scholarly journals Oscillation criterion for two-dimensional dynamic systems on time scales

2012 ◽  
Vol 44 (3) ◽  
pp. 227-232
Author(s):  
Taher Hassan
Keyword(s):  
Dynamic System ◽  
Time Scales ◽  
Dynamic Systems ◽  
Time Scale ◽  
Continuous Functions ◽  
Oscillation Criterion ◽  
Two Dimensional

The purpose of this paper is to prove oscillation criterion for dynamic system \begin{equation*} u^{\Delta }=pv,\qquad v^{\Delta }=-qu^{\sigma }, \end{equation*}% where $p>0$ and $q$ are rd-continuous functions on a time scale such that $% \sup \mathbb{T=\infty }$ without explicit sign assumptions on $q$ and also without restrictive conditions on the time scale $\mathbb{T}.$

scholarly journals A new oscillation criterion for two-dimensional dynamic systems on time scales

2011 ◽  
Vol 42 (2) ◽  
pp. 237-244
Author(s):  
Jia Baoguo
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Continuous Functions ◽  
Oscillation Criterion ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
And Mathematics ◽  
Computers And Mathematics ◽  
Oscillation And Nonoscillation Criteria ◽  
Oscillation And Nonoscillation

Consider the linear dynamic system on time scales\begin{equation}u^\Delta=pv, \quad\quad v^\Delta=-qu^\sigma\end{equation}where $p>0$ and $q$ are rd-continuous functions on a time scale $\mathbb T$ such that $\sup\mathbb T=\infty$. When $p(t)$ is allowed to take on negative values, we establish an oscillation criterion for system (0.1). Our result improves a main result of Fu and Lin [S. C. Fu and M. L. Lin, Oscillation and nonoscillation criteria for linear dynamic systems on time scales, Computers and Mathematics with Applications, 59(2010), 2552-2565].

scholarly journals Controllability and observability of linear impulsive adjoint dynamic system on time scale

2020 ◽  
Vol 51 (3) ◽  
pp. 201-217
Author(s):  
Nusrat Yasmin ◽  
Safia Mirza ◽  
Awais Younus ◽  
Asif Mansoor
Keyword(s):  
Dynamic System ◽  
Time Scales ◽  
Time Scale ◽  
Continuous Case ◽  
Time Varying ◽  
Controllability And Observability

This paper deals with the controllability, observability of the solution of time-varying system on time scales. We obtain new results about controllability and observability and generalize to a time scale some known properties about stability from the continuous case.

scholarly journals Oscillation for a Nonlinear Dynamic System with a Forced Term on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xinli Zhang ◽  
Shanliang Zhu
Keyword(s):  
Dynamic System ◽  
Time Scale ◽  
Nonlinear Dynamic ◽  
Sufficient Conditions ◽  
Nonlinear Dynamic System ◽  
Two Dimensional ◽  
Differential Systems ◽  
Difference Systems ◽  
All Solutions ◽  
Forced Term

We consider a class of two-dimensional nonlinear dynamic system with a forced term on a time scale𝕋and obtain sufficient conditions for all solutions of the system to be oscillatory. Our results not only unify the oscillation of two-dimensional differential systems and difference systems but also improve the oscillation results that have been established by Saker, 2005, since our results are not restricted to the case whereb(t)≠0for allt∈𝕋andg(u)=u. Some examples are given to illustrate the results.

scholarly journals Lyapunov-Type Inequalities for Some Quasilinear Dynamic System Involving the(p1,p2,…,pm)-Laplacian on Time Scales

2011 ◽  
Vol 2011 ◽  
pp. 1-10
Author(s):  
Xiaofei He ◽  
Qi-Ming Zhang
Keyword(s):  
Dynamic System ◽  
Time Scales ◽  
Time Scale ◽  
Arbitrary Time

We establish several new Lyapunov-type inequalities for some quasilinear dynamic system involving the(p1,p2,…,pm)-Laplacian on an arbitrary time scale𝕋, which generalize and improve some related existing results including the continuous and discrete cases.

