Order-by-disorder in classical oscillator systems

We consider classical nonlinear oscillators on hexagonal lattices. When the coupling between the elements is repulsive, we observe coexisting states, each one with its own basin of attraction. These states differ by their degree of synchronization and by patterns of phase-locked motion. When disorder is introduced into the system by additive or multiplicative Gaussian noise, we observe a non-monotonic dependence of the degree of order in the system as a function of the noise intensity: intervals of noise intensity with low synchronization between the oscillators alternate with intervals where more oscillators are synchronized. In the latter case, noise induces a higher degree of order in the sense of a larger number of nearly coinciding phases. This order-by-disorder effect is reminiscent to the analogous phenomenon known from spin systems. Surprisingly, this non-monotonic evolution of the degree of order is found not only for a single interval of intermediate noise strength, but repeatedly as a function of increasing noise intensity. We observe noise-driven migration of oscillator phases in a rough potential landscape.Comment: 12 pages, 13 figures; comments are welcom

Sierpinski signal generates 1/fα1/f^\alpha spectra

We investigate the row sum of the binary pattern generated by the Sierpinski automaton: Interpreted as a time series we calculate the power spectrum of this Sierpinski signal analytically and obtain a unique rugged fine structure with underlying power law decay with an exponent of approximately 1.15. Despite the simplicity of the model, it can serve as a model for $1/f^\alpha$ spectra in a certain class of experimental and natural systems like catalytic reactions and mollusc patterns.Comment: 4 pages (4 figs included). Accepted for publication in Physical Review

Steady Stokes flow with long-range correlations, fractal Fourier spectrum, and anomalous transport

We consider viscous two-dimensional steady flows of incompressible fluids past doubly periodic arrays of solid obstacles. In a class of such flows, the autocorrelations for the Lagrangian observables decay in accordance with the power law, and the Fourier spectrum is neither discrete nor absolutely continuous. We demonstrate that spreading of the droplet of tracers in such flows is anomalously fast. Since the flow is equivalent to the integrable Hamiltonian system with 1 degree of freedom, this provides an example of integrable dynamics with long-range correlations, fractal power spectrum, and anomalous transport properties.Comment: 4 pages, 4 figures, published in Physical Review Letter

Electromagnetic Oscillations in a Driven Nonlinear Resonator: A New Description of Complex Nonlinear Dynamics

Many intriguing properties of driven nonlinear resonators, including the appearance of chaos, are very important for understanding the universal features of nonlinear dynamical systems and can have great practical significance. We consider a cylindrical cavity resonator driven by an alternating voltage and filled with a nonlinear nondispersive medium. It is assumed that the medium lacks a center of inversion and the dependence of the electric displacement on the electric field can be approximated by an exponential function. We show that the Maxwell equations are integrated exactly in this case and the field components in the cavity are represented in terms of implicit functions of special form. The driven electromagnetic oscillations in the cavity are found to display very interesting temporal behavior and their Fourier spectra contain singular continuous components. To the best of our knowledge, this is the first demonstration of the existence of a singular continuous (fractal) spectrum in an exactly integrable system.Comment: 5 pages, 3 figure

Modeling rhythmic patterns in the hippocampus

We investigate different dynamical regimes of neuronal network in the CA3 area of the hippocampus. The proposed neuronal circuit includes two fast- and two slowly-spiking cells which are interconnected by means of dynamical synapses. On the individual level, each neuron is modeled by FitzHugh-Nagumo equations. Three basic rhythmic patterns are observed: gamma-rhythm in which the fast neurons are uniformly spiking, theta-rhythm in which the individual spikes are separated by quiet epochs, and theta/gamma rhythm with repeated patches of spikes. We analyze the influence of asymmetry of synaptic strengths on the synchronization in the network and demonstrate that strong asymmetry reduces the variety of available dynamical states. The model network exhibits multistability; this results in occurrence of hysteresis in dependence on the conductances of individual connections. We show that switching between different rhythmic patterns in the network depends on the degree of synchronization between the slow cells.Comment: 10 pages, 9 figure

Noise Induced Complexity: From Subthreshold Oscillations to Spiking in Coupled Excitable Systems

We study stochastic dynamics of an ensemble of N globally coupled excitable elements. Each element is modeled by a FitzHugh-Nagumo oscillator and is disturbed by independent Gaussian noise. In simulations of the Langevin dynamics we characterize the collective behavior of the ensemble in terms of its mean field and show that with the increase of noise the mean field displays a transition from a steady equilibrium to global oscillations and then, for sufficiently large noise, back to another equilibrium. Diverse regimes of collective dynamics ranging from periodic subthreshold oscillations to large-amplitude oscillations and chaos are observed in the course of this transition. In order to understand details and mechanisms of noise-induced dynamics we consider a thermodynamic limit $N\to\infty$ of the ensemble, and derive the cumulant expansion describing temporal evolution of the mean field fluctuations. In the Gaussian approximation this allows us to perform the bifurcation analysis; its results are in good agreement with dynamical scenarios observed in the stochastic simulations of large ensembles

Optimizing busy time on parallel machines

We consider the following fundamental scheduling problem in which the input consists of n jobs to be scheduled on a set of identical machines of bounded capacity g (which is the maximal number of jobs that can be processed simultaneously by a single machine). Each job is associated with a start time and a completion time, it is supposed to be processed from the start time to the completion time (and in one of our extensions it has to be scheduled also in a continuous number of days, this corresponds to a two-dimensional version of the problem). We consider two versions of the problem. In the scheduling minimization version the goal is to minimize the total busy time of machines used to schedule all jobs. In the resource allocation maximization version the goal is to maximize the number of jobs that are scheduled for processing under a budget constraint given in terms of busy time. This is the first study of the maximization version of the problem. The minimization problem is known to be NP-Hard, thus the maximization problem is also NP-Hard. We consider various special cases, identify cases where an optimal solution can be computed in polynomial time, and mainly provide constant factor approximation algorithms for both minimization and maximization problems. Some of our results improve upon the best known results for this job scheduling problem. Our study has applications in power consumption, cloud computing and optimizing switching cost of optical networks