Noise-induced bifurcations and chaos in the average motion of globally-coupled oscillators

A system of coupled master equations simplified from a model of noise-driven globally coupled bistable oscillators under periodic forcing is investigated. In the thermodynamic limit, the system is reduced to a set of two coupled differential equations. Rich bifurcations to subharmonics and chaotic motions are found. This behavior can be found only for certain intermediate noise intensities. Noise with intensities which are too small or too large will certainly spoil the bifurcations. In a system with large though finite size, the bifurcations to chaos induced by noise can still be detected to a certain degree. Received 6 April 1999 and Received in final form 1 November 1999     

Stochastic resonance in ferromagnetic domain motion

A model for the motion of a single ferromagnetic domain is studied numerically and analytically. A single strip in two dimensions and pinned at two inhomogeneities is considered. We suppose two stable configurations (positively or negatively curved with pinned ends) due to the action of a bistable potential. Further, it is assumed that the domain is driven externally by periodic and noisy magnetic fields. The noise makes the domain able to flip between the two configurations. The small temporally periodic fields synchronize these flippings and the phenomenon of stochastic resonance is observed. The signal to noise ratio of the output is investigated and shows a maximum for a nonvanishing intensity of the applied noise. Its dependency on the stiffness of the domain is studied. Received 14 May 1999 and Received in final form 14 October 1999     

Functional renormalization description of the roughening transition

We reconsider the problem of the static thermal roughening of an elastic manifold at the critical dimension d=2 in a periodic potential, using a perturbative Functional Renormalization Group approach. Our aim is to describe the effective potential seen by the manifold below the roughening temperature on large length scales. We obtain analytically a flow equation for the potential and surface tension of the manifold, valid for low temperatures. On a length scale L, the renormalized potential is made up of a succession of quasi parabolic wells, matching onto one another in a singular region of width for large L. For strong periodic potential, the perturbation theory breaks down, and we argue, based on a variational calculation, that the transition becomes first order. We also obtain numerically the step energy as a function of temperature, and relate our results to the existing experimental data on ⁴He. Finally, we examine the case of a non local elasticity which is realized physically for the contact line. Received 16 April 1999 and Received in final form 11 October 1999     

Periodic orbits in magnetic billiards

We propose a simple method to calculate periodic orbits in two-dimensional systems with no symbolic dynamics. The method is based on a line by line scan of the Poincaré surface of section and is particularly useful for billiards. We have applied it to the Square and Sinai's billiards subjected to a uniform orthogonal magnetic field and we obtained about 2000 orbits for both systems using absolutely no information about their symbolic dynamics. Received 21 September 1999 and Received in final form 13 April 2000     

A stability criterion for financial markets

Using the theory of random cluster models, we give a stability criterion for financial markets with random communications between agents. Received 25 September 1999 and Received in final form 2 October 1999     

Cubic phases of amphiphilic molecular aggregates

Stability and geometry of the lyotropic periodic cubic mesophases are considered in the framework of a general symmetry based phenomenological approach. A limited number of cubic structural types is shown to be formed by amphiphilic molecular aggregates due to the specific nature of self-organizing units. The related thermodynamic models predict topology of phase diagrams and specific features of transitions between isotropic, lamellar, cubic bicontinuous and cubic micellar phases. Received 25 February 1999 and Received in final form 29 June 1999     

Spike-burst and other oscillations in a system composed of two coupled, drastically different elements

The dynamics of a system composed of two nonlinearly coupled, drastically different nonlinear and eventually oscillatory elements is studied. The rich variety of attractors of the system is studied with the help of phase space analysis and Poincare maps. Received 19 March 1999 and Received in final form 1 November 1999     

