L.E. Guerrero

Exact solutions to chaotic and stochastic systems

We investigate functions that are exact solutions to chaotic dynamical systems. A generalization of these functions can produce truly random numbers. For the first time, we present solutions to random maps. This allows us to check, analytically, some recent results about the complexity of random dynamical systems. We confirm the result that a negative Lyapunov exponent does not imply predictability in random systems. We test the effectiveness of forecasting methods in distinguishing between chaotic and random time series. Using the explicit random functions, we can give explicit analytical formulas for the output signal in some systems with stochastic resonance. We study the influence of chaos on the stochastic resonance. We show, theoretically, the existence of a new type of solitonic stochastic resonance, where the shape of the kink is crucial. Using our models we can predict specific patterns in the output signal of stochastic resonance systems. (c) 2001 American Institute of Physics.

A mechanism for randomness

We investigate explicit functions that can produce truly random numbers. We use the analytical properties of the explicit functions to show that certain class of autonomous dynamical systems can generate random dynamics. This dynamics presents fundamental differences with the known chaotic systems. We present real physical systems that can produce this kind of random time-series. We report the results of real experiments with nonlinear circuits containing direct evidence for this new phenomenon. In particular, we show that a Josephson junction coupled to a chaotic circuit can generate unpredictable dynamics. Some applications are discussed.

From exactly solvable chaotic maps to stochastic dynamics

Abstract For a class of nonlinear chaotic maps, the exact solution can be written as Xn=P(θkn), where P(t) is a periodic function, θ is a real parameter and k is an integer number. A generalization of these functions: Xn=P(θzn), where z is a real parameter, can be proved to produce truly random sequences. Using different functions P(t) we can obtain different distributions for the random sequences. Similar results can be obtained with functions of type Xn=h[f(n)], where f(n) is a chaotic function and h(t) is a noninvertible function. We show that a dynamical system consisting of a chaotic map coupled to a map with a noninvertible nonlinearity can generate random dynamics. We present physical systems with this kind of behavior. We report the results of real experiments with nonlinear circuits and Josephson junctions. We show that these dynamical systems can produce a type of complexity that cannot be observed in common chaotic systems. We discuss applications of these phenomena in dynamics-based computation.

Chaos-induced true randomness

We investigate functions of type Xn=P(θzn), where P(t) is a periodic function, θ and z are real parameters. We show that these functions produce truly random sequences. We prove that a class of autonomous dynamical systems, containing nonlinear terms described by periodic functions of the variables, can generate random dynamics. We generalize these results to dynamical systems with nonlinearities in the form of noninvertible functions. Several examples are studied in detail. We discuss how the complexity of the dynamics depends on the kind of nonlinearity. We present real physical systems that can produce random time-series. We report the results of real experiments using nonlinear circuits with noninvertible I–V characteristics. In particular, we show that a Josephson junction coupled to a chaotic circuit can generate unpredictable dynamics.

Topological Defects with Long-Range Interactions

We investigate a modified sine-Gordon equation which possesses soliton solutions with long-range interaction. We introduce a generalized version of the Ginzbug-Landau equation which supports long-range topological defects in D = 1 and D > 1. The interaction force between the defects decays so slowly that it is possible to enter the non-extensivity regime. These results can be applied to non-equilibrium systems, pattern formation and growth models.

Controlling soliton explosions

We investigate the dynamics of solitons in generalized Klein-Gordon equations in the presence of nonlinear damping and spatiotemporal perturbations. We will present different mechanisms for soliton explosions. We show (both analytically and numerically) that some space-dependent perturbations or nonlinear damping can make the soliton internal mode unstable leading to soliton explosion. We will show that, in some cases, while some conditions are satisfied, the soliton explodes becoming a permanent, extremely complex, spatiotemporal dynamics. We believe these mechanisms can explain some of the phenomena that recently have been reported to occur in excitable media. We present a method for controlling soliton explosions.

How to excite the internal modes of sine-Gordon solitons

Abstract We investigate the dynamics of the sine-Gordon solitons perturbed by spatiotemporal external forces. We prove the existence of internal (shape) modes of sine-Gordon solitons when they are in the presence of inhomogeneous space-dependent external forces, provided some conditions (for these forces) hold. Additional periodic time-dependent forces can sustain oscillations of the soliton width. We show that, in some cases, the internal mode even can become unstable, causing the soliton to decay in an antisoliton and two solitons. In general, in the presence of spatiotemporal forces the soliton behaves as a deformable (non-rigid) object. A soliton moving in an array of inhomogeneities can also present sustained oscillations of its width. There are very important phenomena (like the soliton–antisoliton collisions) where the existence of internal modes plays a crucial role. We show that, under some conditions, the dynamics of the soliton shape modes can be chaotic. A short report of some of our results has been published in [Phys. Rev. E 65 (2002) 065601(R)].

Spatiotemporal Chaotic Dynamics of Solitons with Internal Structure in the Presence of Finite-Width Inhomogeneities

Abstract We present an analytical and numerical study of the Klein–Gordon kink-soliton dynamics in inhomogeneous media. In particular, we study an external field that is almost constant for the whole system but that changes its sign at the center of coordinates and a localized impurity with finite-width. The soliton solution of the Klein–Gordon-like equations is usually treated as a structureless point-like particle. A richer dynamics is unveiled when the extended character of the soliton is taken into account. We show that interesting spatiotemporal phenomena appear when the structure of the soliton interacts with finite-width inhomogeneities. We solve an inverse problem in order to have external perturbations which are generic and topologically equivalent to well-known bifurcation models and such that the stability problem can be solved exactly. We also show the different quasiperiodic and chaotic motions the soliton undergoes as a time-dependent force pumps energy into the traslational mode of the kink and relate these dynamics with the excitation of the shape modes of the soliton.

Long-range interacting solitons: pattern formation and nonextensive thermostatistics

The nonlinear Klein–Gordon equation with a different potential that satisfies the degeneracy properties discussed in this paper possesses solitonic solutions that interact with long-range forces. We generalize the Ginzburg–Landau equation in such a way that the topological defects supported by this equation present long-range interaction both in D=1 and D>1. Finally, we construct a system of two equations with two complex order parameters for which the interaction forces between the topological defects decay so slowly that the system enters the nonextensivity regime.

Quasiperiodic route to soft turbulence in long Josephson junctions

We present a numerical study of the onset of a turbulent-like regime in long Josephson junctions. We show that the local generation of different linear combinations of frequencies in the quasiperiodic regime leads to the breakdown of coherence of the spatiotemporal profile.