Chaotic Transition Research Articles

Correlation function studies on the Domany–Kinzel cellular automaton

The Domany–Kinzel cellular automaton is a simple and yet very rich model to study phase transitions in nonequilibrium systems. This model exhibits three characteristic phases: frozen, active and chaotic. In this paper we discuss the behavior of the equal-time two-point correlation functions and that of the associated correlation lengths as one crosses the phase boundary both for the frozen–active and active–chaotic transitions. We have investigated in detail how the correlation lengths diverge as one approaches the phase boundary from both sides. The divergence of the correlation length coupled with the previous studies on the divergence of the susceptibility, suggests that the fluctuation–dissipation theorem holds true in the Domany–Kinzel cellular automaton model. Time dependence of the correlation functions is also discussed.

Optical sequence generation based on collision avoidance using chaotic mode transitions

An adaptive optical sequence generation scheme has been described in a nonlinear optical ring which exhibits chaotic dynamics. The adaptation is based on avoidance of collisions between the ring output and signals from another source. It is shown that collision-avoiding sequences can be autonomously selected via chaotic mode transitions induced by the feedback of collision signals. Experimental results demonstrate that the method can be successful for various collision signal patterns with different periods and pulsewidths. The mechanism of inducing chaos with collision signals is examined both experimentally and with numerical simulations and the results verify the effectiveness of the proposed adaptation method. A possible application of the proposed signal generation scheme in intelligent optical communication networks is mentioned.

Transition to chaos in the conservative four-wave parametric interactions

In this work, we analyze the transition from regular to chaotic states in the parametric four-wave interactions. The temporal evolution describing the coupling of two sets of three-waves with quadratic nonlinearity is considered. This system is shown to undergo a chaotic transition via the separatrix chaos scenario, where a soliton-like solution (separatrix) that is found for the integrable (perfect matched) case becomes irregular as a small mismatch is turned on. As the mismatch is increased the separatrix chaotic layer spreads along the phase space, eventually engrossing most part of it. This scenario is typical of low-dimensional Hamiltonian systems.

Chaotic and hexagonal spontaneous pattern formation in the cross section of a laser beam in a defocusing Kerr-like film with single feedback mirror

We present a comparative experimental study on the spontaneous formation of patterns in the transverse cross-section of a laser beam traversing a thin slice of self-focusing and self-defocusing Kerr-like media in the one feedback mirror arrangement. In both cases, at critical laser intensity, hexagon to chaotic pattern transition was observed.

Adaptive optical sequence generation for collisionavoidance using chaos

An adaptive optical sequence generation scheme has been proposed which uses chaotic dynamics in a nonlinear optical ring. The adaptation is based on avoidance of collisions between the ring output and signals from another source. Experimental results show that collision-avoiding sequences can be autonomously selected via chaotic transitions which are induced by the feedback of collision signals.

SYMMETRY-RESTORING CRISES, PERIOD-ADDING AND CHAOTIC TRANSITIONS IN THE CUBIC VAN DER POL OSCILLATOR

A generalization of the van der Pol oscillator, in which a cubic non-linearity in the restoring force is considered, has been studied. For very small values of the cubic parameter, different chaotic transitions take place, via period-doubling, saddle-node bifurcations and crises, which restore the symmetry of the chaotic attractor. For larger values of the parameter a period-adding sequence of saddle-node bifurcations and the scaling laws that rule them are found. When negative values of the cubic parameter are considered, birfucations via hysteresis following a period-adding sequence are found. Finally, it has been found that there is a strong sensitivity to very small modifications of some system parameters.

Application of nonlinear dynamic system approach to oscillations of mercury arc in external magnetic field

Oscillations of an arc burning in mercury vapours and exposed to an external axial magnetic field were recorded. The fast Fourier transform, phase portrait construction and attractor dimension calculations show that the experimental results present a nonlinear dynamic system with a rich scenario of various metastable states and transitions between them. The system develops spontaneously with time mainly through a formation of new subharmonic frequencies. Among the different transition types especially the chaotic transitions may result in relatively long intervals of chaotic behaviour which are interesting from the point of view of the nonlinear system studies.

