A structural approach to control of transitional waves of nonlinear diffusion‐reaction systems: Theory and possible applications to bio‐medicine

Abstract

This paper has considered a novel approach to structural recognition and control of nonlinear reaction-diffusion systems (systems with density dependent diffusion). The main consistence of the approach is interactive variation of the nonlinear diffusion and sources structural parameters that allows to implement a qualitative control and recognition of transitional system conditions (transients). The method of inverse solutions construction allows formulating the new analytic conditions of compactness and periodicity of the transients that is also available for nonintegrated systems. On the other hand, using of energy conservations laws, allows transfer to nonlinear dynamics models that gives the possiblity to apply the modern deterministic chaos theory (particularly the Feigenboum's universal constants and scenario of chaotic transitions)

Nonlinear reaction‐diffusion models of self‐organization and deterministic chaos: Theory and possible applications to description of electrical cardiac activity and cardiovascular circulation

“…are fulfilled, which corresponds to the assumption that m > 0, the finite-localized (compact) waves are formed [10]. From our point of view, the strong nonlinear phenomena of the steady state localized structure forming in the heart electrical potential distribution can be considered as a powerful interblocking factor that may cause acute heart failure, arrhythmia, and a transition to venticular fibrillation.…”

“…It can be shown that for wide classes of structural functions f ,g, for instance, for even functions, the solutions of problem (3.1) determined by formula (3.9) are by three parameters dependent periodic function. On some critical values of these parameters and structural functions coefficients these solutions transfer to transitional waves (transients) of the two following types: compact solutions of the solitons type (finite-localized transients: compactons or peakons [11,15]) and compact shock waves that, in contrast to the traditional transients, have finite-transitional time or periodic structure (see the recent author's paper [10]).…”

“…We suppose that f (u) = a + a 1 u + bu 2 10 Nonlinear reaction-diffusion models Below we will consider the localized solutions of (3.7) (see the conditions of its formation, e.g., in [10]). It is evident from (3.7) that the conditions of the solution localization are the equality E = 0 and the requirement that the number ϕ 0 = 0 is two-multiple root of the equation…”