Transformations of linear Hamiltonian systems preserving oscillatory behaviour. (English) Zbl 0764.34026

Došlý, Ondřej

Arch. Math., Brno 27b, 211-219 (1991).

Relations between oscillatory behaviour of the linear Hamiltonian system (1) \(Jx'=A(t)x\) and the linear Hamiltonian system (2) \(Jw'=B(t)w\) which is related to (1) via the transformation \(x=R(t)w\), are studied. Here \(A\): \(I=[a,\infty)\to R^{2n\times 2n}\) is a symmetric matrix, \(J=\left({0\atop -I_ n} {I_ n\atop 0}\right)\), \(I_ n\) is the identity \(n\times n\) matrix and \(R(t)\in C^ 1(I)\) is \(2n\times 2n\) \(J\)- unitary matrix, i.e. \(R^ T(t)JR(t)=J\). The main result of the paper generalizes the well-known duality in oscillatory behaviour of mutually reciprocal Hamiltonian systems. In [S. Staněk and the reviewer, Arch. Math., Brno 22, 55-59 (1986; Zbl 0644.34029)] the differential equation for \(z(t)=\alpha(t)y(t)+\beta(t)y'(t)\), where \(y(t)\) is solution of the scalar equation \(y''+a(t)y'+b(t)y=0\), is derived. The generalization of this result to the second order system is done here, too.

Reviewer: J.Vosmanský (Brno)

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

oscillatory behaviour; linear Hamiltonian system; transformation; duality; mutually reciprocal; Hamiltonian systems

Citations:

Zbl 0644.34029