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SIAM J. on Applied Dynamical Systems

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2006

Volume 5, Issue 4, pp. 529-807


Phase Boundaries as Electrically Induced Phosphenes

Jonathan D. Drover and G. Bard Ermentrout

SIAM J. Appl. Dyn. Syst. 5, pp. 529-551 (23 pages) | Cited 1 time

Online Publication Date: October 04, 2006

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A model is presented of experiments where electrical stimulation of the eye of human subjects results in the perception of evenly spaced lines, or phosphenes. The model is a two‐dimensional grid of integrate‐and‐fire oscillators that captures the important experimental characteristics of line‐creation when a sinusoidal current injection is used. The spatio‐temporal behavior of the lines, once formed, is also reproduced. A reduced model consisting of an evolution/convolution equation on the real line is analyzed, and it is shown that stationary solutions with arbitrarily located discontinuities exist and are linearly stable. Traveling waves are numerically shown to exist when the coupling is both sufficiently strong and biased, which accounts for the movement of the lines in the experiments.

Existence and Wandering of Bumps in a Spiking Neural Network Model

Carson C. Chow and S. Coombes

SIAM J. Appl. Dyn. Syst. 5, pp. 552-574 (23 pages) | Cited 6 times

Online Publication Date: November 03, 2006

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We study spatially localized states of a spiking neuronal network populated by a pulse‐coupled phase oscillator known as the lighthouse model. We show that in the limit of slow synaptic interactions in the continuum limit the dynamics reduce to those of the standard Amari model. For nonslow synaptic connections we are able to go beyond the standard firing rate analysis of localized solutions, allowing us to explicitly construct a family of coexisting one‐bump solutions and then track bump width and firing pattern as a function of system parameters. We also present an analysis of the model on a discrete lattice. We show that multiple width bump states can coexist, and uncover a mechanism for bump wandering linked to the speed of synaptic processing. Moreover, beyond a wandering transition point we show that the bump undergoes an effective random walk with a diffusion coefficient that scales exponentially with the rate of synaptic processing and linearly with the lattice spacing.

Pulses in Nonlinearly Coupled Schrödinger Equations I. A Homoclinic Flip Bifurcation

Russell K. Jackson

SIAM J. Appl. Dyn. Syst. 5, pp. 575-597 (23 pages) | Cited 1 time

Online Publication Date: November 14, 2006

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In this work, we describe a new mechanism for the generation of multipulse solutions in a class of nonlinearly coupled Schrödinger equations. Many novel pulses have been observed in such systems both numerically and experimentally, but, until now, an understanding of their origins has been lacking. The particular bifurcation studied here is spurred by the passage through degeneracy of a one‐component pulse in orbit‐flip configuration. We provide a straightforward geometric analysis, demonstrating the production not only of a multicomponent 1–pulse nearby the original one‐component pulse, but also of an entire family of alternating $N$‐pulses, for all positive integers $N$.

Three Is a Crowd: Solitary Waves in Photorefractive Media with Three Potential Wells

Todd Kapitula, P. G. Kevrekidis, and Zhigang Chen

SIAM J. Appl. Dyn. Syst. 5, pp. 598-633 (36 pages) | Cited 12 times

Online Publication Date: November 14, 2006

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In this paper we analytically, numerically, and experimentally study the dynamics of waves in photorefractive media in the presence of a potential with three wells. The results contained herein are also immediately applicable to the study of Bose–Einstein condensates in the weak interaction limit. Motivated by the recent theoretical and experimental efforts in the case of two wells, we systematically analyze the ways in which the bifurcation analysis of steady states and the stability picture are modified in the presence of a third potential well. In particular, it is shown that the presence of a third well causes all bifurcations to be of saddle‐node type. Our analytical results are based on a Lyapunov–Schmidt reduction in the modes of the underlying linear problem. We corroborate the analytical predictions with numerical results which are based on fixed point methods. Finally, we illustrate how these findings may be related to experimental observations obtained in strontium‐barium‐niobate crystals.

Geometric Relations of Absolute and Essential Spectra of Wave Trains

Jens D. M. Rademacher

SIAM J. Appl. Dyn. Syst. 5, pp. 634-649 (16 pages) | Cited 2 times

Online Publication Date: November 22, 2006

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We analyze geometric relations of absolute and essential spectra for certain linear operators on the real line with periodic coefficients. These spectra correspond to accumulation sets of eigenvalues for increasing domain length under separated and periodic boundary conditions, respectively. The main result shows that critical isolated sets of essential spectra contain absolute spectra and yields an algorithm for its numerical computation. Linearizations of reaction diffusion systems in wave trains are used as an illustration, and we present a detailed numerical study of absolute and essential spectra for a wave train in the Schnakenberg model.

