Marcus Pivato

Interior symmetry and local bifurcation in coupled cell networks

A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells and edges that preserves all internal dynamics and all couplings. It is well known that symmetry can lead to patterns of synchronized cells, rotating waves, multirhythms, and synchronized chaos. Recently, the introduction of a less stringent form of symmetry, the ‘symmetry groupoid’, has shown that global group-theoretic symmetry is not the only mechanism that can create such states in a coupled cell system. The symmetry groupoid consists of structure-preserving bijections between certain subsets of the cell network, the input sets. Here, we introduce a concept intermediate between the groupoid symmetries and the global group symmetries of a network: ‘interior symmetry’. This concept is closely related to the groupoid structure, but imposes stronger constraints of a group-theoretic nature. We develop the local bifurcation theory of coupled cell systems possessing interior symmetries, by analogy with symmetric bifurcation theory. The main results are analogues for ‘synchrony-breaking’ bifurcations of the Equivariant Branching Lemma for steady-state bifurcation, and the Equivariant Hopf Theorem for bifurcation to time-periodic states.

Conservation laws in cellular automata

If is a discrete Abelian group and a finite set, then a cellular automaton (CA) is a continuous map :? that commutes with all -shifts. If :?, then, for any a, we define ?(a) = ?x(ax) (if finite); is conserved by if ? is constant under the action of . We characterize such conservation laws in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all one-dimensional CA exhibiting a given conservation law.

Limit measures for affine cellular automata II

Let \mathbb{M} be a monoid (e.g. \mathbb{N} , \mathbb{Z} , or \mathbb{M}^D ), and \mathcal{A} an abelian group. \mathcal{A}^\mathbb{M} is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism \mathfrak{F}:\mathcal{A}^\mathbb{M}\longrightarrow\mathcal{A}^\mathbb{M} that commutes with all shift maps. Let \mu be a (possibly non-stationary) probability measure on \mathcal{A}^\mathbb{M} ; we develop sufficient conditions on \mu and \mathfrak{F} so that the sequence \{\mathfrak{F}^N\mu\}_{N=1}^\infty weak* converges to the Haar measure on \mathcal{A}^\mathbb{M} in density (and thus, in Cesaro average as well). As an application, we show that, if \mathcal{A}=\mathbb{Z}_{/p} ( p prime), \mathfrak{F} is any ‘non-trivial’ LCA on \mathcal{A}^{(\mathbb{Z}^D)} , and \mu belongs to a broad class of measures (including most Bernoulli measures (for D \geq 1 ) and ‘fully supported’ N -step Markov measures (when D=1 )), then \mathfrak{F}^N\mu weak* converges to the Haar measure in density.

Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis

A new method is developed for estimating the spectral measure of a multivariate stable probability measure, by representing the measure as a sum of spherical harmonics.

The ergodic theory of cellular automata

Ergodic theory is the study of how a dynamical system transforms the information encoded in an invariant probability measure. This article reviews the major recent results in the ergodic theory of cellular automata.

Asymptotic randomization of sofic shifts by linear cellular automata

Let

The Condorcet set: Majority voting over interconnected propositions

{\mathbb{M}}={\mathbb{Z}}^D

Twofold optimality of the relative utilitarian bargaining solution

be a

Formal utilitarianism and range voting

D

Algebraic invariants for crystallographic defects in cellular automata

-dimensional lattice, and let