Brownian particles far from equilibrium

We study a model of Brownian particles which are pumped with energy by means of a non-linear friction function, for which different types are discussed. A suitable expression for a non-linear, velocity-dependent friction function is derived by considering an internal energy depot of the Brownian particles. In this case, the friction function describes the pumping of energy in the range of small velocities, while in the range of large velocities the known limit of dissipative friction is reached. In order to investigate the influence of additional energy supply, we discuss the velocity distribution function for different cases. Analytical solutions of the corresponding Fokker-Planck equation in 2d are presented and compared with computer simulations. Different to the case of passive Brownian motion, we find several new features of the dynamics, such as the formation of limit cycles in the four-dimensional phase-space, a large mean squared displacement which increases quadratically with the energy supply, or non-equilibrium velocity distributions with crater-like form. Further, we point to some generalizations and possible applications of the model. Received 24 November 1999     

Instabilities in population dynamics

Biologists have long known that the smaller the population, the more susceptible it is to extinction from various causes. Biologists define minimum viable population size (MVP), which is the critical population size, below which the population has a very small chance to survive. There are several theoretical models for predicting the probability that a small population will become extinct. But these models either embody unrealistic assumptions or lead to currently unresolved mathematical problems. In other popular models of population dynamics, like the logistic model, MVP does not exist. In this paper we find the existence of such a critical concentration in a simple model of evolution. We solve this model by a mean field theory and show, in one and two dimensions, the existence of the critical adaptation and concentration below which a population dies out. We also show that, like in the logistic model, above the critical value a population reaches its carrying capacity. Moreover, in the two-dimensional case we find - the so common in biological models - periodic solutions and their biffurcations. Received 15 February 2000     

External fluctuations in front dynamics with inertia: The overdamped limit

We study the dynamics of fronts when both inertial effects and external fluctuations are taken into account. Stochastic fluctuations are introduced as multiplicative white noise arising from a control parameter of the system. Contrary to the non-inertial (overdamped) case, we find that important features of the system, such as the velocity selection picture, are not modified by the noise. We then compute the overdamped limit of the underdamped dynamics in a more careful way, finding that it does not exhibit any effect of noise either. Our result poses the question as to whether or not external noise sources can be measured in physical systems of this kind. Received 2 July 1999 and Received in final form 25 November 1999     

A neural network model composed of two-state and three-state neurons

A neural network model composed of two-state (1 and -1) and three-state (1, 0 and -1) neurons is proposed. The two-state neurons are connected with the three-state ones only and vice versa. We derive dynamic equations for the model under the assumption of non-symmetrical dilution of connections. A zero-noise phase diagram is obtained and a region in which two fixed point solutions can coexist is found. Basins of attraction for the solutions are also investigated. Received 26 October 1998 and Received in final form 12 February 1999     

Quantum motion of three trapped ions in one dimension

The hyperspherical-coordinate approach is employed to a one-dimensional model of three ions in a Paul trap. It is shown that the eigen wave functions have well-defined nodal structure indicating a near separability in the hyperspherical coordinates, then two approximate good quantum numbers are introduced to classify the eigenstates. Three important classical periodic motions, including the breathing motion and the (distorted-)symmetric or anti-symmetric stretching motion, are found to dominate the wave function distribution. Received: 10 February 1999 / Received in final form: 25 March 1999     

Dynamics of spatio-temporal defects in the Taylor-Dean system

The dynamics of defects in a pattern of traveling inclined rolls has been investigated. Two regimes were identified in the neighborhood of defects: a diffusive regime, with a negative phase diffusion coefficient, and a coalescence regime in which the phase gradient diverges in time following a power law behavior. The observed periodic nucleation of defects is related to the frequency inhomogeneity induced by the disymmetry of the wave amplitude. Amplitude holes have been observed in the secondary modulated pattern. Received 23 February 1999 and Received in final form 26 May 1999     

A cellular automata model for highway traffic

We present results relative to a simple cellular automata model without periodic boundary conditions for an highway with on-ramps. Simulations performed with this model reproduce experimental phenomena observed in traffic such as free flow, synchronized flow, congested flow, lane inversion, forward and backward propagating waves. On-ramps play the important role of nucleation points for the dynamic features of traffic. Received 4 February 2000 and Received in final form 5 May 2000     

