Michal Pospíšil

Representation of solutions of delayed difference equations with linear parts given by pairwise permutable matrices via Z -transform

In the present paper, a system of nonhomogeneous linear difference equations with any finite number of constant delays and linear parts given by pairwise permutable matrices is considered. Representation of its solution is derived in a form of a matrix polynomial using the Z -transform. So the recent results for one and two delays, and an inductive formula for multiple delays are unified. The representation is suitable for theoretical as well as practical computations.

Representation of a Solution of the Cauchy Problem for an Oscillating System with Multiple Delays and Pairwise Permutable Matrices

Nonhomogeneous system of linear differential equations of second order with multiple different delays and pairwise permutable matrices defining the linear parts is considered. Solution of corresponding initial value problem is represented using matrix polynomials.

Travelling Waves in Nonlinear Magnetic Metamaterials

In this article, a model of one-dimensional metamaterial formed by a discrete array of nonlinear resonators is considered. The existence and uniqueness results of periodic and asymptotic travelling waves of the system are presented. The existence and the stability of asymptotic waves are also computed and discussed numerically.

Bifurcation from single periodic orbit in discontinuous autonomous systems

This article is devoted to the study of bifurcations of periodic solutions for discontinuous autonomous systems from single periodic solutions of unperturbed discontinuous equations. In addition, local asymptotic properties of derived perturbed periodic solutions are also investigated. An example of planar discontinuous ordinary differential equations is given to illustrate the theory.

Relative Controllability of Neutral Differential Equations with a Delay

The problem of relative controllability for linear systems of neutral differential equations with a delay is investigated. An equivalent condition of Kalman type is proved. All the control function...

Persistence of periodic orbits in periodically forced impact systems

This paper is devoted to the study of persistence of forced periodic solutions for impact systems from single periodic solutions of unperturbed impact equations. An example of planar discontinuous ordinary differential equations is given to illustrate the theory.

Observability of difference equations with a delay

Initial value problem for n-dimensional difference equation with a delay and a control is considered. Its solution is represented using a discrete delayed matrix exponential and a necessary and sufficient condition is stated for the observability of this system.

On Relative Controllability of Delayed Difference Equations with Multiple Control Functions

An n-dimensional linear difference equation with a delay and a vector control function is considered. An equivalent condition for relative controllability is stated, and a complete characterization of control functions is given. Moreover, a sufficient condition for relative controllability of weakly nonlinear difference equation is proved.

Dynamics of generalized PT-symmetric dimers with time-periodic gain–loss

A parity-time (PT)-symmetric system with periodically varying-in-time gain and loss modeled by two coupled Schrödinger equations (dimer) is studied. It is shown that the problem can be reduced to a perturbed pendulum-like equation. This is done by finding two constants of motion. Firstly, a generalized problem using Melnikov-type analysis and topological degree arguments is studied for showing the existence of periodic (libration), shift- periodic (rotation), and chaotic solutions. Then these general results are applied to the PT-symmetric dimer. It is interestingly shown that if a sufficient condition is satisfied, then rotation modes, which do not exist in the dimer with constant gain–loss, will persist. An approximate threshold for PT-broken phase corresponding to the disappearance of bounded solutions is also presented. Numerical study is presented accompanying the analytical results.

Bifurcation of travelling waves in implicit nonlinear lattices: applications in metamaterials

We consider implicit nonlinear lattice equations modelling one-dimensional metamaterials formed by a discrete array of nonlinear split-ring resonators. We study the existence and bifurcation of localised excitations and use the results to prove the existence of periodic travelling waves in the presence of small damping and travelling drive. Two different systems are considered, with each of them admitting either homoclinic or heteroclinic solutions.