An interation method for studying the bifurcation of solutions of the nonlinear equations,L(?)u+?R(?,u)=0

Demoulin, Yves-Marie J.; Chen, Y. M.

doi:10.1007/bf01409990

Cited by 4 publications

(2 citation statements)

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“…We seek a branch (y(., e), 2(e)) of nontrivial solutions of (2.1) near (0, 0) in the form y(t, ~) = e ~b(t) + ~2 w(t, ~), 6) for IE] <%, w and r/ continuous in e, and w orthogonal to qS, that is (w, q~)=0. Note that e= (y, qS), so (2.6) says that the nullspace of (2.4) is "tangent" to the branch y(-, e) at e = 0.…”

Section: Q(y)--89mentioning

confidence: 98%

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Numerical solution of bifurcation problems for ordinary differential equations

Langford

1977

Numer. Math.

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Summary. An iterative algorithm has been developed for the numerical solution of bifurcation problems involving ordinary differential equations. It is based on a new method for computing generalized inverses, and the classical Liapunov-Schmidt theory. A novel formulation of the Hopf bifurcation theorem makes possible the treatment of both simple bifurcation and Hopf bifurcation by the same iterative approach. The method has been applied to the Euler rod buckling problem as an example of simple bifurcation, and to a biological feedback-inhibition control system, as an example of Hopf bifurcation. The results exhibit accuracy and efficiency in a sizable neighborhood of the bifurcation point, and in the second example reveal interesting behaviour not shown by the Poincar6-Hopf asymptotic solution or by previous numerical solutions.

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Section: Q(y)--89mentioning

confidence: 98%

“…The idea of the iteration scheme was presented by Keller in [15], and was refined in [17]. Similar schemes have been described in [6,26,30]. Section 3 describes the numerical implementation of the theory of Section 2.…”

Section: Introductionmentioning

confidence: 96%