Finite-size effects in the self-organized critical forest-fire model

We study finite-size effects in the self-organized critical forest-fire model by numerically evaluating the tree density and the fire size distribution. The results show that this model does not display the finite-size scaling seen in conventional critical systems. Rather, the system is composed of relatively homogeneous patches of different tree densities, leading to two qualitatively different types of fires: those that span an entire patch and those that do not. As the system size becomes smaller, the system contains less patches, and finally becomes homogeneous, with large density fluctuations in time. Received 24 April 1999 and Received in final form 26 October 1999     

Conserved dynamics and interface roughening in spontaneous imbibition: A critical overview

Imbibition phenomena have been widely used experimentally and theoretically to study the kinetic roughening of interfaces. We critically discuss the existing experiments and some associated theoretical approaches on the scaling properties of the imbibition front, with particular attention to the conservation law associated to the fluid, to problems arising from the actual structure of the embedding medium, and to external influences such as evaporation and gravity. Our main conclusion is that the scaling of moving interfaces includes many crossover phenomena, with competition between the average capillary pressure gradient and its fluctuations setting the maximal lengthscale for roughening. We discuss the physics of both pinned and moving interfaces and the ability of the existing models to account for their properties. Received 17 February 1999 and Received in final form 24 November 1999     

Thermal diffusion of sine-Gordon solitons

We analyze the diffusive motion of kink solitons governed by the thermal sine-Gordon equation. We analytically calculate the correlation function of the position of the kink center as well as the diffusion coefficient, both up to second-order in temperature. We find that the kink behavior is very similar to that obtained in the overdamped limit: There is a quadratic dependence on temperature in the diffusion coefficient that comes from the interaction among the kink and phonons, and the average value of the wave function increases with due to the variance of the centers of individual realizations and not due to kink distortions. These analytical results are fully confirmed by numerical simulations. Received 4 October 1999 and Received in final form 3 February 2000     

Resonance effects in nonlinear lattices

We study a class of one-dimensional nonlinear lattices with nearest-neighbour interactions described by a potential of the binomial type. This potential contains a free parameter which can be chosen to reproduce a variety of models, such as the Toda, the Fermi-Pasta-Ulam and the Coulomb-like lattices. Carrying out essentially numerical experiments, the effects of soliton propagation on a lattice with defects are investigated. In particular, the properties of the localized mode, generated by the propagation of the soliton through the defect, are discussed with respect to the defect mass and the potential parameter, in the light of a simple theoretical model. Furthermore, an interesting phenomenon is observed: the amplitude of the speed of the mass defect shows a sequel of resonance peaks in terms of the mass defect. The positions of these peaks appear to be independent of the potential parameter. Received 16 August 1999 and Received in final form 3 February 2000     

Lattice model for translational and rotational motions based on generalised diffusion equations: I. General formalism

A simple lattice model based on generalised diffusion equations and Gaussian statistics, aimed at describing diffusive translational and rotational motions, is presented. It is shown that it allows the generation of correlation functions relevant to spectroscopic techniques that are very similar to those experimentally observed in a large variety of complex systems. For some ranges of values of the model parameters, these functions, which can be expressed in closed mathematical forms, can be approximately represented by the sum of two exponentials or by “stretched" exponentials. Received 17 September 1999 and Received in final form 10 February 2000     

Learning, competition and cooperation in simple games

The minority model was introduced to study the competition between agents with limited information. It has the remarkable feature that, as the number of strategies available to the agents increases, the collective gain made by the agents is reduced. This crowd effect arises from the fact that only a minority can profit at each moment, while all agents make their choices using the same input. We show that the properties of the model change drastically if the agents make choices based on their individual stories, keeping all remaining rules unaltered. This variation reduces the intrinsic frustration of the model, and improves the tendency towards cooperation and self organization. We finally study the stable mixing of individual and collective behavior. Received 30 June 1999 and Received in final form 27 September 1999