Efficient topological chaos embedded in the blinking vortex system

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston-Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system.     

Entanglement production in quantized chaotic systems

Quantum chaos is a subject whose major goal is to identify and to investigate different quantum signatures of classical chaos. Here we study entanglement production in coupled chaotic systems as a possible quantum indicator of classical chaos. We use coupled kicked tops as a model for our extensive numerical studies. We find that, in general, chaos in the system produces more entanglement. However, coupling strength between two subsystems is also a very important parameter for entanglement production. Here we show how chaos can lead to large entanglement which is universal and describable by random matrix theory (RMT). We also explain entanglement production in coupled strongly chaotic systems by deriving a formula based on RMT. This formula is valid for arbitrary coupling strengths, as well as for sufficiently long time. Here we investigate also the effect of chaos on the entanglement production for the mixed initial state. We find that many properties of the mixed-state entanglement production are qualitatively similar to the pure state entanglement production. We however still lack an analytical understanding of the mixed-state entanglement production in chaotic systems.     

Feigenbaum universality in string theory

Brane-like vertex operators, defining backgrounds with ghost-matter mixing in NSR superstring theory, play an important role in the world-sheet formulation of D-branes and M theory as creation operators for extended objects in the second quantized formalism. In this paper, we show that the dilaton beta function in ghost-matter mixing backgrounds becomes stochastic. The renormalization group (RG) equations in ghost-matter mixing backgrounds lead to non-Markovian Fokker-Planck equations whose solutions describe superstrings in curved space-times with brane-like metrics. We show that the Feigenbaum universality constant δ=4.669..., describing transitions from order to chaos in a huge variety of dynamical systems, appears analytically in these RG equations. We find that the appearance of this constant is related to the scaling of relative space-time curvatures at fixed points of the RG flow. In this picture, the fixed points correspond to the period doubling of Feigenbaum iteration schemes.     

Ghost-matter mixing and feigenbaum universality in string theory

Branelike vertex operators, defining backgrounds with ghost-matter mixing in Neveu-Schwarz-Ramond superstring theory, play an important role in a world-sheet formulation of D branes and M theory, being creation operators for extended objects in the second quantized formalism. We show that the dilaton beta function in ghost-matter mixing backgrounds becomes stochastic. The renormalization group (RG) equations in ghost-matter mixing backgrounds lead to non-Markovian Fokker-Planck equations whose solutions describe superstrings in curved spacetimes with branelike metrics. We show that the Feigenbaum universality constant δ=4.669 ..., describing transitions from order to chaos in a huge variety of dynamical systems, appears analytically in these RG equations. We find that the appearance of this constant is related to the scaling of relative spacetime curvatures at fixed points of the RG flow. In this picture, the fixed points correspond to the period doubling of Feigenbaum iterational schemes.

     

Suppression of chaos and other dynamical transitions induced by intercellular coupling in a model for cyclic AMP signaling in Dictyostelium cells

The effect of intercellular coupling on the switching between periodic behavior and chaos is investigated in a model for cAMP oscillations in Dictyostelium cells. We first analyze the dynamic behavior of a homogeneous cell population which is governed by a three-variable differential system for which bifurcation diagrams are obtained as a function of two control parameters. We then consider the mixing of two populations behaving in a chaotic and periodic manner, respectively. Cells are coupled through the sharing of a common chemical intermediate, extracellular cAMP, which controls its production and release by the cells into the extracellular medium; the dynamics of the mixed suspension is governed by a five-variable differential system. When the two cell populations differ by the value of a single parameter which measures the activity of the enzyme that degrades extracellular cAMP, the bifurcation diagram established for the three-variable homogeneous population can be used to predict the dynamic behavior of the mixed suspension. The analysis shows that a small proportion of periodic cells can suppress chaos in the mixed suspension. Such a fragility of chaos originates from the relative smallness of the domain of aperiodic oscillations in parameter space. The bifurcation diagram is used to obtain the minimum fraction of periodic cells suppressing chaos. These results are related to the suppression of chaos by the small-amplitude periodic forcing of a strange attractor. Numerical simulations further show how the coupling of periodic cells with chaotic cells can produce chaos, bursting, simple periodic oscillations, or a stable steady state; the coupling between two populations at steady state can produce similar modes of dynamic behavior.     

