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 iteration processes on the basis of linear approximation of the processes. The basic element of this analysis is the geomet...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
1997-01-01
|
| Series: | Discrete Dynamics in Nature and Society |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S102602269700006X |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850171091239043072 |
|---|---|
| author | M. Sonis |
| author_facet | M. Sonis |
| author_sort | M. Sonis |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a48a4823ff404c2fb3b31ffc22b65352 |
| institution | OA Journals |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 1997-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a48a4823ff404c2fb3b31ffc22b653522025-08-20T02:20:21ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X1997-01-0111455610.1155/S102602269700006XLinear bifurcation analysis with applications to relative socio-spatial dynamicsM. Sonis0Department of Geography, Bar-Ilan University, Ramat-Gan 52900, IsraelThe 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.http://dx.doi.org/10.1155/S102602269700006XControl of bifurcationsDiscrete non-linear dynamicsDiscrete relative m-population/n-location Socio-spatial dynamics. |
| spellingShingle | M. Sonis Linear bifurcation analysis with applications to relative socio-spatial dynamics Discrete Dynamics in Nature and Society Control of bifurcations Discrete non-linear dynamics Discrete relative m-population/n-location Socio-spatial dynamics. |
| title | Linear bifurcation analysis with applications to relative socio-spatial dynamics |
| title_full | Linear bifurcation analysis with applications to relative socio-spatial dynamics |
| title_fullStr | Linear bifurcation analysis with applications to relative socio-spatial dynamics |
| title_full_unstemmed | Linear bifurcation analysis with applications to relative socio-spatial dynamics |
| title_short | Linear bifurcation analysis with applications to relative socio-spatial dynamics |
| title_sort | linear bifurcation analysis with applications to relative socio spatial dynamics |
| topic | Control of bifurcations Discrete non-linear dynamics Discrete relative m-population/n-location Socio-spatial dynamics. |
| url | http://dx.doi.org/10.1155/S102602269700006X |
| work_keys_str_mv | AT msonis linearbifurcationanalysiswithapplicationstorelativesociospatialdynamics |