Discrete Dynamics in Nature and Society
Volume 1 (1997), Issue 1, Pages 45-56

Linear bifurcation analysis with applications to relative socio-spatial dynamics

M. Sonis

Department of Geography, Bar-Ilan University, Ramat-Gan 52900, Israel

Received 9 October 1996

Copyright © 1997 M. Sonis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 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.