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SIAM J. on Applied Dynamical Systems

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2006

Volume 5, Issue 3, pp. 365-527


Accurately Model the Kuramoto--Sivashinsky Dynamics with Holistic Discretization

T. MacKenzie and A. J. Roberts

SIAM J. Appl. Dyn. Syst. 5, pp. 365-402 (38 pages) | Cited 2 times

Online Publication Date: August 07, 2006

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We analyze the nonlinear Kuramoto--Sivashinsky equation to develop accurate discretizations modeling its dynamics on coarse grids. The analysis is based upon center manifold theory, so we are assured that the discretization accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing isolating internal boundaries which are later removed. Comprehensive numerical solutions and simulations show that the holistic discretizations excellently reproduce the steady states and the dynamics of the Kuramoto--Sivashinsky equation. The Kuramoto--Sivashinsky equation is used as an example to show how holistic discretization may be successfully applied to fourth-order, nonlinear, spatio-temporal dynamical systems. This novel center manifold approach is holistic in the sense that it treats the dynamical equations as a whole, not just as the sum of separate terms.

The Moment Map: Nonlinear Dynamics of Density Evolution via a Few Moments

D. Barkley, I. G. Kevrekidis, and A. M. Stuart

SIAM J. Appl. Dyn. Syst. 5, pp. 403-434 (32 pages) | Cited 1 time

Online Publication Date: August 07, 2006

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We explore situations in which certain stochastic and high-dimensional deterministic systems behave effectively as low-dimensional dynamical systems. We define and study moment maps, maps on spaces of low-order moments of evolving distributions, as a means of understanding equation-free multiscale algorithms for these systems. The moment map itself is deterministic and attempts to capture the implied probability distribution of the dynamics. By choosing situations where the low-dimensional dynamics can be understood a priori, we evaluate the moment map. Despite requiring the evolution of an ensemble to define the map, this can be an efficient numerical tool, as the map opens up the possibility of bifurcation analyses and other high level tasks being performed on the system. We demonstrate how nonlinearity arises in these maps and how this results in the stabilization of metastable states. Examples are shown for a hierarchy of models, ranging from simple stochastic differential equations to molecular dynamics simulations of a particle in contact with a heat bath.

Numerical Continuation of Symmetric Periodic Orbits

Claudia Wulff and Andreas Schebesch

SIAM J. Appl. Dyn. Syst. 5, pp. 435-475 (41 pages) | Cited 7 times

Online Publication Date: August 07, 2006

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The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for symmetric dynamical systems. However, there are hardly any results on the numerical computation of those bifurcations yet. In this paper we show how spatio-temporal symmetries of periodic orbits can be exploited numerically. We describe methods for the computation of symmetry breaking bifurcations of periodic orbits for free group actions and show how bifurcations increasing the spatio-temporal symmetry of periodic orbits (including period halving bifurcations and equivariant Hopf bifurcations) can be detected and computed numerically. Our pathfollowing algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization.

Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region

John M. Neuberger, Nándor Sieben, and James W. Swift

SIAM J. Appl. Dyn. Syst. 5, pp. 476-507 (32 pages)

Online Publication Date: September 21, 2006

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We apply the gradient Newton–Galerkin algorithm (GNGA) of Neuberger and Swift to find solutions to a semilinear elliptic Dirichlet problem on the region whose boundary is the Koch snowflake. In a recent paper, we described an accurate and efficient method for generating a basis of eigenfunctions of the Laplacian on this region. In that work, we used the symmetry of the snowflake region to analyze and postprocess the basis, rendering it suitable for input to the GNGA. The GNGA uses Newton’s method on the eigenfunction expansion coefficients to find solutions to the semilinear problem. This article introduces the bifurcation digraph, an extension of the lattice of isotropy subgroups. For our example, the bifurcation digraph shows the 23 possible symmetry types of solutions to the PDE, and the 59 generic symmetry‐breaking bifurcations among these symmetry types. Our numerical code uses continuation methods and follows branches created at symmetry‐breaking bifurcations, and so the human user does not need to supply initial guesses for Newton’s method. Starting from the known trivial solution, the code automatically finds at least one solution with each of the symmetry types that we predict can exist. Such computationally intensive investigations necessitated the writing of automated branch following code, whereby symmetry information was used to reduce the number of computations per GNGA execution and to make intelligent branch following decisions at bifurcation points.

Stability of Coupled Map Networks with Delays

Fatihcan M. Atay and Özkan Karabacak

SIAM J. Appl. Dyn. Syst. 5, pp. 508-527 (20 pages) | Cited 10 times

Online Publication Date: September 26, 2006

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We consider networks of coupled scalar maps, with weighted connections which may include a time delay, and study the stability of equilibria with respect to the delays and connection structure. We prove that the largest eigenvalue of the graph Laplacian determines the effect of the connection topology on stability. The stability region in the parameter plane shrinks with increasing values of the largest eigenvalue, or of the time delay of the same parity. In particular, all bipartite graphs have an identical stability region, regardless of the delay or graph size, which is also the smallest stability region among those of all graphs. Furthermore, for certain parameter ranges, unstable (and possibly chaotic) maps can be stabilized via diffusive coupling with an odd time delay, provided that the network does not have a nontrivial and connected bipartite component. On the other hand, stabilization is not possible for even values of the delay or for bipartite networks.
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