Chain Of Nonlinear Oscillators Research Articles

Localization of nonlinear excitations in curved waveguides and chains of nonlinear oscillators

It is shown that the interplay of curvature and nonlinearity in systems with finite curvature: bent waveguides, curved chains of nonlinear oscillators, etc can lead to the qualitative effects, such as symmetry breaking of the nonlinear excitations and their trapping by the bending. The finite curvature of the waveguide with infinite hard walls (Dirichlet boundary conditions) provides a stabilizing effect on otherwise unstable localized states of repelling nonlinear Schrodinger excitations. The number of quanta which the curved waveguide can bind monotonically increases when the radius of curvature decreases. In the waveguides with Neumann boundary conditions at the confining walls the curved region might manifest itself as a two-hump potential barrier with interbarrier space acting as a potential valley. A threshold character of the scattering process, i.e. transmission, trapping, or reflection of the moving nonlinear excitation passing through the bending, is demonstrated.

Noise accelerates synchronization of coupled nonlinear oscillators

For a chain of homogeneous nonlinear oscillators starting from different initial phases, a certain amount of time is required for the system to evolve to complete phase synchronization. The effect of independent noise in such a system was investigated, and an optimal noise intensity was found that minimized the average synchronization time. Both threshold noise and connection noise show similar effects. The features of the phenomenon and the underlying mechanism are discussed through the analysis of a two-unit system and the numerical studies of chains up to 30 units in length.

Systems-level modeling of neuronal circuits for leech swimming

This paper describes a mathematical model of the neuronal central pattern generator (CPG) that controls the rhythmic body motion of the swimming leech. The systems approach is employed to capture the neuronal dynamics essential for generating coordinated oscillations of cell membrane potentials by a simple CPG architecture with a minimal number of parameters. Based on input/output data from physiological experiments, dynamical components (neurons and synaptic interactions) are first modeled individually and then integrated into a chain of nonlinear oscillators to form a CPG. We show through numerical simulations that the values of a few parameters can be estimated within physiologically reasonable ranges to achieve good fit of the data with respect to the phase, amplitude, and period. This parameter estimation leads to predictions regarding the synaptic coupling strength and intrinsic period gradient along the nerve cord, the latter of which agrees qualitatively with experimental observations.

Asymptotic Resonance, Interaction of Modes and Subharmonic Bifurcation

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam β-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.

Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach

Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.

Numerical computation of travelling breathers in Klein–Gordon chains

We numerically study the existence of travelling breathers in Klein–Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions having the property of being exactly translated by p sites along the chain after a fixed propagation time T (these solutions generalize the concept of solitary waves for which p = 1 ). In the case of even on-site potentials, the existence of small amplitude travelling breathers superposed on a small oscillatory tail has been proved recently [G. James, Y. Sire, Travelling breathers with exponentially small tails in a chain of nonlinear oscillators, Commun. Math. Phys., 2005, in press (available online at http://www.springerlink.com)], the tail being exponentially small with respect to the central oscillation size. In this paper, we compute these solutions numerically and continue them into the large amplitude regime for different types of even potentials. We find that Klein–Gordon chains can support highly localized travelling breather solutions superposed on an oscillatory tail. We provide examples where the tail can be made very small and is difficult to detect at the scale of central oscillations. In addition, we numerically observe the existence of these solutions in the case of non-even potentials.

Localized waves in nonlinear oscillator chains

This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.

Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators

We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schrodinger equation (M. Remoissenet, Phys. Rev. B33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlinearity17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchgassner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss, K. Kirchgassner, Commun. Math. Phys.211, 439–464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchgassner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schrodinger approximation.

Global spatiotemporal order and induced stochastic resonance due to a locally applied signal.

We study the phenomenon of spatiotemporal stochastic resonance (STSR) in a chain of diffusively coupled bistable oscillators. In particular, we examine the situation in which the global STSR response is controlled by a locally applied signal and reveal a wave-front propagation. In order to deepen the understanding of the system dynamics, we introduce, on the time scale of STSR, the study of the effective statistical renormalization of a generic lattice system. Using this technique we provide a criterion for STSR, and predict and observe numerically a bifurcationlike behavior that reflects the difference between the most probable value of the local quasiequilibrium density and its mean value. Our results, tested with a chain of nonlinear oscillators, appear to possess some universal qualities and may stimulate a deeper search for more generic phenomena.

Universality of anomalous one-dimensional heat conductivity.

In one and two dimensions, transport coefficients may diverge in the thermodynamic limit due to long-time correlation of the corresponding currents. The effective asymptotic behavior is addressed with reference to the problem of heat transport in one-dimensional crystals, modeled by chains of classical nonlinear oscillators. Extensive accurate equilibrium and nonequilibrium numerical simulations confirm that the finite-size thermal conductivity diverges with system size L as kappa proportional to L alpha. However, the exponent alpha deviates systematically from the theoretical prediction alpha=1/3 proposed in a recent paper [O. Narayan and S. Ramaswamy, Phys. Rev. Lett. 89, 200601 (2002)].

