LOCBIF - the Interactive LOCal BIFurcation AnalyzerAn 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
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
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.
version of LOCBIF