For real t and a vector function y(t), we study the model problem
where \(y(t) \in I R^{n}\), \(f(y,\lambda ) \in I R^{n}\), and λ is a real parameter. The overdot refers to the derivative of y(t). Stationary (constant) solutions satisfy
Equations (1) or (2) may result from discretizations of other equations. Solutions of (1) or (2), if they exist, depend on λ. For convenience write Y : = (y, λ). Solutions of (2), f(Y ) = 0, form curves in \(I R^{n+1}\). Provided these continua satisfy the full-rank condition
they can be extended. (The subscripts in (3) denote first-order partial derivatives.) Similarly, in a proper sense, periodic solutions of (1) form continua. The continua of solutions are called branches. For a graphical illustration of branches (Fig. 1), depict a scalar measure [y] of the vector yover...
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.J.: AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations. http://cmvl.cs.concordia.ca (1997)
Govaerts, W.: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia (2000)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields. Springer, New York (1983)
Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, New York (1998)
Seydel, R.: Practical Bifurcation and Stability Analysis, 3rd edn. Springer, New York (2010)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Seydel, R. (2015). Bifurcations: Computation. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_384
Download citation
DOI: https://doi.org/10.1007/978-3-540-70529-1_384
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70528-4
Online ISBN: 978-3-540-70529-1
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering