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Asymptotic Behavior Of Solutions Of Dynamic Equations
Sigrun Bodine,M. Bokhner,D. Lutts +2 more
- Vol. 1
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TLDR
In this article, an asymptotic representation for a fundamental solution matrix for scalar linear dynamic systems on time scales is given, which is a generalization of the usual exponential function.Abstract:
We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.read more
Citations
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Book
Nonoscillation and Oscillation Theory for Functional Differential Equations
TL;DR: In this paper, the qualitative theory of differential equations with or without delays is summarized, collecting recent oscillation studies important to applications and further developments in mathematics, physics, engineering, and biology.
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Asymptotic expansions and analytic dynamic equations
Martin Bohner,D. A. Lutz +1 more
TL;DR: In this article, the authors consider a linear dynamic equation on a time scale together with a perturbation term and show that, if certain exponential dichotomy conditions are satisfied, then for any solution of the perturbed equation there exists a solution of an unperturbed equation that asymptotically differs from the solution of a perturbed one no more than the order of the term.
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A view of dynamic derivatives on time scales from approximations
TL;DR: In this article, the authors explore connections between existing dynamic and conventional derivatives over the time scales used and show that while first-order dynamic derivatives provide acceptable accuracies in approximating conventional derivatives, second-order dynamical derivatives are in general inconsistent with conventional derivatives.
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Asymptotic behavior of solutions of perturbed linear difference systems
Guojing Ren,Yuming Shi,Yi Wang +2 more
TL;DR: The Hartman-Wintner theorem, the Harris-Lutz theorem, and the Eastham theorem were shown to be asymptotic results for perturbed linear difference systems in this article.
Oscillation and Asymptotic Behavior of Solutions of Certain Third-Order Nonlinear Delay Dynamic Equations
TL;DR: In this article, the authors deal with the oscillation and asymptotic behavior of solutions of the third-order nonlinear delay dynamic equation and show that the solution can be solved in time O(n).
References
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Book
Dynamic Equations on Time Scales: An Introduction with Applications
Martin Bohner,Allan Peterson +1 more
TL;DR: The Time Scales Calculus as discussed by the authors is a generalization of the time-scales calculus with linear systems and higher-order linear equations, and it can be expressed in terms of linear Symplectic Dynamic Systems.
Journal ArticleDOI
The asymptotic solution of linear differential systems
TL;DR: In this article, the authors considered the system on an interval [a, ∞], where A is a constant nUn matrix with n distinct eigenvalues and either (2a) R(x)→0 as x→∞ and R'x) is L(a,∞).
Book
The asymptotic solution of linear differential systems : applications of the Levinson theorem
TL;DR: Asymptotically diagonal systems -the Levinson theorem, coefficient matrices of Jordan type two-term differential equations -the Liouville-Green asymptotic formulae, application of the Hartman-Wintner Theorem, equations of Euler type equations of self-adjoint type resonance and non-resonance as discussed by the authors.
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Asymptotic Representation of Solutions of Perturbed Systems of Linear Difference Equations
TL;DR: In this paper, the authors asymptotically represent solutions of linear difference equations by transforming them into so-called L-diagonal form, and two properties are then responsible for the equivalence of an L-Diagonal form to a diagonal one: a dichotomy condition on the diagonal part, and a growth condition on perturbation term.