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LOCBIF - the Interactive LOCal BIFurcation Analyzer

An interactive tool for bifurcation analysis of non-linear ordinary differential equations ODE's and maps
by A. Khibnik, Y.A. Kuznetsov, V.Levitin and E.V. Nikolaev

What is LOCBIF?

LOCBIF is an interactive tool for bifurcation analysis of non-linear ordinary differential equations (ODE's) and iterated maps which depend upon parameters. With LOCBIF you can explore interactively
1) existence, stability and bifurcations of equilibrium and periodic solutions of autonomous ODE's,
2) fixed points and periodic orbits (cycles) of iterated maps,
3) periodic solutions of periodically forced non autonomous ODE's.

LOCBIF is based on a predictorнcorrector continuation procedure of computing curves of structurallyн unstable (critical) solutions in a proper phaseнparameн ter space. Projection of these curves onto the parameter space determines the boundaries of the existence and stability of the equilibria and cycles. It can handle up to ten phase variables and ten parameters. LOCBIF combines modern results on normal form and local bifurcation theory with interactive computer graphics, resulting in a unique easyнtoнuse environment for the analysis of nonнlinear dynamical systems.

Range of Applications

LOCBIF has been successfully applied to the analysis of many non-linear dynamical models from
1) chemical and electronic engineering,
2) ecosystem and disease modelling,
3) neural networks,
4) adaptive control and economics modelling.
LOCBIF has been used to support several undergraduate and graduate courses in Dynamical System Theory at a number of universities throughout Europe.

What can LOCBIF do for you?

LOCBIF plots two-dimensional projections of the orbits, equilibrium and bifurcation curves in the course of the computation which may be stepwise or automatic. Scales, projections and other graphical attributes can be defined interactively. LOCBIF mainн tains an archive of models and allows you to specify a new ODE or a new map during a session in a simple Pascalнlike language. It compiles the equations by means of a builtнin onнline compiler. Computed curves may be stored in several forms and then may be extracted, plotted and used for further computations.

Which Bifurcations are Supported?

For autonomous ODE's, LOCBIF computes orbits by numerical integration using one of the standard ODE solvers (including a stiff one). It continues equilibrium curves in one active parameter and detects fold (saddleнnode) and Hopf bifurcations. These bifurcaн tions can be continued in two active parameters, and the following higherнorder singularities can be detected: generalized Hopf bifurcation, Bogdanov- Takens bifurcation, cusp, foldнHopf and double Hopf singularities. Continuation of the most of these bifurcations in three active parameters and detection and continuation of some codimension three bifurcations is also supported. While continuing a periodic solution (cycle), LOCBIF detects the fold, periodнdoubling and the secondary Hopf (torus generation) bifurcations and allows one to continue them, as well as some other higherнorder singularities. The mathematical background of LOCBIF is presented in the paper by A.I. Khibnik, Yu.A. Kuznetsov, V.V. Levitin, E.V. Nikolaev, ``Continuation techniques and interactive software for bifurcation analysis of ODE's and iterated maps'', Physica D 62, 1993, 360-371.


LOCBIF is a result of several year evolution. The development of LOCBIF was initiated at the Research Computing Centre of the former USSR Academy of Sciences and then continued at the Politecnico di Milano and Katholieke Universiteit Leuven. Its first version has been presented in 1989. LOCBIF combines numerical continuation algorithms developed by Alexander Khibnik and Eugene Nikolaev with a modern interface designed by Yuri Kuznetsov, who started and coordinated the project, and the builtнin compiler by Victor Levitin.


Old MS-DOS version of LOCBIF

CAN Distribution of Software Products

Article "LOCBIF the Interactive LOCal BIFurcation Analyzer" by David Knapp

Article "A Comparison of LOCBIF and AUTO" by Andrew M. Edwards

Brief Intro to LOCBIF as part of the Windows version of XPP

Computational Resources from sci.nonlinear FAQ by Jim Meiss

Recent improved version of LOCBIF - CONTENT

AUTO is a software for continuation and bifurcation problems in ordinary differential equations developed by Eusebius Doedel.

Andrew B. Ryzhkov Page

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