19 research outputs found
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Existence and stability of multiple spot solutions for the gray-scott model in R^2
We study the Gray-Scott model in a bounded two dimensional domain and establish the existence and stability of {\bf symmetric} and {\bf asymmetric} multiple spotty patterns. The Green's function and its derivatives
together with two nonlocal eigenvalue problems
both play a major role in the analysis.
For symmetric spots, we establish a threshold behavior for stability:
If a certain inequality for the parameters holds
then we get stability, otherwise we get instability of multiple spot solutions.
For asymmetric spots, we show that they can be stable within a narrow parameter range
Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator
A generalized hyperbolic perturbation method for heteroclinic solutions is presented for strongly nonlinear self-excited oscillators in the more general form of xā
ā
+g(x)=Éf(Ī¼,x,xā
)xĀ·Ā·+g(x)=Éf(Ī¼,x,xĀ·). The advantage of this work is that heteroclinic solutions for more complicated and strong nonlinearities can be analytically derived, and the previous hyperbolic perturbation solutions for Duffing type oscillator can be just regarded as a special case of the present method. The applications to cases with quadratic-cubic nonlinearities and with quintic-septic nonlinearities are presented. Comparisons with other methods are performed to assess the effectiveness of the present method.postprin
On a Hypercycle System with Nonlinear Rate
We study an (N+1)-hypercyclical reaction-diffusion system with nonlinear reaction rate n.
It is shown that there exists a critical threshold N_0 such that for N\leq N_0 the system is stable while
for N> N_0 it becomes unstable. It is also shown that for large reaction rate
n, N_0 remains a constant: in fact for n \geq n_0 \sim 3.35, N_0=5 and for n < n_0 \sim 3.35,
N_0=4. Some more general reaction-diffusion systems of N+1 equations are also considered
Rigorous numerics for symmetric connecting orbits: Even homoclinics of the Gray-Scott equation
In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray-Scott equation
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Critical Threshold and Stability of Cluster Solutions for Large Reaction-Diffusion Systems in R
We study a large reaction-diffusion system which arises in
the modeling of catalytic networks and describes the emerging of cluster states.
We construct single cluster solutions on the real line
and then establish their stability or instability in terms of the number N of components and the connection matrix.
We provide a rigorous analysis around the single cluster solutions, which is new for systems of this kind.
Our results show that for N\leq 4 the hypercycle system is linearly stable while
for N\geq 5
the hypercycle system is linearly unstable
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Existence and stability of multiple spot solutions for the Gray-Scott model in R^2$
Existence and Stability of Multiple Spot Solutions for the Gray-Scott Model in In this paper, we rigorously
prove the existence and stability of multiple spot patterns for the Gray-Scott system in a two dimensional domain
which are far from
spatial homogeneity.
The Green's function and its derivatives
together with two nonlocal eigenvalue problems
both play a major role in the analysis.
We establish a threshold behavior for stability:
If a certain inequality for the parameters holds
then we get stability, otherwise we get instability of multiple spot solutions.
The exact asymptotics of the critical thresholds are obtained
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Asymmetric spotty patterns for the Gray-Scott model in R^2
In this paper, we rigorously
prove the existence and stability of asymmetric spotty patterns for the Gray-Scott model in a bounded two dimensional domain.
We show that given any two positive integers k_1,\,k_2,
there are asymmetric solutions with k_1 large spots (type A) and k_2 small spots (type B).
We also give conditions for their location and calculate their heights.
Most of these asymmetric solutions are shown
to be unstable. However, in a narrow range of parameters,
asymmetric solutions may be stable