• Corpus ID: 117173067

Lectures on Numerical Methods in Bifurcation Problems

@inproceedings{Keller1988LecturesON,
  title={Lectures on Numerical Methods in Bifurcation Problems},
  author={Herbert B. Keller},
  year={1988}
}
  • H. Keller
  • Published 20 January 1988
  • Mathematics
These lectures introduce the modern theory and practical numerical methods for continuation of solutions of nonlinear problems depending upon parameters. Bifurcations are one of the many types of singularities that occur along such solution paths and their computation and methods for switching branches are treated. Homotopy methods and degree theory are introduced as are global Newton methods, constructive determination of Brouwer fixed points, periodic solutions of O.D.E.s, Hopf bifurcations… 
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References

SHOWING 1-10 OF 35 REFERENCES
Folds in Solutions of Two Parameter Systems and Their Calculation. Part I
This paper is concerned with paths of turning points in solutions of nonlinear systems having two parameters. It is well known that these paths are solutions of a particular extended system of
Bifurcation in difference approximations to two-point boundary value problems
Numerical methods for bifurcation problems of the form (*) LyAXf(y), By =0, where f(0) = 0 and f'(0) * 0, are considered. Here y is a scalar function, X is a real scalar, L is a linear differential
Elementary stability and bifurcation theory
Asymptotic solutions of evolution problems bifurcation and stability of steady solutions of evolution equations in one dimension imperfection theory and isolated solutions which perturb bifurcation
Topics in stability and bifurcation theory
Nonlinear elliptic boundary value problems of second order.- Functional analysis.- Bifurcation at a simple eigenvalue.- Bifurcation of periodic solutions.- The mathematical problems of hydrodynamic
Widely convergent method for finding multiple solutions of simultaneous nonlinear equations
A new method has been developed for solving a system of nonlinear equations g(x) = 0. This method is based on solving the related system of differential equations dg/dt±g(x)= 0 where in the sign is
Steady State and Periodic Solution Paths: their Bifurcations and Computations
TLDR
This work presents a brief account of the theory and numerical methods for the analysis and Solution of nonlinear autonomous differential equations of the form d w = f ( w,lambda, alpha ) to f ( B_1, R 2) to B_2.
The Hopf Bifurcation and Its Applications
The goal of these notes is to give a reasonably complete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to specific problems, including
Quasi-Newton Methods, Motivation and Theory
This paper is an attempt to motivate and justify quasi-Newton methods as useful modifications of Newton''s method for general and gradient nonlinear systems of equations. References are given to
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