Chaotic synchronization and evolution of optical phase in a bidirectional solid-state ring laser

We present results on experimental and theoretical studies of chaos in a solid-state ring laser with periodic pump modulation. We show that the synchronized chaos in the counter-propagating waves is observed for the values of pump modulation frequency fp satisfying the inequality f1 < fp < f2. The boundaries of this region, f1 and f2, depend on the pump-modulation depth. Inside the region of synchronized chaos we study not only dynamics of amplitudes of the counter-propagating waves but also the optical phases of them by mixing the fields of the counter-propagating waves and recording the intensity of the mixed signal. We demonstrate experimentally that in the regime of synchronized chaos the regular phase jumps appear during intervals between adjacent chaotic pulses. We improve the standard semi-classical model of a SSRL and consider an effect of spontaneous emission noise on the temporal evolution of intensities and phase dynamics in the regime of synchronized chaos. It is shown that at the parameters of the experimentally studied laser the noise strongly affects the temporal dependence of amplitudes of the counter-propagating waves.     

Application of Greene's method and the MacKay residue criterion to the double pendulum

The double pendulum is a non-integrable Hamiltonian system which exhibits the scenario of transition to global chaos via the decay of a golden mean KAM torus. We apply Greene's method and the MacKay residue criterion and compute the threshold to global chaos. We find that MacKay's method is superior to Greene's since it requires much less numerical work but nevertheless gives accurate results.     

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.     

Virtual photon effects on classical chaos in a periodically kicked Rydberg atom system

We study classical chaos in the system of a two-level Rydberg atom interacting with a pulsed standing microwave. This model approaches the form of an atom optics realization of a usual delta-kicked rotor under the rotating-wave approximation (RWA). We find that the non-energy-conserving processes or virtual photon processes neglected in the RWA have a strong effect on the classical chaos, which can enhance, reduce and even completely suppress the chaos under certain kicked conditions. The system displays non-KAM dynamical behavior for rational and irrational kicks.     

Experimental evidence for molecular chaos in granular gases

We present measurements showing the presence and the absence of molecular chaos in a two-layer vertically vibrated granular media where a plate drives a horizontal layer of massive grains, which, in turn, drives a second horizontal layer of lighter grains above the first. In the first layer driven by the plate, the velocities are spatially correlated. In the second layer, we find uncorrelated velocities consistent with the presence of molecular chaos. In this experiment, energy injection that is randomized in both space and time throughout the shaking cycle is necessary for observing molecular chaos and "kinetic theory"-like behavior. At higher densities, excluded volume effects force velocity correlations in the system which is no longer "gaslike" in behavior.     

Chaotic temperature dependence in a model of spin glasses

We address the problem of chaotic temperature dependence in disordered glassy systems at equilibrium by following states of a random-energy random-entropy model in temperature; of particular interest are the crossings of the free-energies of these states. We find that this model exhibits strong, weak or no temperature chaos depending on the value of an exponent. This allows us to write a general criterion for temperature chaos in disordered systems, predicting the presence of temperature chaos in the Sherrington-Kirkpatrick and Edwards-Anderson spin glass models, albeit when the number of spins is large enough. The absence of chaos for smaller systems may justify why it is difficult to observe chaos with current simulations. We also illustrate our findings by studying temperature chaos in the naıve mean field equations for the Edwards-Anderson spin glass. Received 27 March 2002 Published online 19 July 2002     

Optical chaos in multiple degenerate four-wave mixing

Spatial evolution has been studied of an ensemble of equal-frequency light waves involved in several four-wave mixing processes under linear photoabsorption. It is shown that this system may exhibit spatial chaos caused by competition of optical mixing processes with strong energy exchange between the waves. For pulsed radiation, the temporal envelopes of interacting waves undergo modulations which may also be of chaotic character. Also discussed are possible variants of identifying chaos in systems of interacting light waves experimentally and the role of accompanying photoprocesses (photoabsorption, self-action of radiation).     

