under license by Gordon and Breach Science Publishers Printed in India Linear Bifurcation Analysis with Applications to Relative Socio-Spatial Dynamics

The objective of this research is the elaboration of elements of linear bifurcation analysis for the description the qualitative properties of orbits of the discrete autonomous itera-tion 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