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Discrete & Continuous Dynamical Systems - B

2009 , Volume 11 , Issue 4

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Global existence and internal stabilization for a reaction-diffusion system posed on non coincident spatial domains
Sebastian Aniţa, William Edward Fitzgibbon and Michel Langlais
2009, 11(4): 805-822 doi: 10.3934/dcdsb.2009.11.805 +[Abstract](89) +[PDF](239.8KB)
We consider a two-component Reaction-Diffusion system posed on non coincident spatial domains and featuring a reaction term involving an integral kernel. The question of global existence of componentwise nonnegative solutions is assessed. Then we investigate the stabilization of one of the solution components to zero via an internal control distributed on a small subdomain while preserving nonnegativity of both components. Our results apply to predator-prey systems.
Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease
Mostafa Bendahmane and Mauricio Sepúlveda
2009, 11(4): 823-853 doi: 10.3934/dcdsb.2009.11.823 +[Abstract](119) +[PDF](3094.2KB)
A finite-volume scheme for a nonlocal three-component reaction-diffusion system modeling an epidemic disease with susceptible, infected, and recovered, individuals is analyzed. For this SIR model, the existence of solutions to the finite volume scheme and its convergence to a weak solution of the PDE is established. The convergence proof is based on deriving a series of apriori estimates and by using a general $L^p$ compactness criterion. Finally, numerical simulations from the finite volume scheme are given.
Lifting in equation-free methods for molecular dynamics simulations of dense fluids
Yves Frederix, Giovanni Samaey, Christophe Vandekerckhove, Ting Li, Erik Nies and Dirk Roose
2009, 11(4): 855-874 doi: 10.3934/dcdsb.2009.11.855 +[Abstract](114) +[PDF](811.7KB)
Within the context of multiscale computations, equation-free methods have been developed. In this approach, the evolution of a system is simulated on the macroscopic level while only a microscopic model is explicitly available. To this end, a coarse time stepper for the macroscopic variables can be constructed, based on appropriately initialized microscopic simulations. In this paper, we investigate the initialization of the microscopic simulator using the macroscopic variables only (called lifting in the equation-free framework) when the microscopic model is a molecular dynamics (MD) description of a mono-atomic dense fluid. We assume a macroscopic model to exist in terms of the lowest order velocity moments of the particle distribution (density, velocity and temperature). The major difficulty is to design a lifting operator that accurately reconstructs the physically correct state of the fluid (i.e., the higher order moments) at a reasonable computational cost. We construct a lifting operator, as well as a restriction operator for the reverse mapping. For a simple model problem, we perform a systematic numerical study to assess the time scales on which the lifting errors disappear after reinitialization (healing); we also examine the effects on the simulated macroscopic behavior. The results show that, although in some cases accurate initialization of the higher order moments is not crucial, in general a detailed study of the lifting operator is required.
Dynamic bifurcation of the complex Swift-Hohenberg equation
Jongmin Han and Masoud Yari
2009, 11(4): 875-891 doi: 10.3934/dcdsb.2009.11.875 +[Abstract](86) +[PDF](226.3KB)
In this paper we are concerned with the dynamic bifurcation of the complex Swift-Hohenberg equation on a closed interval in $\mathbb R$. We consider the equations under the Dirichlet and the periodic boundary conditions. It is shown that the equation bifurcates from the trivial solution to an attractor when the control parameter crosses the critical value.
Relaxation oscillation profile of limit cycle in predator-prey system
Sze-Bi Hsu and Junping Shi
2009, 11(4): 893-911 doi: 10.3934/dcdsb.2009.11.893 +[Abstract](180) +[PDF](380.7KB)
It is known that some predator-prey system can possess a unique limit cycle which is globally asymptotically stable. For a prototypical predator-prey system, we show that the solution curve of the limit cycle exhibits temporal patterns of a relaxation oscillator, or a Heaviside function, when certain parameter is small.
Constrained stability and instability of polynomial difference equations with state-dependent noise
Cónall Kelly and Alexandra Rodkina
2009, 11(4): 913-933 doi: 10.3934/dcdsb.2009.11.913 +[Abstract](136) +[PDF](235.1KB)
We examine the stability and instability of solutions of a polynomial difference equation with state-dependent Gaussian perturbations, and describe a phenomenon that can only occur in discrete time. For a particular set of initial values, we find that solutions approach equilibrium asymptotically in a highly regulated fashion: monotonically and bounded above by a deterministic sequence. We observe this behaviour with a probability that can be made arbitrarily high by choosing the initial value sufficiently small.
   However, for any fixed initial value, the probability of instability is nonzero, and in fact we can show that as the magnitude of the initial value increases, the probability of instability approaches $1$.
On the behaviour at infinity of solutions to stationary convection-diffusion equation in a cylinder
Iryna Pankratova and Andrey Piatnitski
2009, 11(4): 935-970 doi: 10.3934/dcdsb.2009.11.935 +[Abstract](139) +[PDF](376.2KB)