scholarly journals On Inequalities of Lyapunov for Two-Dimensional Nonlinear Dynamic Systems on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Qiao-Luan Li ◽  
Wing-Sum Cheung ◽  
Xu-Yang Fu
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Nonlinear Dynamic ◽  
Type Equation ◽  
Nonlinear Dynamic Systems ◽  
Two Dimensional

We establish some new Lyapunov-type inequalities for two-dimensional nonlinear dynamic systems on time scales. As for application, boundedness of the Emden-Fowler-type equation is proved.

scholarly journals Oscillation of Two-Dimensional Neutral Delay Dynamic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Xinli Zhang ◽  
Shanliang Zhu
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Positive Real ◽  
Neutral Delay ◽  
All Solutions ◽  
Special Case

We consider a class of nonlinear two-dimensional dynamic systems of the neutral type(x(t)-a(t)x(τ1(t)))Δ=p(t)f1(y(t)),yΔ(t)=-q(t)f2(x(τ2(t))).We obtain sufficient conditions for all solutions of the system to be oscillatory. Our oscillation results whena(t)=0improve the oscillation results for dynamic systems on time scales that have been established by Fu and Lin (2010), since our results do not restrict to the case wheref(u)=u. Also, as a special case when𝕋=ℝ, our results do not requireanto be a positive real sequence. Some examples are given to illustrate the main results.

scholarly journals Oscillation of symplectic dynamic systems

2004 ◽  
Vol 46 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Martin Bohner ◽  
Ondřej Došlý
Keyword(s):  
Dynamic System ◽  
Dynamic Systems ◽  
Time Scale ◽  
Unperturbed System ◽  
Perturbation Matrix ◽  
Perturbed System ◽  
Oscillatory Properties

AbstractWe investigate oscillatory properties of a perturbed symplectic dynamic system on a time scale that is unbounded above. The unperturbed system is supposed to be nonoscillatory, and we give conditions on the perturbation matrix, which guarantee that the perturbed system becomes oscillatory. Examples illustrating the general results are given as well.

scholarly journals Oscillation for Higher Order Dynamic Equations on Time Scales

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Taixiang Sun ◽  
Qiuli He ◽  
Hongjian Xi ◽  
Weiyong Yu
Keyword(s):  
Time Scales ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Higher Order ◽  
Dynamic Equations ◽  
Positive Integers

We investigate the oscillation of the following higher order dynamic equation:{an(t)[(an-1(t)(⋯(a1(t)xΔ(t))Δ⋯)Δ)Δ]α}Δ+p(t)xβ(t)=0, on some time scaleT, wheren≥2,ak(t)  (1≤k≤n)andp(t)are positive rd-continuous functions onTandα,βare the quotient of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

scholarly journals Symmetries for Nonconservative Field Theories on Time Scale

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 552
Author(s):  
Octavian Postavaru ◽  
Antonela Toma
Keyword(s):  
Time Scales ◽  
Dynamic Systems ◽  
Time Scale ◽  
Selection Criteria ◽  
Noether Symmetry ◽  
Field Theories ◽  
Conserved Quantities ◽  
Hamilton’S Principle ◽  
Lagrange Equations ◽  
Infinitesimal Transformations

Symmetries and their associated conserved quantities are of great importance in the study of dynamic systems. In this paper, we describe nonconservative field theories on time scales—a model that brings together, in a single theory, discrete and continuous cases. After defining Hamilton’s principle for nonconservative field theories on time scales, we obtain the associated Lagrange equations. Next, based on the Hamilton’s action invariance for nonconservative field theories on time scales under the action of some infinitesimal transformations, we establish symmetric and quasi-symmetric Noether transformations, as well as generalized quasi-symmetric Noether transformations. Once the Noether symmetry selection criteria are defined, the conserved quantities for the nonconservative field theories on time scales are identified. We conclude with two examples to illustrate the applicability of the theory.

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