Sshaped bifurcations in a twodimensional Hamiltonian system
We study the solutions to the following Dirichlet boundary problem: d 2x(t) dt2 + λ f(x(t)) = 0, where x ∈ R, t ∈ R, λ ∈ R+, with boundary conditions: x(0) = x(1) = A ∈ R. Especially we focus on varying the parameters λ and A in the case where the phase plane representation of the equation contains...
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Format:  Serial 
Published: 
2021

Series:  Electronic journal of qualitative theory of differential equations

Kulcsszavak:  Hamiltonrendszer, Bifurkáció 
doi:  10.14232/ejqtde.2021.1.49 
Online Access:  http://acta.bibl.uszeged.hu/73701 
Summary:  We study the solutions to the following Dirichlet boundary problem: d 2x(t) dt2 + λ f(x(t)) = 0, where x ∈ R, t ∈ R, λ ∈ R+, with boundary conditions: x(0) = x(1) = A ∈ R. Especially we focus on varying the parameters λ and A in the case where the phase plane representation of the equation contains a saddle loop filled with a period annulus surrounding a center. We introduce the concept of mixed solutions which take on values above and below x = A, generalizing the concept of the wellstudied positive solutions. This leads to a generalization of the socalled period function for a period annulus. We derive expansions of these functions and formulas for the derivatives of these generalized period functions. The main result is that under generic conditions on f(x) socalled Sshaped bifurcations of mixed solutions occur. As a consequence there exists an open interval for sufficiently small A for which λ can be found such that three solutions of the same mixed type exist. We show how these concepts relate to the simplest possible case f(x) = x(x + 1) where despite its simple form difficult open problems remain. 

Physical Description:  38 
ISSN:  14173875 