Motion Planning for an Articulated Body in a Perfect Planar Fluid

Juan B. Melli, Clarence W. Rowley, and Dzhelil S. Rufat

SIAM J. Appl. Dyn. Syst. 5, pp. 650-669 (20 pages) | Cited 13 times

Online Publication Date: November 22, 2006

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Understanding fish‐like locomotion as a result of internal shape changes may result in improved underwater propulsion mechanisms. We use a geometric framework to consider the simplified problem of an articulated two‐dimensional body in a potential flow. This paper builds upon the current geometric theory by showing that although the group of Euclidean transformations is non‐Abelian, certain tools available for Abelian groups may still be exploited, making use of the semidirect‐product structure of this group. In particular, the holonomy in the rotation component may be explicitly computed as a function of the area enclosed by a path in shape space. We use this tool to develop open‐loop gaits for an articulated body with two shape variables, using plots of the curvature of the mechanical connection, which relates motion in the shape space to motion of the overall body. Results from numerical computations of the mechanical connection are compared to theoretical results assuming the joints are hydrodynamically decoupled. Finally, we consider a simple method for trajectory tracking in the plane, using a one‐parameter family of gaits.

Neural Fields with Distributed Transmission Speeds and Long‐Range Feedback Delays

Fatihcan M. Atay and Axel Hutt

SIAM J. Appl. Dyn. Syst. 5, pp. 670-698 (29 pages) | Cited 15 times

Online Publication Date: December 01, 2006

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We introduce distributed axonal transmission speeds and a long‐range constant feedback loop into the standard neural field model. We analyze the stability of spatially homogeneous equilibrium solutions for general connectivity kernels. By studying reduced models based on the assumption of small delays, we determine the effects of the delays on the stability and bifurcations. We show in a reduced model that delayed excitatory feedback generally facilitates stationary bifurcations and Turing patterns, while suppressing the bifurcation of periodic solutions and traveling waves. The reverse conclusion holds for inhibitory feedback. In case of oscillatory bifurcations, the variance of the distributed propagation and feedback delays affects the frequency of periodic solutions and the phase speed of traveling waves. Moreover, we give a nonlinear analysis of traveling fronts and find that distributed transmission speeds can maximize the front speed.

Delayed‐Mutual Coupling Dynamics of Lasers: Scaling Laws and Resonances

T. W. Carr, I. B. Schwartz, Min‐Young Kim, and Rajarshi Roy

SIAM J. Appl. Dyn. Syst. 5, pp. 699-725 (27 pages) | Cited 5 times

Online Publication Date: December 05, 2006

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We consider a model for two lasers that are mutually coupled optoelectronically by modulating the pump of one laser with the intensity deviations of the other. Signal propagation time in the optoelectronic loop causes a significant delay leading to the onset of oscillatory output. Multiscale perturbation methods are used to describe the amplitude and period of oscillations as a function of the coupling strength and delay time. For weak coupling the oscillations have the laser’s relaxation period, and the amplitude varies as the one‐fourth power of the parameter deviations from the bifurcation point. For order‐one coupling strength the period is determined as multiples of the delay time, and the amplitude varies with a square‐root power law. Because we allow for independent control of the individual coupling constants, for certain parameter values there is an atypical amplitude‐resonance phenomena. Finally, our theoretical results are consistent with recent experimental observations when the inclusion of a low‐pass filter in the coupling loop is taken into account.

The Kelvin–Helmholtz Instability of Momentum Sheets in the Euler Equations for Planar Diffeomorphisms

Robert I. McLachlan and Stephen R. Marsland

SIAM J. Appl. Dyn. Syst. 5, pp. 726-758 (33 pages) | Cited 2 times

Online Publication Date: December 05, 2006

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The Euler equations that describe geodesics on the group of diffeomorphisms of the plane admit singular solutions in which the momentum is concentrated on curves, the so‐called momentum sheets analogous to vortex sheets in the Euler fluid equations. We study the stability of straight and circular momentum sheets for a large family of metrics. We prove that straight sheets moving normally to themselves under an $H^1$ metric, corresponding to peakons for the one‐dimensional (1D) Camassa–Holm equation, are linearly stable in Eulerian coordinates, suffering only a weak instability of Lagrangian particle paths, while most other cases are unstable but well‐posed. Expanding circular sheets are algebraically unstable for all metrics. The evolution of the instabilities are followed numerically, illustrating several typical dynamical phenomena of momentum sheets.