Classical and Quantum Transitions to Chaos for a Family of Periodically Driven Hamiltonians

The classical and quantum behaviors of a family of periodically driven hamiltonians are compared. By the resonance overlap criterion the perturbation field strengthεcfor the classical chaotic transition is estimated. The quantum spectral transition is analyzed in terms of the photonic approximation. From the validity of the photonic approximation and quantum delocalization condition the functional dependence ofεcis recovered in a purely quantum way.

Regular and chaotic quantum motions

In the framework of Bohm's interpretation of quantum mechanics, particles are subject to both classical and quantum forces and actually follow physical trajectories. A natural definition of chaos in quantum mechanics is therefore possible and it is measured by a quantum Lyapunov exponent. In order to test the effectiveness of this new definition of quantum chaoticity, we have comparatively studied a classical model and a truncated N-level quantum model of a hydrogen atom acted upon by an external electromagnetic field. We find direct evidence for the existence of quantum chaotic motions, for a chaotic transition and for a quantum attenuation of classical chaos in a broad range of external field amplitudes.

Chaotic transitions in resonant asteroidal dynamics

The utilization of chaotic dynamics approaches allowed the identification of many modes of motion in resonant asteroidal dynamics. As these dynamical systems are not integrable, the motion modes are not separated and one orbit may transit from one mode to another. In some cases, as in the 3/1 resonance, these transitions may lead, in a relatively short time scale, to eccentricities so high that the asteroid may approach the Sun and be destroyed. In the 2/1 and 3/2 resonances these transitions are much slower and only indirect estimations of the time which is needed for a generic asteroid to leave the resonance are possible. It may reach hundreds of million years in the more robust regions of the 2/1 resonance and a time of the order of billions of years in those of the 3/2 resonance. These values are consistent with the observed depletion of the 2/1 resonance (only a few asteroids known while almost 60 asteroids are known in the 3/2 resonance).

Parametrically excited non-linear surface waves and chaos in magnetic fluids

The resonant excitation of weakly non-linear surface waves in a magnetic fluid subject to a periodically oscillating magnetic field is examined using the Melnikov method. A Hamiltonian formulation results in transverse homoclinic orbits leading to a chaotic transition. A necessary condition for the existence of chaos in magnetic fluids is obtained.

Transition to chaos in a shell model of turbulence

We study a shell model for the energy cascade in three-dimensional turbulence by varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. When the control parameter ϵ related to the strength of backward energy transfer is small enough, the dynamical system has a stable fixed point corresponding to the Kolmogorov scaling. By using the bi-orthogonal decomposition, the transition to chaos is shown to follow the Ruelle-Takens scenario. For ϵ > 0.3953… there exists a strange attractor which remains close to the Kolmogorov fixed point. The intermittency of the chaotic evolution and of the scaling can be described by an intermittent one-dimensional map. We introduce a modified shell model which has a good scaling behaviour also in the infrared region. We study the multifractal properties of this model for large number of shells and for values of ϵ slightly above the chaotic transition. In this case by making a local analysis of the scaling properties in the inertial range we found that the multifractal corrections seem to become weaker and weaker approaching the viscous range.

Transition to chaos in a shell model of turbulence

Abstract We study a shell model for the energy cascade in three-dimensional turbulence by varying the coefficients of the non-linear terms in such a way that the fundamental symmetries of Navier-Stokes are conserved. When the control parameter ϵ related to the strength of backward energy transfer is small enough, the dynamical system has a stable fixed point corresponding to the Kolmogorov scaling. By using the bi-orthogonal decomposition, the transition to chaos is shown to follow the Ruelle-Takens scenario. For ϵ > 0.3953… there exists a strange attractor which remains close to the Kolmogorov fixed point. The intermittency of the chaotic evolution and of the scaling can be described by an intermittent one-dimensional map. We introduce a modified shell model which has a good scaling behaviour also in the infrared region. We study the multifractal properties of this model for large number of shells and for values of ϵ slightly above the chaotic transition. In this case by making a local analysis of the scaling properties in the inertial range we found that the multifractal corrections seem to become weaker and weaker approaching the viscous range.

Assessment of behavioral chaos with a focus on transitions in depression.

