An important case of hybrid systems are the rectangular automata. First, rectangular dynamics can naturally and arbitrarily closely approximate more general, nonlinear dynamics. Second, rectangular automata are the most general type of hybrid systems for which model checking |in particular, Ltl model checking| is decidable. However, on one hand, the original proofs of decidability did not sugge...

This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a “dynamical” sense. This means precisely that they (semi-) conjugate a given continuous (or measure-preserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings c...

Gradient Symbolic Computation is proposed as a means of solving discrete global optimization problems using a neurally plausible continuous stochastic dynamical system. Gradient symbolic dynamics involves two free parameters that must be adjusted as a function of time to obtain the global maximizer at the end of the computation. We provide a summary of what is known about the GSC dynamics for s...

Beat-to-beat variations in heart period provide information on cardiovascular control and are closely linked to variations in arterial pressure and respiration. Joint symbolic analysis of heart period, systolic arterial pressure and respiration allows for a simple description of their shared short-term dynamics that are governed by cardiac baroreflex control and cardiorespiratory coupling. In t...

A piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks. A symbolic representation of the dynamics in this equation is given as a directed graph on an N-dimensional hypercube. This provides a formal link with discrete neural networks such as the original Hopfield models. Analytic criteria are given to establish steady states and limit cycle osci...

The elliptic isosceles restricted three body problem (REI3BP) models the motion of a massless under influence Newtonian gravitational force caused by two other bodies called primaries. primaries masses $m_{1}=m_{2}$ move along degenerate Keplerian collision orbit (on line) their attraction, whereas third, particle, moves on plane perpendicular to line and passing through center mass By symmetry...