arXiv subjects

F. Zertuche

Publications and source records attributed to F. Zertuche.

6 recordsLinked to original sources

arXiv2006

Discrete Dynamical Systems Embedded in Cantor Sets

While the notion of chaos is well established for dynamical systems on manifolds, it is not so for dynamical systems over discrete spaces with $ N $ variables, as binary neural networks and cellular automata. The main difficulty is the choice of a suitable topology to study the limit $N\to\infty$. By embedding the discrete phase space into a Cantor set we provided a natural setting to define topological entropy and Lyapunov exponents through the concept of error-profile. We made explicit calculations both numerical and analytic for well known discrete dynamical models.

nlin.CD↗

arXiv2003

Quantum Dynamical Entropies in Discrete Classical Chaos

We discuss certain analogies between quantization and discretization of classical systems on manifolds. In particular, we will apply the quantum dynamical entropy of Alicki and Fannes to numerically study the footprints of chaos in discretized versions of hyperbolic maps on the torus.

math-ph↗

arXiv1998

Discrete Chaos

We propose a theory of deterministic chaos for discrete systems, based on their representations in binary state spaces $ Ω$, homeomorphic to the space of symbolic dynamics. This formalism is applied to neural networks and cellular automata; it is found that such systems cannot be viewed as chaotic when one uses the Hamming distance as the metric for the space. On the other hand, neural networks with memory can in principle provide examples of discrete chaos; numerical simulations show that the orbits on the attractor present topological transitivity and a dimensional phase space reduction. We compute this by extending the methodology of Grassberger and Procaccia to $ Ω$. As an example, we consider an asymmetric neural network model with memory which has an attractor of dimension $D_a = 2$ for $N = 49$.

chao-dyn↗

arXiv1995

Chaotic Dynamics of Binary Systems

We propose a theory of chaos for discrete systems, based on their representation in a space of ``binary histories'', $ {\cal B^{\infty}} $. We show that $ {\cal B^{\infty}} $ is a metrizable Cantor set which embeds the attractor $Λ$, itself also a Cantor set.

chao-dyn↗

arXiv1994

Recognition of Temporal Sequences of Patterns

Using an asymmetric associative network with synchronous updating, it is possible to recall a sequence of patterns. To obtain a stable sequence generation with a large storage capacity, we introduce a threshold that eliminates the contribution of weakly correlated patterns. For this system we find a set of evolution equations for the overlaps of the states with the patterns to be recognized. We solve these equations in the limit of the stationary cycle, and obtain the critical value of the capacity as a function of the threshold and temperature. Finally, a numerical simulation is made, confirming the theoretical results.

comp-gas↗

arXiv1994

Homotopy Invariants and Time Evolution in (2+1)-Dimensional Gravity

We establish the relation between the ISO(2,1) homotopy invariants and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2,1) cycle conditions, are equivalent to the ISO(2,1) cycle conditions for the representa- tions of the fundamental group in ISO(2,1). Also, the symplectic structure on the space of invariants is closely related to that of the polygon representation. We choose one of the polygon variables as internal time and compute the Hamiltonian, then perform the Hamilton-Jacobi transformation explicitly. We make contact with other authors' results for g = 1 and g = 2 (N = 0).

gr-qc↗