Trigonometric transformations of symplectic difference systems
type: Item
instance of: scholarly article
title: Trigonometric transformations of symplectic difference systems
zbMATH Open document ID: 0956.39011
author: Ondřej Došlý
author: Martin Bohner
published in: Journal of Differential Equations
publication date: 2001-03-12T00:00:00Z
review text: The authors deal with the symplectic difference system \[ z_{k+1} ={\mathcal S}_kz_k, \tag{S} \] where \(z_k\in\mathbb{R}^{2n}\) and for each \(k\in\mathbb{Z}\), \({\mathcal S}_k\in \mathbb{R}^{2n\times 2n}\) satisfies the relation \({\mathcal S}_k^T{\mathcal I} {\mathcal S}_k={\mathcal I}\), where \({\mathcal I}=\left( \begin{smallmatrix} 0 & I\\ -I & 0 \end{smallmatrix} \right)\), \(I\) being the identity matrix. System (S) is called a trigonometric symplectic difference system if in addition to the above hypotheses \({\mathcal I}^T{\mathcal S}_k {\mathcal I}={\mathcal S}_k\) for \(k\in\mathbb{Z}\). The main result of the paper shows that any symplectic difference system (S) can be transformed into a trigonometric symplectic difference system, using a transformation which preserves the oscillatory properties of the original equation. This result can be regarded as a discrete version of an earlier result of the second author concerning Hamiltonian differential systems. For certain trigonometric difference systems the authors establish a necessary and sufficient condition to be nonoscillatory, and this result is applied to the oscillation theory of symplectic difference and Hamiltonian systems.
Mathematics Subject Classification ID: 39A12
Mathematics Subject Classification ID: 39A11
Mathematics Subject Classification ID: 37J10
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1454729
zbMATH Keywords: symplectic difference system
zbMATH Keywords: trigonometric system
zbMATH Keywords: oscillation
zbMATH Keywords: linear Hamiltonian difference system
reviewed by: Mihály Pituk
MaRDI profile type: Publication
OpenAlex ID: W2110051764
cites work: Q4716258
cites work: Q3128681
cites work: Q4940907
cites work: A Prufer Transformation for Matrix Differential Equations
cites work: Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions
cites work: Disconjugacy and transformations for symplectic systems
cites work: Disconjugacy, disfocality, and oscillation of second order difference equations
cites work: Phase matrix of linear differential systems
cites work: On transformations of self-adjoint linear differential systems and their reciprocals
cites work: Transformations of linear Hamiltonian difference systems and some of their applications
cites work: Q4940914
cites work: Q4040931
cites work: \(C\)-disfocality for linear Hamiltonian difference systems
cites work: Oscillation of second order linear matrix difference equations
cites work: Sturmian theory for ordinary differential equations
DOI: 10.1006/JDEQ.1999.3728
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description: scientific article; zbMATH DE number 1454729