Computing the location and the direction of bifurcation

Abstract

Abstract. We consider positive solutions of the Dirichlet problemdepending on a positive parameter λ. Each solution u(x) is an even function, and hence it is uniquely identified by α = u(0). We present a formula, which allows to compute all α's where a turn may occur, and then we give another formula, which allows to compute the direction of the turn. As an application, we present a computer assisted proof of the exact bifurcation diagram in case f (u) is any cubic with real and distinct roots. Another application is a computer assisted proof of a conjecture by S.-H. Wang [10], related to gas combustion.

Computing the location and the direction of bifurcation for sign changing solutions

“…If a turn does occur, we give another formula allowing to compute the direction of the turn. Our results generalize those in P. Korman, Y. Li and T. Ouyang [6], where positive solutions were considered. We give similar results for Neumann problem.…”

A global solution curve for a class of periodic problems, including the pendulum equation

“…u(0), uniquely identifies both the value of the parameter λ and the solution u(x), see e.g. [9]. The above result gives an exact analogy for the periodic solutions.…”

Verification of bifurcation diagrams for polynomial-like equations

“…In [10] we provided a necessary and sufficient condition on for the solution to be singular and thus a necessary condition for the turning point to occur:…”

“…< b 2n−1 < b 2n , and a positive parameter . In case n = 1, i.e., when f (u) is a cubic, this problem was studied in [12,13], where time-maps were used, and in our papers [7][8][9][10], where we used bifurcation approach. We present here two computer-assisted approaches to the exact multiplicity of positive solutions.…”

“…We wish to validate this bifurcation diagram, i.e., to prove its correctness. The tools come from our recent paper [10], where we presented a formula, which allows one to compute all 's where a turn may occur, and another formula, which allows to compute the direction of the turn, and to prove that the turning point is non-degenerate, i.e., it persists under small perturbations.…”