This article is the second of three in a special section on chaos theory and psychological assessment. The purpose of this article is to address implications for the measurement of some unstable, transitional human behaviors. One early hallmark of behavioral assessment is the idiographic measurement of temporal, causal, and target factors (e.g., M. R. Goldfried & R. N. Kent, 1972). This is the same basic measurement approach used in chaos theory testing. In addition, behavioral theories have posited nonlinear relations among variables (e.g., J. D. Cone, 1981). However, theory testing often involves investigation of linear relations among episodic and averaged indexes of the variables (e.g., M. J. Marr, 1989). The study of unstable behaviors and chaotic transitions could revive the use of unaveraged single-subject, time series measurement approaches. A rationale for a subtype of chaotic depression (e.g., L. Glass & M. C. Mackey, 1988) and its measurement is offered

Phenomenological description of the optical field chaos in storage ring free-electron lasers.

The problem of optical field chaos in a storage ring free-electron laser oscillator has been discussed by using a phenomenological model. The result of theoretical analysis and numerical simulation shows that the variation of laser intensity versus time can be both periodic and chaotic when there is a weak gain modulation in the optical cavity. As time increases, the leading Lyapunov characteristic exponent of the system goes to a negative and a positive real number, respectively. Further research is carried out, and a chaotic transition via period-doubling bifurcation has been found when there are variations in the modulation parameters. The behavior of the system with two different kinds of gain modulation is also discussed.

Self-sustained oscillations and chaotic transitions in current-carrying thin HTSC-films cooled by boiling nitrogen

An increase of the relaxation oscillations in the nonlinear system of a thin film of a high-temperature superconductor (HTSC) connected to an external electric circuit with an inductance, boiling nitrogen, has been shown experimentally. With increasing dc bias voltage one could observe an upset of the periodic oscillations and a transition to chaotic ones. Also, relaxation oscillations and chaos transitions have been studied experimentally in the case when a dc bias voltage was added to a harmonic component.

Chaotic transitions in a short-pulse FEL oscillator

Considering the slippage and the cavity detuning, we have investigated the bifurcation and the chaotic transitions in a FEL oscillator. In describing the nonlinear dissipative dynamics of the FEL oscillator, we have used the properly defined branching parameters, which are closely related to the physical parameters. Throughout this study, we have found three well-known chaotic transition routes via period-doubling cascade, intermittency, and quasiperiodicity, which are confirmed by the qualitative and quantitative analyses.

Chaotic transitions of convection rolls in a rapidly rotating annulus

Drifting convection rolls in a rapidly rotating cylindrical annulus with conical endwalls exhibit different transitional modes to chaotic flows at different Prandtl numbers. Three transition sequences for Prandtl numbers 0.3, 1.0 and 7.0 are studied for a moderately large Coriolis parameter and a wavenumber near the critical value using an initial-value code. As the Rayleigh number increases, each transition sequence first leads to a vacillating flow, and then to an aperiodic flow, the route of which is Prandtl-number dependent. From the low Prandtl number to the high Prandtl number, the transitions take different routes of torus folding, period doubling, and mode-locking intermittency.

Oscillation mode selection using bifurcation of chaotic mode transitions in a nonlinear ring resonator

This paper demonstrates the effectiveness of an adaptive parametric control method for searching and switching among a large number of multistable oscillation modes using chaotic mode transitions. The adaptive control is used to select nonlinear oscillation modes in an electro-optic ring resonator. In the adaptive control scheme, the result of a simple test of resonator output is fed back to a single parameter, pump laser power, governing bifurcation to and from chaos. The test is the presence or absence of a target code in the oscillation waveform of the resonator output. Chaotic mode transition phenomena, called chaotic itinerancy, are investigated in terms of code dynamics, and the results are used to determine the optimal parameters for the adaptive control.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Stabilization of quasiperiodic orbits for line-coupled diode resonator systems.

Chaotic transitions in line-coupled diode resonator systems were experimentally found to follow the Curry-Yorke model. [The Structure of Attractors in Dynamical Systems (Springer, Berlin, 1977), p. 48]. The standard diode model from the computer program spice was used to simulate these systems; the complete Lyapunov spectra from simulated systems are in good agreement with the spectra computed from the experimental time series. By applying the proportional-feedback technique to these systems, we can stabilize periodic orbits of increasing periods as well as unstable quasiperiodic orbits densely covering a torus.