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SIAM J. on Applied Dynamical Systems

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2010

Volume 9, Issue 4, pp. 1135-1347


Dynamical Solutions of Singular Wave Equations Modeling Electrostatic MEMS

Yujin Guo

SIAM J. Appl. Dyn. Syst. 9, pp. 1135-1163 (29 pages) | Cited 1 time

Online Publication Date: October 07, 2010

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We study a fourth-order singular wave equation involving a singular nonlinear term ${\frac{\lambda}{(1-u)^2}}$ in a bounded domain of ${\mathbb{R}^N}$. This equation models a simple electrostatic microelectromechanical system (MEMS) device consisting of a thin elastic plate with boundary supported at 0 above a rigid ground plate located at 1. Here $u$ is modeled to describe the dynamical deflection of the elastic plate. When a voltage—represented here by $\lambda$—is applied, the elastic plate deflects towards the ground plate, and snap-through (quenching) may occur when it exceeds a certain critical value $\lambda^*$ (the pull-in voltage), creating a so-called pull-in instability, which greatly affects the design of many devices. For $1\leq N\leq3$, analytic results show that there exist $0<\lambda_1\leq\lambda^*<\infty$ such that for $0\leq\lambda<\lambda_1$ the elastic plate globally exists and exponentially converges to a regular steady state, while for $\lambda>\lambda^*$ the elastic plate quenches at finite time.

Adaptive Set-Oriented Computation of Topological Horseshoe Factors in Area and Volume Preserving Maps

J. D. Mireles James

SIAM J. Appl. Dyn. Syst. 9, pp. 1164-1200 (37 pages) | Cited 2 times

Online Publication Date: October 14, 2010

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We describe an automatic chaos verification scheme, based on set-oriented numerical methods, which is especially well suited to the study of area- and volume preserving diffeomorphisms. The novel feature of the scheme is an iterative algorithm for approximating connecting orbits between collections of hyperbolic fixed and periodic points with greater and greater accuracy. The algorithm is geometric rather than graph theoretic in nature and, in contrast to existing methods, does not require the computation of chain recurrent sets. We provide several example computations in dimensions two and three.

Continuation-based Computation of Global Isochrons

Hinke M. Osinga and Jeff Moehlis

SIAM J. Appl. Dyn. Syst. 9, pp. 1201-1228 (28 pages) | Cited 1 time

Online Publication Date: November 02, 2010

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Isochrons are foliations of phase space that extend the notion of phase of a stable periodic orbit to the basin of attraction of this periodic orbit. Each point in the basin of attraction lies on only one isochron, and two points on the same isochron converge to the periodic orbit with the same phase. Global isochrons, that is, isochrons extended into the full basin of attraction rather than just a neighborhood of the periodic orbit, can form remarkable foliations. For example, accumulations of all isochrons can occur in arbitrarily small regions of phase space; the limit of such an accumulation is called the phaseless set, which lies on the boundary of the basin of attraction of the periodic orbit. Since global isochrons must typically be approximated numerically, such complicated geometries are often difficult to realize for actual examples. Indeed, the computation of global isochrons can be challenging, particularly for systems with multiple time scales. We present a novel method for computing isochrons via the continuation of a two-point boundary value problem, which is particularly effective for systems with multiple time scales. We use this method to compute global isochrons for a two-dimensional reduced Hodgkin–Huxley model and illustrate that the one-dimensional isochrons for a planar multiple-time-scale system can accumulate in the interior of the basin of attraction of the periodic orbit in a way similar to two-dimensional isochrons accumulating on the boundary of a three-dimensional basin of attraction.

Observing Infinite-dimensional Dynamical Systems

Jessica Lin and William Ott

SIAM J. Appl. Dyn. Syst. 9, pp. 1229-1243 (15 pages)

Online Publication Date: November 04, 2010

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We study the extent to which properties of infinite-dimensional dynamical systems can be accurately detected by examining observations of such systems. Let $H$ be a separable Hilbert space. Let $f:H\to H$ be a map, and let $A\subset H$ be a compact set satisfying $f(A)=A$. We prove that, for almost every (in the sense of prevalence) continuous observable $\varphi:H\to\mathbb{R}^{M}$, if $f$ induces a map $\bar{f}$ satisfying $\bar{f}\circ\varphi=\varphi\circ f$ on $A$ and if this induced map has certain properties, then the observable $\varphi$ is one-to-one on $A$, and therefore the dynamics of $f$ on $A$ are topologically conjugate to those of $\bar{f}$ on $\varphi(A)$.

