistable ranges of spots and stripes" investigates the complex dynamics of pattern formation in two-dimensional bistable reaction-diffusion systems. Utilizing the continuation and bifurcation software {\tt pde2path}, the authors explore various branches of solutions, including localized patterns such as spots and stripes, and the phenomenon of snaking behavior in parameter spaces. The framework primarily focuses on the Ginzburg-Landau reduction to approximate the bifurcation structures and behaviors within these reaction-diffusion systems.

Key Contributions and Findings

1. Localized Patterns and Snaking Behavior: The paper demonstrates the emergence of localized patterns such as spots integrated within stripes and vice versa in a two-dimensional domain. A particularly notable feature is the observation of homoclinic snaking, where these patterns exhibit oscillations in parameter space, reminiscent of known behaviors in simpler, one-dimensional systems like the Swift-Hohenberg equation.
2. Numerical Continuation Approach: By employing {\tt pde2path}, the authors achieve numerical continuation of pattern solutions, allowing for precise tracking and characterization of bifurcation branches between distinct patterns. This methodology unveils transitions between multi-dimensional configurations and reveals how parameters influence these dynamic behaviors.
3. Ginzburg-Landau Formalism: The Ginzburg-Landau reduction serves as the theoretical backbone of the analysis, providing a robust framework for predicting where branches of stationary fronts might occur. Through this approach, the paper identifies Maxwell points, marking transitional values in parameter space that serve as attractors for pattern transitions.
4. Comparison with Analytical Results: Extensive comparison between numerical data and theoretical insights derived from the Ginzburg-Landau reduction highlights both the strengths and limitations of current models in capturing the complexity of multi-dimensional pattern formation.

Numerical and Theoretical Insights

The numerical simulations conducted reveal intricate bifurcation diagrams with multiple branching mechanisms. The snaking phenomena observed are attributed to the subcritical bifurcation characteristics inherent to the Ginzburg-Landau system, providing a rich tapestry of dynamic states within the reaction-diffusion model. Notably, the paper achieves this exploration without the need for scaling assumptions traditionally employed in one-dimensional cases.

Moreover, theoretical considerations underscore the role of Maxwell points in homoclinic and heteroclinic connections, which predict the presence of standing fronts. These predictions are juxtaposed against empirical findings, affirming that snaking behavior emerges predominantly in parameter regions characterized by strong subcriticality.

Implications and Future Directions

This research enriches our understanding of pattern formations in applied mathematics and theoretical physics, especially in complex, higher-dimensional feedback systems. The implications extend to disciplines involving chemical reactions, biological pattern formation, and ecological systems experiencing bistable dynamics.

Future research could explore exploring the impacts of domain size and boundary conditions more extensively, as these are identified as pivotal factors influencing the dynamical behaviors of localized patterns and their stability. Additionally, incorporating more comprehensive beyond-all-order asymptotic approaches might further refine our grasp of snaking phenomena and extend these insights to broader classes of reaction-diffusion systems beyond the quadratic-cubic and cubic-quintic Swift-Hohenberg paradigms.

In conclusion, the paper provides a significant, rigorous exploration of pattern dynamics in bistable reaction-diffusion systems, offering both a numerical and theoretical lens to examine complex bifurcations and pattern formations in multidimensional spaces. Through this dual approach, it paves the way for future advancements in mathematical models that aim to describe naturally occurring instabilities and transitions more accurately.