9 research outputs found
Butterfly catastrophe for fronts in a three-component reaction-diffusion system
Abstract We study the dynamics of front solutions in a three-component reaction-diffusion system via a combination of geometric singular perturbation theory, Evans function analysis and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically
Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system
This manuscript extends the analysis of a much studied singularly perturbed
three-component reaction-diffusion system for front dynamics in the regime
where the essential spectrum is close to the origin. We confirm a conjecture
from a preceding paper by proving that the triple multiplicity of the zero
eigenvalue gives a Jordan chain of length three. Moreover, we simplify the
center manifold reduction and computation of the normal form coefficients by
using the Evans function for the eigenvalues. Finally, we prove the unfolding
of a Bogdanov-Takens bifurcation with symmetry in the model. This leads to
stable periodic front motion, including stable traveling breathers, and these
results are illustrated by numerical computations.Comment: 39 pages, 7 figure
Traveling pulse solutions in a three-component FitzHugh-Nagumo Model
We use geometric singular perturbation techniques combined with an action
functional approach to study traveling pulse solutions in a three-component
FitzHugh--Nagumo model. First, we derive the profile of traveling -pulse
solutions with undetermined width and propagating speed. Next, we compute the
associated action functional for this profile from which we derive the
conditions for existence and a saddle-node bifurcation as the zeros of the
action functional and its derivatives. We obtain the same conditions by using a
different analytical approach that exploits the singular limit of the problem.
We also apply this methodology of the action functional to the problem for
traveling -pulse solutions and derive the explicit conditions for existence
and a saddle-node bifurcation. From these we deduce a necessary condition for
the existence of traveling -pulse solutions. We end this article with a
discussion related to Hopf bifurcations near the saddle-node bifurcation
Butterfly catastrophe for fronts in a three-component reaction-diffusion system
We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically
A geometric approach to stationary defect solutions in one space dimension
Analysis and Stochastic