Jesús Urías

Multimodal synchronization of chaos

An elementary notion of master-slave synchronization that accepts multimodal synchronization is introduced. We prove rigorously that the attractor of a coupled pair in a regime of multimodal synchronization is the graph of a multivalued function. Our framework provides the theoretical basis for some practical tools for detection of multimodal synchrony in experiments. Results are illustrated with the analysis of experiments with coupled electronic oscillators.

A cryptosystem based on cellular automata

Cryptosystems for binary information are based on two primitives: an indexed family of permutations of binary words and a generator of pseudorandom sequences of indices. A very efficient implementation of the primitives is constructed using the phenomenon of synchronization in cellular automata. (c) 1998 American Institute of Physics.

Fractal Dimensions for Poincaré Recurrences

1. Introduction Part 1: Fundamentals 2. Symbolic Systems 3. Geometric Constructions 4. Spectrum of Dimensions for Recurrences Part II: Zero-Dimensional Invariant Sets 5. Uniformly Hyperbolic Repellers 6. Non-Uniformly Hyperbolic Repellers 7. The Spectrum for a Sticky Set 8. Rhythmical Dynamics Part III: One-Dimensional Systems 9. Markov Maps of the Interval 10. Suspended Flows Part IV: Measure Theoretical Results 11. Invariant Measures 12. Dimensional for Measures 13. The Variational Principle Part V: Physical Interpretation and Applications 14. Intuitive Explanation 15. Hamiltonian Systems 16. Chaos Synchronization Part VI: Appendices 17. Some Known Facts About Recurrences 18. Birkhoffs Individual Theorem 19. The SMB Theorem 20. Amalgamation and Fragmentation Index

Sensitive dependence on initial conditions for cellular automata

The property of sensitive dependence on intial conditions is the basis of a rigorous mathematical construction of local maximum Lyapunov exponents for cellular automata. The maximum Lyapunov exponent is given by the fastest average velocity of either the left or right propagating damage fronts. Deviations from the long term behavior of the finite time Lyapunov exponents due to generation of information are quantified and could be used for the characterization of the space time complexity of cellular automata. (c) 1997 American Institute of Physics.

Synchronization of cellular automaton pairs

The phenomenon of synchronization in pairs of cellular automata coupled in a driver-replica mode is studied. Necessary and sufficient conditions for synchronization in linear cellular automaton pairs are given. The couplings that make a pair synchronize are determined for all linear elementary cellular automata. (c) 1998 American Institute of Physics.

Symbolic dynamics for sticky sets in Hamiltonian systems

Hamiltonian systems, possessing an infinite hierarchy of islands-around-islands structure, have sticky sets, sets of all limiting points of islands of stability. A class of symbolic systems, called multipermutative, is introduced to model the dynamics in the sticky (multifractal) sets. Every multipermutative system is shown to consist of a collection of minimal subsystems that are topologically conjugate to adding machines. These subsystems are uniquely ergodic. Sufficient and necessary conditions of topological conjugacy are given. A subclass of sticky sets is constructed for which Hausdorff dimension is found and multifractal decomposition is described.

An algebraic measure of complexity

Abstract Discrete dynamical systems are defined as a set S of states provided with an evolution operator U: S → S; evolution proceeds in discrete time steps. A reading operator σ acting on S that generates a group (σ), is introduced and determines what are the state aspects that are observable. State configurations are thus defined as sets of states that cannot be resolved by σ. The set of configurations is ordered in complexity layers, defined as classes of configurations that are stabilized by isomorphic subgroups of (σ). Layers are indexed by the quotient (σ)/H, where H⊂(σ), up to an isomorphism, stabilizes the configurations in a given layer. Generally, the algebraic complexity is defined as (σ)/H, while for a finite (σ) a complexity number, κσ(x)=[(σ):Gx], is defined, where x ϵ S, and Gx⊂(σ) stabilizes x. Irreversibility of evolution implies that Gx⊂GU.x or, for finite (σ), Δκσ≥0, at every time step in evolution, provided that [σ, U] = 0. This effect of complexity degredation gives the dynamical system the property of self-similarity. Translation complexity is used to study one-dimensional cellular automata.

On the wavelet formalism for multifractal analysis.

It is proved that the multifractal characterizations of diametrically regular measures that are provided by the wavelet and by the Hentschel-Procaccia formalisms are identical. (c) 2001 American Institute of Physics.

Traveling patterns in cellular automata

A method to identify the invariant subsets of bi-infinite configurations of cellular automata that propagate rigidly with a constant velocity nu is described. Causal traveling configurations, propagating at speeds not greater than the automaton range, mid R:numid R:</=r, are considered. The sets of traveling configurations are presented by finite automata and its topological entropy is calculated. When the invariant subset of traveling configurations has nonzero topological entropy, the dynamics is dominated by the interaction of domains, composed of traveling patterns of finite size. The sets of traveling patterns and domains are presented by finite automata. End-resolving CA are shown to always have sets of traveling configurations that are spatially periodic with zero entropy, except possibly for traveling configurations at top speed. The elementary CA are examined exhaustively along these lines. (c) 1996 American Institute of Physics.

Householder factorizations of unitary matrices

A method to construct all representations of finite dimensional unitary matrices as the product of Householder reflections is given. By arbitrarily severing the state space into orthogonal subspaces, the method may, e.g., identify the entangling and single-component quantum operations that are required in the engineering of quantum states of composite (multipartite) systems. Earlier constructions are shown to be extreme cases of the unifying scheme that is presented here.