Lag synchronization and scaling of chaotic attractor in coupled system

We report a design of delay coupling for lag synchronization in two unidirectionally coupled chaotic oscillators. A delay term is introduced in the definition of the coupling to target any desired lag between the driver and the response. The stability of the lag synchronization is ensured by using the Hurwitz matrix stability. We are able to scale up or down the size of a driver attractor at a response system in presence of a lag. This allows compensating the attenuation of the amplitude of a signal during transmission through a delay line. The delay coupling is illustrated with numerical examples of 3D systems, the Hindmarsh-Rose neuron model, the R\"ossler system and a Sprott system and, a 4D system. We implemented the coupling in electronic circuit to realize any desired lag synchronization in chaotic oscillators and scaling of attractors.Comment: 10 pages, 7 figure

Chaotic universe in the z=2 Hovava-Lifshitz gravity

The deformed z=2 Horava-Lifshitz gravity with coupling constant w leads to a nonrelativistic "mixmaster" cosmological model. The potential of theory is given by the sum of IR and UV potentials in the ADM Hamiltonian formalism. It turns out that adding the UV-potential cannot suppress chaotic behaviors existing in the IR-potential.Comment: 7 pages, 5 figures, version to appear in PR

Front Propagation in Chaotic and Noisy Reaction-Diffusion Systems: a Discrete-Time Map Approach

We study the front propagation in Reaction-Diffusion systems whose reaction dynamics exhibits an unstable fixed point and chaotic or noisy behaviour. We have examined the influence of chaos and noise on the front propagation speed and on the wandering of the front around its average position. Assuming that the reaction term acts periodically in an impulsive way, the dynamical evolution of the system can be written as the convolution between a spatial propagator and a discrete-time map acting locally. This approach allows us to perform accurate numerical analysis. They reveal that in the pulled regime the front speed is basically determined by the shape of the map around the unstable fixed point, while its chaotic or noisy features play a marginal role. In contrast, in the pushed regime the presence of chaos or noise is more relevant. In particular the front speed decreases when the degree of chaoticity is increased, but it is not straightforward to derive a direct connection between the chaotic properties (e.g. the Lyapunov exponent) and the behaviour of the front. As for the fluctuations of the front position, we observe for the noisy maps that the associated mean square displacement grows in time as $t^{1/2}$ in the pushed case and as $t^{1/4}$ in the pulled one, in agreement with recent findings obtained for continuous models with multiplicative noise. Moreover we show that the same quantity saturates when a chaotic deterministic dynamics is considered for both pushed and pulled regimes.Comment: 11 pages, 11 figure

The Control of Dynamical Systems - Recovering Order from Chaos -

Following a brief historical introduction of the notions of chaos in dynamical systems, we will present recent developments that attempt to profit from the rich structure and complexity of the chaotic dynamics. In particular, we will demonstrate the ability to control chaos in realistic complex environments. Several applications will serve to illustrate the theory and to highlight its advantages and weaknesses. The presentation will end with a survey of possible generalizations and extensions of the basic formalism as well as a discussion of applications outside the field of the physical sciences. Future research avenues in this rapidly growing field will also be addressed.Comment: 18 pages, 9 figures. Invited Talk at the XXIth International Conference on the Physics of Electronic and Atomic Collisions (ICPEAC), July 22-27, 1999 (Sendai, Japan

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 Poincar\'e sections. We find that the NUT parameter (magnetic mass) attenuates the presence of chaos.Comment: 10 pages, 5 Postscript figure

Relaxation Time of Quantized Toral Maps

We introduce the notion of the relaxation time for noisy quantum maps on the 2d-dimensional torus - a generalization of previously studied dissipation time. We show that relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers the semiclassical limit ($\hbar$ -> 0) together with the limit of small noise strength (\ep -> 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical regime $\hbar1) in which classical and quantum relaxation times share the same asymptotics: in this regime, a quantized Anosov map relaxes to equilibrium fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantum-classical correspondence for noisy maps on the torus, up to times logarithmic in$\hbar^{-1}$. On the other hand, we show that in the ``quantum regime''$\ep$<<$\hbar$ << 1, quantum and classical relaxation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized ``Arnold's cat'' maps), we obtain the exact asymptotics of the quantum relaxation time and precise the regime of correspondence between quantum and classical relaxations.Comment: LaTeX, 27 pages, former term dissipation time replaced by relaxation time, new introduction and reference

Reduction of dimension for nonlinear dynamical systems

We consider reduction of dimension for nonlinear dynamical systems. We demonstrate that in some cases, one can reduce a nonlinear system of equations into a single equation for one of the state variables, and this can be useful for computing the solution when using a variety of analytical approaches. In the case where this reduction is possible, we employ differential elimination to obtain the reduced system. While analytical, the approach is algorithmic, and is implemented in symbolic software such as {\sc MAPLE} or {\sc SageMath}. In other cases, the reduction cannot be performed strictly in terms of differential operators, and one obtains integro-differential operators, which may still be useful. In either case, one can use the reduced equation to both approximate solutions for the state variables and perform chaos diagnostics more efficiently than could be done for the original higher-dimensional system, as well as to construct Lyapunov functions which help in the large-time study of the state variables. A number of chaotic and hyperchaotic dynamical systems are used as examples in order to motivate the approach.Comment: 16 pages, no figure