Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2503.05736

Data-driven continuation of patterns and their bifurcations

Wenjun Zhao·Samuel Maffa·Björn Sandstede
View original source ↗Read source document ↗Export metadata ↓

Abstract

Patterns and nonlinear waves, such as spots, stripes, and rotating spirals, arise prominently in many natural processes and in reaction-diffusion models. Our goal is to compute boundaries between parameter regions with different prevailing patterns and waves. We accomplish this by evolving randomized initial data to full patterns and evaluate feature functions, such as the number of connected components or their area distribution, on their sublevel sets. The resulting probability measure on the feature space, which we refer to as pattern statistics, can then be compared at different parameter values using the Wasserstein distance. We show that arclength predictor-corrector continuation can be used to trace out transition and bifurcation curves in parameter space by maximizing the distance of the pattern statistics. The utility of this approach is demonstrated through a range of examples involving homogeneous states, spots, stripes, and spiral waves.

Explore related subjects

nlin.PSmath.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Wenjun Zhao, Samuel Maffa, Björn Sandstede. 2025-02-19. Data-driven continuation of patterns and their bifurcations. https://arxiv.org/abs/2503.05736

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Duck hunting with quantum mechanics

We bridge two sides of singular perturbation theory: the classical theory of slow-fast systems and the semi-classical approach to quantum mechanical systems. For a specific but physically important class of dynamical systems, we show that purely classical and exotic objects, so-called canard solutions, are shadows of instantons in the corresponding quantum system. We demonstrate that canard solutions exist in a domain of parameter space whose boundaries are determined by an instanton action. We illustrate our statements analytically for the relevant example, the overdamped Josephson junction, and confirm them numerically. For the Josephson junction, the canard window is the exponentially narrow gap between consecutive Shapiro steps.

nlin.PS↗
arXiv2026

Rolls and Snaking in a Swift-Hohenberg Equation with Non-smooth Nonlinearity

We study rolls and homoclinic snaking in a variation of the one-dimensional Swift-Hohenberg equation, whose standard forms are prototypical order-parameter models for pattern formation in the sciences. Motivated by classes of differential equation models that involve continuous non-smooth low order nonlinear terms, we replace the standard quadratic-cubic nonlinearity by $ν|u|^α-u^3$, $α\in [1,2]$ with $ν> 0$. In the vicinity of zero, for $α<2$ this nonlinearity falls outside the scope of classical Taylor expansion and bifurcation analysis. Our partially analytical and partially numerical results highlight that the non-smooth term modifies the criticality of pattern-forming bifurcations and alters the associated branches of solutions. In particular, $α\in(1,2)$ implies subcriticality of roll bifurcations for any $ν>0$. At $α=1$ differentiability is lost, which has a strong impact on the bifurcations of sign-changing rolls including the disappearance of homoclinic snaking. Homoclinic snaking thus emerges non-smoothly as $α$ increases from $α=1$, and persists when retaining an additional, e.g., quadratic term.

nlin.PS↗