Fractal entropy of a chain of nonlinear oscillators.

We study the time evolution of a chain of nonlinear oscillators. We focus on the fractal features of the spectral entropy and analyze its characteristic intermediate time scales as a function of the nonlinear coupling. A Brownian motion is recognized with an analytic power-law dependence of its diffusion coefficient on the coupling.

Burst dynamics under mixed NMDA and AMPA drive in the models of the lamprey spinal CPG

The spinal CPG of the lamprey is modeled using a chain of nonlinear oscillators. Each oscillator represents a small neuron population capable of bursting under mixed NMDA and AMPA drive. Parameters of the oscillator are derived from detailed conductance-based neuron models. Analysis and simulations of dynamics of a single oscillator, a chain of locally coupled excitatory oscillators and a chain of two pairs of excitatory and inhibitory oscillators in each segment are done. The roles of asymmetric couplings and additional rostral drive for generation of a traveling wave with one cycle per chain length in a realistic frequency range are studied.

Effects of disorder on the wave front depinning transition in spatially discrete systems.

Pinning and depinning of wave fronts are ubiquitous features of spatially discrete systems describing a host of phenomena in physics, biology, etc. A large class of discrete systems is described by overdamped chains of nonlinear oscillators with nearest-neighbor coupling and subject to random external forces. The presence of weak randomness shrinks the pinning interval and it changes the critical exponent of the wave front depinning transition from 1/2 to 3/2. This effect is derived by means of a recent asymptotic theory of the depinning transition, extended to discrete drift-diffusion models of transport in semiconductor superlattices and is confirmed by numerical calculations.

Wave front depinning transition in discrete one-dimensional reaction-diffusion systems.

Pinning and depinning of wave fronts are ubiquitous features of spatially discrete systems describing a host of phenomena in physics, biology, etc. A large class of discrete systems is described by overdamped chains of nonlinear oscillators with nearest-neighbor coupling and controlled by constant external forces. A theory of the depinning transition for these systems, including scaling laws and asymptotics of wave fronts, is presented and confirmed by numerical calculations.

Effects of finite curvature on soliton dynamics in a chain of non-linear oscillators

We consider a curved chain of non-linear oscillators and show thatthe interplay of curvature and non-linearity leads to a number ofqualitative effects. In particular, the energy of non-linearlocalized excitations centred on the bending decreases whencurvature increases, i.e. bending manifests itself as a trap forexcitations. Moreover, the potential of this trap is a double-well one, thus leading to a symmetry-breakingphenomenon: a symmetric stationary state may become unstable andtransform into an energetically favourable asymmetric stationarystate. The essentials of symmetry breaking are examinedanalytically for a simplified model. We also demonstrate athreshold character of the scattering process, i.e. transmission,trapping, or reflection of the moving non-linear excitation passingthrough the bending.

Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics

We continue the study of a model for heat conduction [6] consisting of a chain of non-linear oscillators coupled to two Hamiltonian heat reservoirs at different temperatures. We establish existence of a Liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state. Ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite-dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators.

Curvature-induced symmetry breaking in nonlinear Schrodinger models

We consider a curved chain of nonlinear oscillators and show that the interplay of curvature and nonlinearity leads to a symmetry breaking when an asymmetric stationary state becomes energetically more favorable than a symmetric stationary state. We show that the energy of localized states decreases with increasing curvature, i.e., bending is a trap for nonlinear excitations. A violation of the Vakhitov-Kolokolov stability criterion is found in the case where the instability is due to the softening of the Peierls internal mode.

Travelling Waves in a Chain¶of Coupled Nonlinear Oscillators

In a chain of nonlinear oscillators, linearly coupled to their nearest neighbors, all travelling waves of small amplitude are found as solutions of finite dimensional reversible dynamical systems. The coupling constant and the inverse wave speed form the parameter space. The groundstate consists of a one-parameter family of periodic waves. It is realized in a certain parameter region containing all cases of light coupling. Beyond the border of this region the complexity of wave-forms increases via a succession of bifurcations. In this paper we give an appropriate formulation of this problem, prove the basic facts about the reduction to finite dimensions, show the existence of the ground states and discuss the first bifurcation by determining a normal form for the reduced system. Finally we show the existence of nanopterons, which are localized waves with a noncancelling periodic tail at infinity whose amplitude is exponentially small in the bifurcation parameter.

Complexity and Scaling of Frequency Plateaus in Chains of Coupled Nonlinear Oscillators

Complexity and Scaling of Frequency Plateaus in Chains of Coupled Nonlinear Oscillators

Complexity and Scaling of Frequency Plateaus in Chains of Coupled Nonlinear Oscillators

Chains of coupled phase oscillators with a linear gradient of natural frequencies are studied for the case of symmetric coupling functions. The frequency diagram, which is the plot of average frequencies against the coupling strength, is demonstrated to have a complex structure with some scaling properties, revealing a rich variety of behavior of frequency plateaus. In particular, a power law is found which governs the growth of frequency plateaus as the coupling is intensified.