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Numerical Bifurcation Analysis

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Mathematics of Complexity and Dynamical Systems

Article Outline

Glossary

Definition of the Subject

Introduction

Continuation and Discretization of Solutions

Normal Forms and the Center Manifold

Continuation and Detection of Bifurcations

Branch Switching

Connecting Orbits

Software Environments

Future Directions

Bibliography

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Abbreviations

Dynamical system:

A rule for time evolution on a state space. The term system will be used interchangeably. Here a system is a family given by an ordinary differential equation (ODE) depending on parameters.

Equilibrium:

A constant solution of the system, for given parameter values.

Limit cycle:

An isolated periodic solution of the system, for given parameter values.

Bifurcation:

A qualitative change in the dynamics of a dynamical system produced by changing its parameters. Bifurcation points are the critical parameter combinations at which this happens for arbitrarily small parameter perturbations.

Normal form:

A simplified model system for the analysis of a certain type of bifurcation.

Codimension:

The minimal number of parameters needed to perturb a family of systems in a generic manner.

Defining system:

A set of suitable equations so that the zero set corresponds to a bifurcation of a certain type or to a particular solution of the system. Also called defining function or equation.

Continuation:

A numerical method suited for tracing one‐dimensional manifolds, curves (here called branches) of solutions for a defining system while one or more parameters are varied.

Test function:

A function designed to have a regular zero at a bifurcation. During continuation a test function can be monitored to detect bifurcations.

Branch switching:

Several branches of different codimension can emanate from a bifurcation point. Switching from the computation of one branch to an other requires appropriate procedures.

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Mathematics and Computer Science, University of Twente, Enschede, The Netherlands

    Hil Meijer

  2. Department of Electronics and Information, Politecnico di Milano, Milano, Italy

    Fabio Dercole

  3. Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada

    Bart Oldeman

Authors
  1. Hil Meijer
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  2. Fabio Dercole
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  3. Bart Oldeman
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Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Meijer, H., Dercole, F., Oldeman, B. (2012). Numerical Bifurcation Analysis. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_71

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