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SIAM J. on Applied Dynamical Systems

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Volume 1

Issue 2 | 2002 | pp. 175-302

Issue 1 | 2002 | pp. 1-174

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2002

Volume 1, Issue 2, pp. 175-302

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Select this Article FREE		Chaotic Synchronization in Coupled Map Lattices with Periodic Boundary Conditions Wen-Wei Lin and Yi-Qian Wang SIAM J. Appl. Dyn. Syst. 1, pp. 175-189 (15 pages) \| Cited 4 times Online Publication Date: August 07, 2006 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, we consider a lattice of the coupled logistic map with periodic boundary conditions. We prove that synchronization occurs in the one-dimensional lattice with lattice size n=4 for any $\gamma$ in the chaotic regime $[\gamma_{\infty}\approx 3.57, 4]$. It is worthwhile to emphasize that, despite of the fact that there is a rigorous proof for synchronization in many systems with continuous time, almost nothing is rigorously proved for the systems with discrete time.

Select this Article FREE		A Multiparameter, Numerical Stability Analysis of a Standing Cantilever Conveying Fluid Nawaf M. Bou-Rabee, Louis A. Romero, and Andrew G. Salinger SIAM J. Appl. Dyn. Syst. 1, pp. 190-214 (25 pages) \| Cited 3 times Online Publication Date: August 07, 2006 Full Text: \| Download PDF
	+ -	Show Abstract In this paper, we numerically examine the stability of a standing cantilever conveying fluid in a multiparameter space. Based on nonlinear beam theory, our mathematical model turns out to be replete with exciting behavior, some of which was totally unexpected and novel, and some of which confirm our intuition as well as the work of others. The numerical bifurcation results obtained from applying the Library of Continuation Algorithms (LOCA) reveal a plethora of one, two, and higher codimension bifurcations. For a vertical or standing cantilever beam, bifurcations to buckled solutions (via symmetry breaking) and oscillating solutions are detected as a function of gravity and the fluid-structure interaction. The unfolding of these results as a function of the orientation of the beam compared to gravity is also revealed.

Select this Article FREE		Attracting Fixed Points for the Kuramoto--Sivashinsky Equation: A Computer Assisted Proof Piotr Zgliczynski SIAM J. Appl. Dyn. Syst. 1, pp. 215-235 (21 pages) \| Cited 1 time Online Publication Date: August 07, 2006 Full Text: \| Download PDF
	+ -	Show Abstract We present a computer assisted proof of the existence of several attracting fixed points for the Kuramoto--Sivashinsky equation u_t=(u^2)_x - u_{xx} -\nu u_{xxxx}, \quad u(x,t)=u(x+2\pi,t), \quad u(x,t)=-u(-x,t), where $\nu >0$. The method is general and can be applied to other dissipative PDEs.

Select this Article FREE		Development of Standing-Wave Labyrinthine Patterns Arik Yochelis, Aric Hagberg, Ehud Meron, Anna L. Lin, and Harry L. Swinney SIAM J. Appl. Dyn. Syst. 1, pp. 236-247 (12 pages) \| Cited 17 times Online Publication Date: August 07, 2006 Full Text: \| Download PDF
	+ -	Show Abstract Experiments on a quasi-two-dimensional Belousov--Zhabotinsky (BZ) reaction-diffusion system, periodically forced at approximately twice its natural frequency, exhibit resonant labyrinthine patterns that develop through two distinct mechanisms. In both cases, large amplitude labyrinthine patterns form that consist of interpenetrating fingers of frequency-locked regions differing in phase by $\pi$. Analysis of a forced complex Ginzburg-Landau equation captures both mechanisms observed for the formation of the labyrinths in the BZ experiments: a transverse instability of front structures and a nucleation of stripes from unlocked oscillations. The labyrinths are found in the experiments and in the model at a similar location in the forcing amplitude and frequency parameter plane.

Select this Article FREE		Numerical Bifurcation Analysis for Multisection Semiconductor Lasers Jan Sieber SIAM J. Appl. Dyn. Syst. 1, pp. 248-270 (23 pages) \| Cited 10 times Online Publication Date: August 07, 2006 Full Text: \| Download PDF
	+ -	Show Abstract We investigate the dynamics of a multisection laser implementing a delayed optical feedback experiment where the length of the cavity is comparable to the length of the laser. First, we reduce the traveling-wave model with gain dispersion (a hyperbolic system of PDEs) to a system of ODEs describing the semiflow on a local center manifold. Then we analyze the dynamics of the system of ODEs using numerical continuation methods (AUTO). We explore the plane of the two parameters---feedback phase and feedback strength---to obtain a bifurcation diagram for small and moderate feedback strength. This diagram permits us to understand the roots of a variety of nonlinear phenomena observed numerically and experimentally such as, e.g., self-pulsations, excitability, hysteresis, or chaos, and to locate them in the parameter plane.

Select this Article FREE		Higher Order Modulation Equations for a Boussinesq Equation C. Eugene Wayne and J. Douglas Wright SIAM J. Appl. Dyn. Syst. 1, pp. 271-302 (32 pages) \| Cited 2 times Online Publication Date: August 07, 2006 Full Text: \| Download PDF
	+ -	Show Abstract In order to investigate corrections to the common KdV approximation to long waves, we derive modulation equations for the evolution of long wavelength initial data for a Boussinesq equation. The equations governing the corrections to the KdV approximation are explicitly solvable, and we prove estimates showing that they do indeed give a significantly better approximation than the KdV equation alone. We also present the results of numerical experiments which show that the error estimates we derive are essentially optimal.