Bifurcation and stability: a computational approach

• Published in: Computer physics communications 1991-04, Vol.65 (1), p.299-309
• Main Author: Winters, K.H.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Mathematical methods in physics; Numerical approximation and analysis; Physics
• ISSN: 0010-4655; 1879-2944
• DOI: 10.1016/0010-4655(91)90183-L

Non-linearity gives rise to complex and unexpected behaviour in many physical, chemical and biological systems. The complexity in the steady-state behaviour may be encapsulated in a simple way by representing the various solutions to the governing equations in the form of state diagrams, which show the stability and domains of existence of the alternative solutions that arise at singular points, as a function of some parameter contained in the equations. A computational approach is described for deriving such state diagrams for general non-linear systems defined over spatial domains of arbitrary shape and governed by elliptic partial differential equations. The approach couples together an extended-systems methodology, in which the governing equations are supplemented by conditions satusfied at the singular points, with finite-element methods for solving the resulting extended systems of equations. The application of continuation techniques to the solution of the extended systems, to yield a bifurcation set that shows the dependence of bifurcation points on other parameters, is described. In the same way that the state diagram encapsulates the behaviour as one parameter changes, this bifurcation set captures the even more complex steady-state behaviour that occurs as two (or more) parameters change. In this way the bifurcation structure of the possible solutions is explored in an optimal way in multi-parameter space. The application of the extended-systems approach to predicting the onset of simple time-periodic behaviour at Hopf bifurcations is discussed, and the advantages of this approach over time-dependent methods are highlighted.