A general class of evolutionary equations

Abstract

Using observable quantities and state variable of a dynamical process, a general evolutionary equation is defined which unifies classical ordinary differential equations, partial differential equations, and hereditary systems of retarded and neutral type. Specific illustrations are given using transmission lines nearest neighbor coupled at the boundary and the theory of heat transfer in solids. Some spectral theory for linearization of the equations also is discussed.Za dopomohog spostereΩuvanyx velyçyn ta zminno] stanu dynamiçnoho procesu vyznaçeno zahal\ne evolgcijne rivnqnnq, wo uzahal\ng[ klasyçni zvyçajni dyferencial\ni rivnqnnq, dyferencial\ni rivnqnnq z çastynnymy poxidnymy ta spadkovi systemy iz zapiznennqm i systemy nejtral\noho typu. Navedeno specyfiçni ilgstraci] z vykorystannqm linij transmisi] iz zçeplennqm ,,najblyΩçyx susidiv" na meΩi ta teori] teploperenosu u tverdyx tilax. Rozhlqnuto takoΩ pevnu spektral\nu teorig dlq linearyzaci] rivnqn\. Introduction.Motivated by the fact that a dynamical system may evolve through an observable quantity rather than the state of the system, a general class of evolutionary equations is defined. This class includes standard ordinary and partial differential equations as well as functional differential equations of retarded and neutral type. In this way, the theory serves as a unification of these classical problems.Included in this general formulation is a general theory for the evolution of temperature in a solid material. In the general case, temperature is transmitted as waves with a finite speed of propagation. Special cases include a theory of delayed diffusion.We describe also in some detail a lattice on a circle where each point on the lattice is a transmission line for current and voltage whose dynamics is governed by a linear hyperbolic equation on [0, 1] with dynamic boundary conditons given by the circuitry on the line. The systems are coupled to their nearest neighbor at the end point 1 through resistors. A limiting process letting the distance between the lattice points approach zero leads to an interesting set of partial differential equations on [0, 1] × S 1 with a hyperbolic equation on [0, 1] and a parabolic equation on S 1 . These equations have not been analyzed in detail. However, we can show that the voltage at 1 satisfies a partial neutral functional differential equation. We analyze some properties of these equations including synchronization and the behavior of solutins near periodic orbits.There are other applications which involve partial differential equations on lattices for which the dynamics on each lattice point is governed by a partial differential equation on a bounded domain Ω. These systems could be coupled to neighbors through interaction on some subset of the boundary ∂Ω of Ω. If certain limiting processes are justified, one can obtain a partial differential equation on Ω together with another partial differential equation on another domain Ω 1 (determined by the nature of the lattice). We do not discuss this in the text, but ...