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Generalized reciprocity for self-adjoint linear differential equations

Ondřej Došlý (1995)

Archivum Mathematicum

Let L ( y ) = y ( n ) + q n - 1 ( t ) y ( n - 1 ) + + q 0 ( t ) y , t [ a , b ) , be an n -th order differential operator, L * be its adjoint and p , w be positive functions. It is proved that the self-adjoint equation L * p ( t ) L ( y ) = w ( t ) y is nonoscillatory at b if and only if the equation L w - 1 ( t ) L * ( y ) = p - 1 ( t ) y is nonoscillatory at b . Using this result a new necessary condition for property BD of the self-adjoint differential operators with middle terms is obtained.

Generalized solutions to hybrid dynamical systems

Ricardo G. Sanfelice, Rafal Goebel, Andrew R. Teel (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Several recent results in the area of robust asymptotic stability of hybrid systems show that the concept of a generalized solution to a hybrid system is suitable for the analysis and design of hybrid control systems. In this paper, we show that such generalized solutions are exactly the solutions that arise when measurement noise in the system is taken into account.

Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication

Hongtao Liang, Zhen Wang, Zongmin Yue, Ronghui Lu (2012)

Kybernetika

A fractional differential controller for incommensurate fractional unified chaotic system is described and proved by using the Gershgorin circle theorem in this paper. Also, based on the idea of a nonlinear observer, a new method for generalized synchronization (GS) of this system is proposed. Furthermore, the GS technique is applied in secure communication (SC), and a chaotic masking system is designed. Finally, the proposed fractional differential controller, GS and chaotic masking scheme are...

Generalized synchronization in a system of several non-autonomous oscillators coupled by a medium

Rogério Martins, Gonçalo Morais (2015)

Kybernetika

An abstract theory on general synchronization of a system of several oscillators coupled by a medium is given. By generalized synchronization we mean the existence of an invariant manifold that allows a reduction in dimension. The case of a concrete system modeling the dynamics of a chemical solution on two containers connected to a third container is studied from the basics to arbitrary perturbations. Conditions under which synchronization occurs are given. Our theoretical results are complemented...

Generalized synchronization in the networks with directed acyclic structure

Sergej Čelikovský, Volodymyr Lynnyk, Anna Lynnyk, Branislav Rehák (2023)

Kybernetika

Generalized synchronization in the direct acyclic networks, i.e. the networks represented by the directed tree, is presented here. Network nodes consist of copies of the so-called generalized Lorenz system with possibly different parameters yet mutually structurally equivalent. The difference in parameters actually requires the generalized synchronization rather than the identical one. As the class of generalized Lorenz systems includes the well-known particular classes such as (classical) Lorenz...

Generalized trigonometric functions in complex domain

Petr Girg, Lukáš Kotrla (2015)

Mathematica Bohemica

We study extension of p -trigonometric functions sin p and cos p to complex domain. For p = 4 , 6 , 8 , , the function sin p satisfies the initial value problem which is equivalent to (*) - ( u ' ) p - 2 u ' ' - u p - 1 = 0 , u ( 0 ) = 0 , u ' ( 0 ) = 1 in . In our recent paper, Girg, Kotrla (2014), we showed that sin p ( x ) is a real analytic function for p = 4 , 6 , 8 , on ( - π p / 2 , π p / 2 ) , where π p / 2 = 0 1 ( 1 - s p ) - 1 / p . This allows us to extend sin p to complex domain by its Maclaurin series convergent on the disc { z : | z | < π p / 2 } . The question is whether this extensions sin p ( z ) satisfies (*) in the sense of differential equations in complex domain. This interesting...

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