Branches of index-preserving solutions to systems of second order ODEs
type: Item
instance of: scholarly article
title: Branches of index-preserving solutions to systems of second order ODEs
zbMATH Open document ID: 1190.34015
author: Francesca Dalbono
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2009-11-23T00:00:00Z
review text: Consider a boundary value problem of the form \[ u''+A(t,u(t),\lambda)u(t)=0, u(0)=0=u(T), \] where for every \((t,x,\lambda) \in [0,T] \times {\mathbb R}^n \times {\mathbb R}\), \(A(t,x,\lambda)\) is an \(n\times n\) real symmetric matrix satisfying suitable assumptions. Consider also the auxiliary linear system of the form \[ u''+(K(t)+\eta Id)u=0, \] together with the sequence of its eigenvalues. Moreover, the author uses the notions of index (the number of negative eigenvalues of the auxiliary linear problem) and of nullity (the number of zero eigenvalues of the auxiliary linear problem). Using these concepts, a global bifurcation result for the given nonlinear problem is proved: more precisely, the existence of a global branch of solutions having index equal to zero and nullity equal to one is obtained. An application is performed for the study of branches of odd periodic solutions for some second order system modeling two pendula coupled by a torsion elastic force, in presence of odd periodic disturbances of their lengths.
reviewed by: Anna Capietto
Mathematics Subject Classification ID: 34B09
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5636892
zbMATH Keywords: second order systems
zbMATH Keywords: bifurcation
zbMATH Keywords: index of a solution
MaRDI profile type: Publication
full work available at URL: s00030-009-0019-8
OpenAlex ID: W2041555697
cites work: An existence theorem for periodically perturbed conservative systems
cites work: Preservation of the Maslov index along bifurcating branches of solutions of first order systems in \(\mathbf {R}^{N}\)
cites work: Detecting multiplicity for systems of second-order equations: an alternative approach
cites work: Q3947949
cites work: Principal eigenvalue for weight matrix in elliptic systems
cites work: Multiplicity of solutions of Dirichlet problems associated with second-order equations in ℝ2
cites work: Application of a Lemma on Bilinear Forms to a Problem in Nonlinear Oscillations
cites work: Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems
cites work: Periodic solutions for systems of forced coupled pendulum-like equations
cites work: Bifurcation for an elliptic system coupled in the linear part
cites work: Q4523717
cites work: Branches of forced oscillations for periodically perturbed second-order autonomous ODEs on manifolds
cites work: A Fredholm-like result for strongly nonlinear second order ODE's
cites work: Q3933274
cites work: On the eigenvalue problem for weakly coupled elliptic systems
cites work: Q4774824
cites work: On periodically perturbed conservative systems
cites work: Periodic motions of coupled simple pendulums with periodic disturbances
cites work: Forced second order conservative systems with periodic nonlinearity
cites work: Q3676486
cites work: Multiple solutions for strongly resonant periodic systems
cites work: Some global results for nonlinear eigenvalue problems
cites work: Bifurcation results for quasilinear elliptic systems.
cites work: Periodic Solutions of Perturbed Conservative Systems
cites work: Q3025542
DOI: 10.1007/S00030-009-0019-8
description: scientific article; zbMATH DE number 5636892