Nonlinear phenomena, which are so important in nature and society, are considered here in relation to the world of algorithms and computations. To have a mathematical model for this world, formal computability spaces are introduced. It is demonstrated that the traditional approach to algorithms, which is based on such popular models as Turing machines, results in linear subspaces of the computability space. Nonlinear phenomena appear when we go to the more powerful class of such super-recursive algorithms as inductive Turing machines. It is demonstrated how this nonlinearity imports much higher computing power of inductive Turing machines in comparison with conventional Turing machines. This provides a base to consider problems of chaos, emergent computations and infinity from the algorithmic point of view.

Nonlinear phenomena in spaces of algorithms

A new theoretical foundation for the discrete dynamics of physicochemical systems is presented. Based on the analogy between the π-theorem of the theory of dimensionality, the second law of thermodynamics and the stoichiometry of complex physicochemical reactions, basic dynamic equations and an extreme principle were formulated. The meaning of discrete time and space in the proposed equations is discussed. Some results of numerical calculations are presented to demonstrate the potential of the proposed approach to the mathematical simulation of spatiotemporal physicochemical reaction dynamics.

/pdf/theoretical-foundation-for-the-discrete-dynamics-of-2i4nn26zcr.pdf

Theoretical foundation for the discrete dynamics of physicochemical systems: Chaos, self-organization, time and space in complex systems

for the description the qualitative properties of orbits of the discrete autonomous iteration processes on the basis of linear approximation of the processes. The basic element of this analysis is the geometrical and numerical modification and application of the classical Routhian formalism, which is giving the description of the behavior of the iteration processes near the boundaries of the stability domains of equilibria. The use of the Routhian formalism is leading to the mapping of the domain of stability of equilibria from the space of control bifurcation parameters into the space of orbits of iteration processes. The study of the behavior of the iteration processes near the boundaries of stability domains can be achieved by the converting of coordinates of equilibria into control bifurcation parameters and by the movement of equilibria in the space of orbits. The crossing the boundaries of the stability domain reveals the plethora of the possible ways from stability, periodicity, the Arnold mode-locking tongues and quasi-periodicity to chaos. The numerical procedure of the description of such phenomena includes the spatial bifurcation diagrams in which the bifurcation parameter is the equilibrium itself. In this way the central problem of control of bifurcation can be solved: for each autonomous iteration process with big enough number of external parameters construct the realization of this iteration process with a preset combination of qualitative properties of equilibria. In this study the two-dimensional geometrical and numerical realizations of linear bifurcation analysis is presented in such a form which can be easily extended to multi-dimensional case. Further, a newly developed class of the discrete relative m-population]n-location Socio-Spatial dynamics is described. The proposed algorithm of linear bifurcation analyses is used for the detail analysis of the log-log-linear model of the one population]three location discrete relative dynamics.

/pdf/linear-bifurcation-analysis-with-applications-to-relative-3s6v13p9td.pdf

Linear bifurcation analysis with applications to relative socio-spatial dynamics

The theory of hypernumbers and extrafunctions is a novel approach in functional analysis aimed at problems of mathematical and computational physics. The new technique allows operations with divergent integrals and series and makes it possible to distinct different kinds of convergence and divergence. Although, it resembles nonstandard analysis, there are several distinctions between these theories. For example, while nonstandard analysis changes spaces of real and complex numbers by injecting into them infinitely small numbers and other nonstandard entities, the theory of extrafunctions does not change the inner structure of spaces of real and complex numbers, but adds to them infinitely big and oscillating numbers as external objects. In this paper, we consider a simplified version of hypernumbers, but a more general version of extrafunctions and their extraderivatives in comparison with previous works.

/pdf/theory-of-hypernumbers-and-extrafunctions-functional-spaces-1uyvihwtke.pdf

Theory of hypernumbers and extrafunctions: Functional spaces and differentiation

This paper has considered a novel approach to structural recognition and control of nonlinear reactiondiffusion 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 ofenergy conservations laws, allows transfer to nonlinear dynamics models that gives the possiblity to apply the modem deterministic chaos theory (particularly the Feigenboum’s universal constants and scenario of chaotic transitions).

/pdf/a-structural-approach-to-control-of-transitional-waves-of-2n383zbzde.pdf

A Structural Approach to Control of Transitional Waves of Nonlinear Diffusion-reaction Systems: Theory and Possible Applications to Bio-medicine

The work advances a discrete mathematical model of physico-chemical reactions. Possibilities of the described model application for calculation of physico-chemical systems with complicated behaviour are analyzed.

New Mathematical Model of Physico‐Chemical Dynamics

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New Mathematical Model of Physico‐Chemical Dynamics