Solvable model for spatiotemporal chaos

We show that the dynamical behavior of a coupled map lattice where the individual maps are Bernoulli shift maps can be solved analytically for integer couplings. We calculate the invariant density of the system and show that it displays a nontrivial spatial behavior. We also introduce and calculate a generalized spatiotemporal correlation function

Memory difference control of unknown unstable fixed points: Drifting parameter conditions and delayed measurement

Difference control schemes for controlling unstable fixed points become important if the exact position of the fixed point is unavailable or moving due to drifting parameters. We propose a memory difference control method for stabilization of a priori unknown unstable fixed points by introducing a memory term. If the amplitude of the control applied in the previous time step is added to the present control signal, fixed points with arbitrary Lyapunov numbers can be controlled. This method is also extended to compensate arbitrary time steps of measurement delay. We show that our method stabilizes orbits of the Chua circuit where ordinary difference control fails.Comment: 5 pages, 8 figures. See also chao-dyn/9810029 (Phys. Rev. E 70, 056225) and nlin.CD/0204031 (Phys. Rev. E 70, 046205

Comment on "White-Noise-Induced Transport in Periodic Structures"

In the paper by J.\L uczka {\em et al.} ({\em Europhys. Lett.}, {\bf 31} (1995) 431), the authors reported by rigorous calculation that an additive Poissonian white shot noise can induce a macroscopic current of a dissipative particle in a periodic potential -- even {\em in the absence} of spatial asymmetry of the potential. We argue that their main result is an obvious one caused by the spatially broken symmetry of a probability distribution of the additive noise, unlike the similar result caused by chaotic noise which has a symmetric probability distribution ({\em J.Phys.Soc.Jpn.}, {\bf 63} (1994) 2014).Comment: 2 pages (Latex); submitted to Europhys.Let

Pinning control of spatiotemporal chaos

Linear control theory is used to develop an improved localized control scheme for spatially extended chaotic systems, which is applied to a coupled map lattice as an example. The optimal arrangement of the control sites is shown to depend on the symmetry properties of the system, while their minimal density depends on the strength of noise in the system. The method is shown to work in any region of parameter space and requires a significantly smaller number of controllers compared to the method proposed earlier by Hu and Qu [Phys. Rev. Lett. 72, 68 (1994)]. A nonlinear generalization of the method for a 1D lattice is also presented

Robustness of predator-prey models for confinement regime transitions in fusion plasmas

Energy transport and confinement in tokamak fusion plasmas is usually determined by the coupled nonlinear interactions of small-scale drift turbulence and larger scale coherent nonlinear structures, such as zonal flows, together with free energy sources such as temperature gradients. Zero-dimensional models, designed to embody plausible physical narratives for these interactions, can help to identify the origin of enhanced energy confinement and of transitions between confinement regimes. A prime zero-dimensional paradigm is predator-prey or Lotka-Volterra. Here, we extend a successful three-variable (temperature gradient; microturbulence level; one class of coherent structure) model in this genre [M. A. Malkov and P. H. Diamond, Phys. Plasmas 16, 012504 (2009)], by adding a fourth variable representing a second class of coherent structure. This requires a fourth coupled nonlinear ordinary differential equation. We investigate the degree of invariance of the phenomenology generated by the model of Malkov and Diamond, given this additional physics. We study and compare the long-time behaviour of the three-equation and four-equation systems, their evolution towards the final state, and their attractive fixed points and limit cycles. We explore the sensitivity of paths to attractors. It is found that, for example, an attractive fixed point of the three-equation system can become a limit cycle of the four-equation system. Addressing these questions which we together refer to as “robustness” for convenience is particularly important for models which, as here, generate sharp transitions in the values of system variables which may replicate some key features of confinement transitions. Our results help to establish the robustness of the zero-dimensional model approach to capturing observed confinement phenomenology in tokamak fusion plasmas

Parallels between the dynamics at the noise-perturbed onset of chaos in logistic maps and the dynamics of glass formation

We develop the characterization of the dynamics at the noise-perturbed edge of chaos in logistic maps in terms of the quantities normally used to describe glassy properties in structural glass formers. Following the recognition [Phys. Lett. \textbf{A 328}, 467 (2004)] that the dynamics at this critical attractor exhibits analogies with that observed in thermal systems close to vitrification, we determine the modifications that take place with decreasing noise amplitude in ensemble and time averaged correlations and in diffusivity. We corroborate explicitly the occurrence of two-step relaxation, aging with its characteristic scaling property, and subdiffusion and arrest for this system. We also discuss features that appear to be specific of the map.Comment: Revised version with substantial improvements. Revtex, 8 pages, 11 figure

Bifurcations and Chaos in Time Delayed Piecewise Linear Dynamical Systems

We reinvestigate the dynamical behavior of a first order scalar nonlinear delay differential equation with piecewise linearity and identify several interesting features in the nature of bifurcations and chaos associated with it as a function of the delay time and external forcing parameters. In particular, we point out that the fixed point solution exhibits a stability island in the two parameter space of time delay and strength of nonlinearity. Significant role played by transients in attaining steady state solutions is pointed out. Various routes to chaos and existence of hyperchaos even for low values of time delay which is evidenced by multiple positive Lyapunov exponents are brought out. The study is extended to the case of two coupled systems, one with delay and the other one without delay.Comment: 34 Pages, 14 Figure

Dynamics towards the Feigenbaum attractor

We expose at a previously unknown level of detail the features of the dynamics of trajectories that either evolve towards the Feigenbaum attractor or are captured by its matching repellor. Amongst these features are the following: i) The set of preimages of the attractor and of the repellor are embedded (dense) into each other. ii) The preimage layout is obtained as the limiting form of the rank structure of the fractal boundaries between attractor and repellor positions for the family of supercycle attractors. iii) The joint set of preimages for each case form an infinite number of families of well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps in each of these families can be ordered with decreasing width in accord to power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. v) The power law with log-periodic modulation associated to the rate of approach of trajectories towards the attractor (and to the repellor) is explained in terms of the progression of gap formation. vi) The relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated.Comment: 8 pages, 12 figure

Triggering up states in all-to-all coupled neurons

Slow-wave sleep in mammalians is characterized by a change of large-scale cortical activity currently paraphrased as cortical Up/Down states. A recent experiment demonstrated a bistable collective behaviour in ferret slices, with the remarkable property that the Up states can be switched on and off with pulses, or excitations, of same polarity; whereby the effect of the second pulse significantly depends on the time interval between the pulses. Here we present a simple time discrete model of a neural network that exhibits this type of behaviour, as well as quantitatively reproduces the time-dependence found in the experiments.Comment: epl Europhysics Letters, accepted (2010

Tsallis' q index and Mori's q phase transitions at edge of chaos

We uncover the basis for the validity of the Tsallis statistics at the onset of chaos in logistic maps. The dynamics within the critical attractor is found to consist of an infinite family of Mori's $q$-phase transitions of rapidly decreasing strength, each associated to a discontinuity in Feigenbaum's trajectory scaling function $\sigma$. The value of $q$ at each transition corresponds to the same special value for the entropic index $q$, such that the resultant sets of $q$-Lyapunov coefficients are equal to the Tsallis rates of entropy evolution.Comment: Significantly enlarged version, additional figures and references. To be published in Physical Review