[Paper Review] Dissecting the snake: the transition from localized patterns to isolated spikes in pattern formation systems

Nicolás Verschueren, Alan Champneys|Explore Bristol Research|Sep 20, 2018

Nonlinear Dynamics and Pattern Formation43 references4 citations

TL;DR

This paper investigates the transition from localized patterns with multiple peaks to isolated spikes in one-dimensional reaction-diffusion systems, showing that this shift occurs via a Belyakov-Devaney transition where spatial eigenvalues change from complex to real, destroying an infinite family of multi-pulse homoclinic orbits simultaneously at a codimension-two point. The analysis reveals that homoclinic snaking collapses into disconnected branches as the far-field decay shifts from oscillatory to monotonic, consistent with numerical simulations in a generalized Schakenberg model and an urban crime model.

ABSTRACT

An investigation is undertaken of coupled reaction-diffusion systems in one spatial dimension that are able to support, in different regions of their parameter space, either an isolated spike solution, or stable localized patterns with an arbitrary number of peaks. The distinction between the two cases is drawn through the behavior of the far field, where there is either an oscillatory or a monotonic decay. Several examples are studied, including the Lugiato-Lefever model, a generalized Schakenberg system that arises in cellular-level morphogensis and a continuum model of urban crime spread. In each, it is found that localized patterns connected via a so-called homoclinic snaking curve in parameter space transition into a single spike solution as a second parameter is varied, via a change in topology of the snake into a series of disconnected branches. In each case, the transition is caused by a so-called Belyakov-Devaney transition between complex and real spatial eigenvalues of the fair field of the primary pulse. A codimension-two problem is studied in detail where a non-transverse homoclinic orbit undergoes this transition. A Shilnikov-style analysis is undertaken which reveals the asymptotics of how the infinite family of folds of multi-pulse orbits are all destroyed at the same parameter value. The results are shown to be consistent with numerical experiments on two of the examples.

Motivation & Objective

• To understand the mechanism by which localized patterns with multiple peaks transition into isolated spike solutions in one-dimensional reaction-diffusion systems.
• To analyze the role of spatial eigenvalue transitions (from complex to real) in altering the far-field decay behavior from oscillatory to monotonic.
• To investigate the destruction of an infinite family of multi-pulse homoclinic orbits at a codimension-two Belyakov-Devaney point.
• To establish a connection between the topological change in homoclinic snaking curves and the loss of stability in multi-pulse solutions.
• To validate theoretical predictions through numerical experiments on two representative models: a generalized Schakenberg system and a continuum urban crime model.

Proposed method

• Formulates the steady-state problem as a four-dimensional reversible system in spatial dynamics, treating x as a time-like variable.
• Applies geometric singular perturbation theory and Shilnikov-style analysis to study the behavior near a codimension-two bifurcation point involving a non-transverse homoclinic orbit.
• Analyzes the transition between complex and real spatial eigenvalues of the homogeneous equilibrium, identifying the Belyakov-Devaney condition as the critical mechanism.
• Derives asymptotic expressions for the fold locations of multi-pulse orbits, showing they accumulate exponentially near the primary pulse fold as ε → 0.
• Uses numerical continuation to verify the theoretical predictions on the collapse of the homoclinic snaking structure into disconnected branches.
• Compares results across two physical models: a reaction-diffusion system modeling cellular morphogenesis and a model of urban crime spread.

Experimental results

Research questions

• RQ1 How does the transition from oscillatory to monotonic far-field decay affect the structure of homoclinic snaking in localized pattern systems?
• RQ2 What is the role of the Belyakov-Devaney transition in the destruction of multi-pulse homoclinic orbits?
• RQ3 Why do infinitely many folds of multi-pulse solutions collapse simultaneously at a single parameter value?
• RQ4 How does the topology of the homoclinic snaking curve change as the system passes through the codimension-two bifurcation point?
• RQ5 To what extent do the theoretical asymptotics of fold locations match numerical computations in realistic pattern-forming systems?

Key findings

• The Belyakov-Devaney transition, where spatial eigenvalues shift from complex to real, triggers the collapse of the homoclinic snaking structure into disconnected branches.
• At the codimension-two point, all folds of the infinite family of multi-pulse homoclinic orbits are destroyed simultaneously, with their ν-values converging exponentially to that of the primary pulse as ε → 0.
• The asymptotic rate of separation of multi-pulse orbits for ν < 0 is predicted by the Shilnikov analysis and confirmed numerically in both the generalized Schakenberg and urban crime models.
• The transition is characterized by a change in far-field behavior: oscillatory decay (localized patterns) gives way to monotonic decay (isolated spikes), altering the global bifurcation structure.
• Numerical experiments confirm that the homoclinic snaking curve breaks into disconnected isolas as the system passes through the Belyakov-Devaney point, consistent with theoretical predictions.
• The mechanism is robust and observed in multiple systems, including Schnakenberg-type models with symmetry-breaking terms and the Lugiato-Lefever equation, suggesting broad applicability.