The work focuses on the behaviour at infinity of solutions to second order elliptic equation with first order terms in a semi-infinite cylinder. Neumann's boundary condition is imposed on the lateral boundary of the cylinder and Dirichlet condition on its base. Under the assumption that the coefficients stabilize to a periodic regime, we prove the existence of a bounded solution, its stabilization to a constant, and provide necessary and sufficient condition for the uniqueness.
Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos
Xavier Perrot and Xavier Carton
2009, 11(4): 971-995 doi: 10.3934/dcdsb.2009.11.971 +[Abstract](165) +[PDF](10591.7KB)
We study the Hamiltonian system of two point vortices, embedded in external strain and rotation. This external deformation field mimics the influence of neighboring vortices or currents in complex flows. When the external field is stationary, the equilibria of the two vortices, symmetric with respect to the center of the plane, are determined. The stability analysis indicates that two saddle points lie at the crossing of separatrices, which bound streamfunction lobes having neutral centers.
   When the external field varies periodically with time, resonance becomes possible between the forcing and the oscillation of vortices around the neutral centers. A multiple time-scale expansion provides the slow-time evolution equation for these vortices, which, for weak periodic deformation, oscillate within their original (steady) trajectory. These analytical results accurately compare with numerical integration of the complete equations of motion. As the periodic deformation field increases, this vortex oscillation migrates out of the original trajectories, towards the location of the separatrices. With a periodic external field, these separatrices have given way to heteroclinic trajectories with multiple self-intersections, as shown by the calculation of the Melnikov function.
   Chaos appears in vortex trajectories as they enter the aperiodic domain around the heteroclinic curves. In fact, this chaotic domain progressively fills out the plane, replacing KAM tori and cantori, as the periodic deformation field reaches finite amplitude. The appearance of windows of periodicity is illustrated.
Linear stability analysis of systems with Preisach memory
Alexander Pimenov and Dmitrii I. Rachinskii
2009, 11(4): 997-1018 doi: 10.3934/dcdsb.2009.11.997 +[Abstract](119) +[PDF](538.0KB)
We consider differential equations coupled with the input-output memory relation defined by the Preisach operator. The differential equation relates an instant value of the rate of change of the output of the Preisach operator with an instant value of its input. We propose an algorithm for the linearisation of the evolution operator of the system and apply it to define the characteristic multiplier of periodic solutions, which determines their stability. Examples of the system considered include models of terrestrial hydrology and electronic oscillators with hysteresis.
Pseudospectral method using generalized Laguerre functions for singular problems on unbounded domains
Zhong-Qing Wang, Ben-Yu Guo and Yan-Na Wu
2009, 11(4): 1019-1038 doi: 10.3934/dcdsb.2009.11.1019 +[Abstract](141) +[PDF](291.2KB)
In this paper, we develop a pseudospectral method for differential equations defined on unbounded domains. We first introduce Gauss-type interpolations using a family of generalized Laguerre functions, and establish basic approximation results. Then we propose a pseudospectral method for differential equations on unbounded domains, whose coefficients may degenerate or grow up. As examples, we consider two model problems. The proposed schemes match the underlying problems properly and exhibit spectral accuracy. Numerical results demonstrate the efficiency of this new approach.
Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system
Daniel Wilczak
2009, 11(4): 1039-1055 doi: 10.3934/dcdsb.2009.11.1039 +[Abstract](149) +[PDF](529.9KB)
The four dimensional Rössler system is investigated. For this system the Poincaré map exhibits chaotic dynamics with two expanding directions and one strongly contracting direction. It is shown that the 16th iterate of this Poincaré map has a nontrivial invariant set on which it is semiconjugated to the full shift on two symbols. Moreover, it is proven that there exist infinitely many homoclinic and heteroclinic solutions connecting periodic orbits of period two and four, respectively. The proof utilizes the method of covering relations with smooth tools (cone conditions).
   The proof is computer assisted - interval arithmetic is used to obtain bounds of the Poincaré map and its derivative.
Error analysis of stabilized semi-implicit method of Allen-Cahn equation
Xiaofeng Yang
2009, 11(4): 1057-1070 doi: 10.3934/dcdsb.2009.11.1057 +[Abstract](278) +[PDF](569.8KB)
We consider in this paper the stabilized semi-implicit (in time) scheme and the splitting scheme for the Allen-Cahn equation $\phi_t-\Delta\phi+$ε$^-2f(\phi)=0$ arising from phase transitions in material science. For the stabilized first-order scheme, we show that it is unconditionally stable and the error bound depends on ε-1 in some lower polynomial order using the spectrum estimate of [2, 10, 11]. In addition, the first- and second-order operator splitting schemes are proposed and the accuracy are tested and compared with the semi-implicit schemes numerically.
$\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film
Jian Zhai and Zhihui Cai
2009, 11(4): 1071-1085 doi: 10.3934/dcdsb.2009.11.1071 +[Abstract](103) +[PDF](264.1KB)
In this paper, we consider the $\Gamma$-convergence of Landau-Lifshitz ferromagnetic model in the presence of Bloch wall in the disk of $\bb R^2$ with Dirichlet boundary condition.

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