Oscillations toward the Singularity of Locally Rotationally Symmetric Bianchi Type IX Cosmological Models with Vlasov Matter

Simone Calogero and J. Mark Heinzle

SIAM J. Appl. Dyn. Syst. 9, pp. 1244-1262 (19 pages)

Online Publication Date: November 04, 2010

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We analyze the dynamics of a class of cosmological solutions of the Einstein–Vlasov equations. These equations describe an ensemble of collisionless particles (which represent galaxies or clusters of galaxies) that interact gravitatively through Einstein's equations of general relativity. The cosmological models we consider are spatially homogeneous, of Bianchi type IX, and locally rotationally symmetric (LRS). We prove that generic solutions exhibit an oscillatory behavior close to the singularities (the “big bang” in the past and the “big crunch” in the future); this is in contrast to the behavior of Einstein-vacuum or Einstein–Euler solutions. To establish this result we make use of dynamical systems theory; we introduce dimensionless dynamical variables that are defined on a compact state space. In this formulation the oscillatory behavior of generic solutions is represented by the existence of heteroclinic cycles on the boundary of the state space.

Uniformly Hyperbolic Attractor of the Smale–Williams Type for a Poincaré Map in the Kuznetsov System

Daniel Wilczak

SIAM J. Appl. Dyn. Syst. 9, pp. 1263-1283 (21 pages) | Cited 1 time

Online Publication Date: November 09, 2010

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We propose a general algorithm for computer assisted verification of uniform hyperbolicity for maps which exhibit a robust attractor. The method has been successfully applied to a Poincaré map for a system of coupled nonautonomous van der Pol oscillators. The model equation has been proposed by Kuznetsov [Phys. Rev. Lett., 95 (2005), paper 144101], and the attractor seems to be of the Smale–Williams type.

Model Reduction of the Nonlinear Complex Ginzburg–Landau Equation

Miloš Ilak, Shervin Bagheri, Luca Brandt, Clarence W. Rowley, and Dan S. Henningson

SIAM J. Appl. Dyn. Syst. 9, pp. 1284-1302 (19 pages)

Online Publication Date: November 09, 2010

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Reduced-order models of the nonlinear complex Ginzburg–Landau (CGL) equation are computed using a nonlinear generalization of balanced truncation. The method involves Galerkin projection of the nonlinear dynamics onto modes determined by balanced truncation of a linearized system and is compared to a standard method using projection onto proper orthogonal decomposition (POD) modes computed from snapshots of nonlinear simulations. It is found that the nonlinear reduced-order models obtained using modes from linear balanced truncation capture very well the transient dynamics of the CGL equation and outperform POD models; i.e., a higher number of POD modes than linear balancing modes is typically necessary in order to capture the dynamics of the original system correctly. In addition, we find that the performance of POD models compares well to that of balanced truncation models when the degree of nonnormality in the system, in this case determined by the streamwise extent of a disturbance amplification region, is lower. Our findings therefore indicate that the superior performance of balanced truncation compared to POD/Galerkin models in capturing the input/output dynamics of linear systems extends to the case of a nonlinear system, both for the case of significant transient growth, which represents a basic model of boundary layer instabilities, and for a limit cycle case that represents a basic model of vortex shedding past a cylinder.

Binocular Rivalry in a Competitive Neural Network with Synaptic Depression

Zachary P. Kilpatrick and Paul C. Bressloff

SIAM J. Appl. Dyn. Syst. 9, pp. 1303-1347 (45 pages) | Cited 1 time

Online Publication Date: December 02, 2010

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We study binocular rivalry in a competitive neural network with synaptic depression. In particular, we consider two coupled hypercolums within primary visual cortex (V1), representing orientation selective cells responding to either left or right eye inputs. Coupling between hypercolumns is dominated by inhibition, especially for neurons with dissimilar orientation preferences. Within hypercolumns, recurrent connectivity is excitatory for similar orientations and inhibitory for different orientations. All synaptic connections are modifiable by local synaptic depression. When the hypercolumns are driven by orthogonal oriented stimuli, it is possible to induce oscillations that are representative of binocular rivalry. We first analyze the occurrence of oscillations in a space-clamped version of the model using a fast-slow analysis, taking advantage of the fact that depression evolves much slower than population activity. We then analyze the onset of oscillations in the full spatially extended system by carrying out a piecewise smooth stability analysis of single (winner-take-all) and double (fusion) bumps within the network. Although our stability analysis takes into account only instabilities associated with real eigenvalues, it identifies points of instability that are consistent with what is found numerically. In particular, we show that, in regions of parameter space where double bumps are unstable and no single bumps exist, binocular rivalry can arise as a slow alternation between either population supporting a bump.
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