Chaos and Taub-NUT related spacetimes

The occurrence of chaos for test particles moving in a Taub-NUT spacetime with a dipolar halo perturbation is studied using Poincaré sections. We find that the NUT parameter (magnetic mass) attenuates the presence of chaos.     

Pendulum limit,chaos and phase-locking in the dynamics of ac-driven semiconductor superlattices

We describe a limiting case when nonlinear dynamics of an ac-driven semiconductor superlattice in the miniband transport regime is governed by a periodically forced and damped pendulum equations. We find analytically the conditions for a transition to chaos. With increasing temperature the chaos disappears. We also discuss fractional dc voltage states in a superlattice originating from phase-locked states of the pendulum.     

Dynamical behaviour of a controlled vibro-impact system

In this paper, we give a controlled two-degree-of-freedom （TDOF） vibro-impact system based on the damping control law, and then investigate the dynamical behaviour of this system. According to numerical simulation, we find that this control scheme can suppress chaos to periodic orbit successfully. Furthermore, the feasibility and the robustness of the controller are confirmed, separately. We also find that this scheme cannot only suppress chaos, but also generate chaos in this system.     

Noise-induced enhancement of chemical reactions in nonlinear flows

Motivated by the problem of ozone production in atmospheres of urban areas, we consider chemical reactions of the general type: A+B-->2C, in idealized two-dimensional nonlinear flows that can generate Lagrangian chaos. Our aims differ from those in the existing work in that we address the role of transient chaos versus sustained chaos and, more importantly, we investigate the influence of noise. We find that noise can significantly enhance the chemical reaction in a resonancelike manner where the product of the reaction becomes maximum at some optimal noise level. We also argue that chaos may not be a necessary condition for the observed resonances. A physical theory is formulated to understand the resonant behavior. (c) 2002 American Institute of Physics.     

Controlling chaos: Stabilization of unstable orbits using exponential control for maps and flows

We demonstrate that chaos can be controlled using multiplicative exponential feedback control. Unstable fixed points, unstable limit cycles and unstable chaotic trajectories can all be stabilized using such control which is effective both for maps and flows. The control is of particular significance for systems with several degrees of freedom, as knowledge of only one variable on the desired unstable orbit is sufficient to settle the system onto that orbit. We find in all cases that the transient time is a decreasing function of the stiffness of control. But increasing the stiffness beyond an optimum value can increase the transient time. We have also used such a mechanism to control spatiotemporal chaos is a well-known coupled map lattice model.     

Critical properties of lattices of diffusively coupled quadratic maps

We study the critical properties of lattices of coupled logistic maps in the regime where the individual maps are closely above the onset of chaos. We discuss both spatial and temporal characteristics and especially the link between them. We show that the mutual information function between two points on the lattice decays exponentially with distance. In this way we find support for the relation xi approximately lambda(-1/2) between the coherence length xi and the largest Lyapunov exponent lambda which is further corroborated by a detailed study of the spreading of small perturbations. Finally we study the structure function of the lattice field variable. It shows that at the onset of chaos the lattice remains smooth.     

Two routes to chaos in the fractional Lorenz system with dimension continuously varying

The object of this paper is to reveal the relation between dynamics of the fractional system and its dimension defined as a sum of the orders of all involved derivatives. We take the fractional Lorenz system as example and regard one or three of its orders as bifurcation parameters. In this framework, we compute the corresponding bifurcation diagrams via an optimal Poincaré section technique developed by us and find there exist two routes to chaos when its dimension increases from some values to 3. One is the process of cascaded period-doubling bifurcations and the other is a crisis (boundary crisis) which occurs in the evolution of chaotic transient behavior. We would like to point out that our investigation is the first to find out that a fractional differential equations (FDEs) system can evolve into chaos by the crisis. Furthermore, we observe rich dynamical phenomena in these processes, such as two-stage cascaded period-doubling bifurcations, chaotic transients, and the transition from coexistence of three attractors to mono-existence of a chaotic attractor. These are new and interesting findings for FDEs systems which, to our knowledge, have not been described before.