skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Homoclinic snaking of contact defects in reaction-diffusion equations
Abstract We apply spatial dynamical-systems techniques to prove that certain spatiotemporal patterns in reversible reaction-diffusion equations undergo snaking bifurcations. That is, in a narrow region of parameter space, countably many branches of patterned states coexist that connect at towers of saddle-node bifurcations. Our patterns of interest are contact defects, which are one-dimensional time-periodic patterns with a spatially oscillating core region that at large distances from the origin in space resemble pure temporally oscillatory states and arise as natural analogues of spiral and target waves in one spatial dimension. We show that these solutions lie on snaking branches that have a more complex structure than has been seen in other contexts. In particular, we predict the existence of families of asymmetric traveling defect solutions with arbitrary background phase offsets, in addition to symmetric standing target and spiral patterns. We prove the presence of these additional patterns by reconciling results in classic ordinary differential equation studies with results from the spatial-dynamics study of patterns in partial differential equations and using geometrical information contained in the stable and unstable manifolds of the background wave trains and their natural equivariance structure.  more » « less
Award ID(s):
2038039 2106566
PAR ID:
10646077
Author(s) / Creator(s):
;
Publisher / Repository:
IOP Publishing
Date Published:
Journal Name:
Nonlinearity
Volume:
38
Issue:
11
ISSN:
0951-7715
Format(s):
Medium: X Size: Article No. 115001
Size(s):
Article No. 115001
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, IMA Journal of Applied Mathematics)
    null (Ed.)
    Abstract We study the structure of stationary patterns in bistable lattice dynamical systems posed on rings with a symmetric coupling structure in the regime of small coupling strength. We show that sparse coupling (for instance, nearest-neighbour or next-nearest-neighbour coupling) and all-to-all coupling lead to significantly different solution branches. In particular, sparse coupling leads to snaking branches with many saddle-node bifurcations, while all-to-all coupling leads to branches with six saddle nodes, regardless of the size of the number of nodes in the graph. 
    more » « less
  2. ; ; (, The ANZIAM Journal)
    Abstract For multi-scale differential equations (or fast–slow equations), one often encounters problems in which a key system parameter slowly passes through a bifurcation. In this article, we show that a pair of prototypical reaction–diffusion equations in two space dimensions can exhibit delayed Hopf bifurcations. Solutions that approach attracting/stable states before the instantaneous Hopf point stay near these states for long, spatially dependent times after these states have become repelling/unstable. We use the complex Ginzburg–Landau equation and the Brusselator models as prototypes. We show that there exist two-dimensional spatio-temporal buffer surfaces and memory surfaces in the three-dimensional space-time. We derive asymptotic formulas for them for the complex Ginzburg–Landau equation and show numerically that they exist also for the Brusselator model. At each point in the domain, these surfaces determine how long the delay in the loss of stability lasts, that is, to leading order when the spatially dependent onset of the post-Hopf oscillations occurs. Also, the onset of the oscillations in these partial differential equations is a hard onset. 
    more » « less
  3. ; ; ; ; ; (, IMA Journal of Applied Mathematics)
    null (Ed.)
    Abstract The passive conserved Swift–Hohenberg equation (or phase-field-crystal [PFC] model) describes gradient dynamics of a single-order parameter field related to density. It provides a simple microscopic description of the thermodynamic transition between liquid and crystalline states. In addition to spatially extended periodic structures, the model describes a large variety of steady spatially localized structures. In appropriate bifurcation diagrams the corresponding solution branches exhibit characteristic slanted homoclinic snaking. In an active PFC model, encoding for instance the active motion of self-propelled colloidal particles, the gradient dynamics structure is broken by a coupling between density and an additional polarization field. Then, resting and traveling localized states are found with transitions characterized by parity-breaking drift bifurcations. Here, we briefly review the snaking behavior of localized states in passive and active PFC models before discussing the bifurcation behavior of localized states in systems of (i) two coupled passive PFC models with common gradient dynamics, (ii) two coupled passive PFC models where the coupling breaks the gradient dynamics structure and (iii) a passive PFC model coupled to an active PFC model. 
    more » « less
  4. ; (, IMA Journal of Applied Mathematics)
    null (Ed.)
    Abstract An activator–inhibitor–substrate model of side branching used in the context of pulmonary vascular and lung development is considered on the supposition that spatially localized concentrations of the activator trigger local side branching. The model consists of four coupled reaction–diffusion equations, and its steady localized solutions therefore obey an eight-dimensional spatial dynamical system in one spatial dimension (1D). Stationary localized structures within the model are found to be associated with a subcritical Turing instability and organized within a distinct type of foliated snaking bifurcation structure. This behavior is in turn associated with the presence of an exchange point in parameter space at which the complex leading spatial eigenvalues of the uniform concentration state are overtaken by a pair of real eigenvalues; this point plays the role of a Belyakov–Devaney point in this system. The primary foliated snaking structure consists of periodic spike or peak trains with $$N$$ identical equidistant peaks, $$N=1,2,\dots \,$$, together with cross-links consisting of nonidentical, nonequidistant peaks. The structure is complicated by a multitude of multipulse states, some of which are also computed, and spans the parameter range from the primary Turing bifurcation all the way to the fold of the $N=1$ state. These states form a complex template from which localized physical structures develop in the transverse direction in 2D. 
    more » « less
  5. ; ; ; (, Bulletin of Mathematical Biology)
    Abstract We analyze a spatially extended version of a well-known model of forest-savanna dynamics, which presents as a system of nonlinear partial integro-differential equations, and study necessary conditions for pattern-forming bifurcations. Homogeneous solutions dominate the dynamics of the standard forest-savanna model, regardless of the length scales of the various spatial processes considered. However, several different pattern-forming scenarios are possible upon including spatial resource limitation, such as competition for water, soil nutrients, or herbivory effects. Using numerical simulations and continuation, we study the nature of the resulting patterns as a function of system parameters and length scales, uncovering subcritical pattern-forming bifurcations and observing significant regions of multistability for realistic parameter regimes. Finally, we discuss our results in the context of extant savanna-forest modeling efforts and highlight ongoing challenges in building a unifying mathematical model for savannas across different rainfall levels. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email