Transition from Rotating Waves to Modulated Rotating Waves on the Sphere

Adela N. Comanici

SIAM J. Appl. Dyn. Syst. 5, pp. 759-782 (24 pages) | Cited 2 times

Online Publication Date: December 05, 2006

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In this article, we consider parameter‐dependent systems of reaction‐diffusion equations on the sphere, which are equivariant under the group $SO(3)$ of all rigid rotations on the sphere. It is known that the transition from rotating waves to modulated rotating waves on the sphere can be explained via a supercritical Hopf bifurcation from a rotating wave, $SO(3)$‐symmetry, and finite‐dimensional equivariant center manifold reduction. Using Floquet theory, it is easy to get the decomposition of these modulated rotating waves into the primary frequency vector part and the periodic part. Going further, we use the Baker–Campbell–Hausdorff (BCH) formula in the Lie algebra $so(3)$ to get the closed form of the reduced differential equations on $so(3)$, and then closed formulas for the primary frequency vectors and for the periodic parts associated to the bifurcating modulated rotating waves. As a consequence, we get the explicit characterization of these modulated rotating waves and all possible tip motions on the sphere: quasi‐periodically meandering and slowly drifting. This approach does not treat separately the resonant and nonresonant Hopf bifurcation of a rotating wave on the sphere; the change that appears in the resonant case in the Taylor expansions for the primary frequency vectors is implicitly captured by using the BCH formula in $so(3)$, and it is independent of the normal forms theory developed in [B. Fiedler and D. Turaev, Arch. Ration. Mech. Anal., 145 (1998), pp. 129–159]. When systems with two parameters are involved and the norm of the frequency vector of the initial rotating wave undergoing Hopf bifurcation is a multiple integer of the critical eigenvalue leading to Hopf bifurcation, we give a shorter and more intuitive proof for the following fact stated in [C. Wulff, Doc. Math., 5 (2000), pp. 227–274] for a general Lie group: the primary frequency vectors of a branch of these modulated rotating waves are generically orthogonal to the frequency vector of the initial rotating wave undergoing Hopf bifurcation, and their tip motions are slowly drifting along the equator of the sphere. Due to the computational nature of the BCH formula in $so(3)$, this approach can be translated in a computer‐implemented method which will allow us to better control the tip motions of the modulated rotating waves on the sphere.

Wave Radiation by Balanced Motion in a Simple Model

J. Vanneste

SIAM J. Appl. Dyn. Syst. 5, pp. 783-807 (25 pages) | Cited 2 times

Online Publication Date: December 26, 2006

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We introduce and study a toy model which captures some essential features of wave radiation by slow (or balanced) motion in the atmosphere and the ocean. Inspired by the widely studied five‐component model due to Lorenz, the model describes the coupling of a nonlinear pendulum with linear waves. The waves obey a one‐dimensional linear Klein–Gordon equation, so their dispersion relation is identical to that of inertia‐gravity waves in a rotating shallow‐water fluid. The model is Hamiltonian. We examine two physically relevant asymptotic regimes in which there is some time‐scale separation between the slow pendulum motion and the fast waves: in regime (i), the time‐scale separation breaks down for waves with asymptotically large wavelengths; in regime (ii), the time‐scale separation holds for all wavelengths. We study the generation of waves in each regime using distinct asymptotic methods. In regime (i), long waves are excited resonantly in a manner that is analogous to the Lighthill radiation of sound waves in weakly compressible flows, and to the radiation of gravitational waves by slow mass motion in general relativity. Matched asymptotics provides the functional form of the waves radiated, and leads, at higher order, to a closed model describing the pendulum dynamics while accounting for the dissipative effect of wave radiation. In regime (ii), an exponentially accurate slow manifold can be defined, and the waves radiated are exponentially small. They are captured using an exponential‐asymptotic technique combining complex‐time matching with Borel summation. The asymptotic results obtained in each regime are tested against numerical simulations of the model.
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