Directional complexity and entropy for lift mappings

We introduce and study the notion of a directional complexity and entropy for maps of degree 1 on the circle. For piecewise affine Markov maps we use symbolic dynamics to relate this complexity to the symbolic complexity. We apply a combinatorial machinery to obtain exact formulas for the directional entropy, to find the maximal directional entropy, and to show that it equals the topological entropy of the map. Keywords: Rotation interval, Space-time window, Directional complexity, Directional entropy;Comment: 19p. 3 fig, Discrete and Continuous Dynamical Systems-B (Vol. 20, No. 10) December 201

Space-time directional Lyapunov exponents for cellular automata

Space-time directional Lyapunov exponents are introduced. They describe the maximal velocity of propagation to the right or to the left of fronts of perturbations in a frame moving with a given velocity. The continuity of these exponents as function of the velocity and an inequality relating them to the directional entropy is proved

Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincare section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking.Comment: To be published in CHAO

A new formalism for the estimation of the CP-violation parameters

In this paper, we use the time super-operator formalism in the 2-level Friedrichs model \cite{fried} to obtain a phenomenological model of mesons decay. Our approach provides a fairly good estimation of the CP symmetry violation parameter in the case of K, B and D mesons. We also propose a crucial test aimed at discriminating between the standard approach and the time super-operator approach developed throughout the paper

Multidimensional Gaussian sums arising from distribution of Birkhoff sums in zero entropy dynamical systems

A duality formula, of the Hardy and Littlewood type for multidimensional Gaussian sums, is proved in order to estimate the asymptotic long time behavior of distribution of Birkhoff sums $S_n$ of a sequence generated by a skew product dynamical system on the $\mathbb{T}^2$ torus, with zero Lyapounov exponents. The sequence, taking the values $\pm 1$, is pairwise independent (but not independent) ergodic sequence with infinite range dependence. The model corresponds to the motion of a particle on an infinite cylinder, hopping backward and forward along its axis, with a transversal acceleration parameter $\alpha$. We show that when the parameter $\alpha /\pi$ is rational then all the moments of the normalized sums $E((S_n/\sqrt{n})^k)$, but the second, are unbounded with respect to n, while for irrational $\alpha /\pi$, with bounded continuous fraction representation, all these moments are finite and bounded with respect to n.Comment: To be published in J. Phys.

Fractalization of Torus Revisited as a Strange Nonchaotic Attractor

Fractalization of torus and its transition to chaos in a quasi-periodically forced logistic map is re-investigated in relation with a strange nonchaotic attractor, with the aid of functional equation for the invariant curve. Existence of fractal torus in an interval in parameter space is confirmed by the length and the number of extrema of the torus attractor, as well as the Fourier mode analysis. Mechanisms of the onset of fractal torus and the transition to chaos are studied in connection with the intermittency.Comment: Latex file ( figures will be sent electronically upon request):submitted to Phys.Rev. E (1996

Towards generalized measures grasping CA dynamics

In this paper we conceive Lyapunov exponents, measuring the rate of separation between two initially close configurations, and Jacobians, expressing the sensitivity of a CA's transition function to its inputs, for cellular automata (CA) based upon irregular tessellations of the n-dimensional Euclidean space. Further, we establish a relationship between both that enables us to derive a mean-field approximation of the upper bound of an irregular CA's maximum Lyapunov exponent. The soundness and usability of these measures is illustrated for a family of 2-state irregular totalistic CA

Chaotic oscillations in a map-based model of neural activity

We propose a discrete time dynamical system (a map) as phenomenological model of excitable and spiking-bursting neurons. The model is a discontinuous two-dimensional map. We find condition under which this map has an invariant region on the phase plane, containing chaotic attractor. This attractor creates chaotic spiking-bursting oscillations of the model. We also show various regimes of other neural activities (subthreshold oscillations, phasic spiking etc.) derived from the proposed model