Branches of index-preserving solutions to systems of second order ODEs
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.
scientific article; zbMATH DE number 5636892
scientific article