Introduction to dynamical systems (original) (raw)
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Studying discrete dynamical systems through differential equations
Journal of Differential Equations, 2008
In this paper we consider dynamical systems generated by a diffeomorphism F defined on U an open subset of R n , and give conditions over F which imply that their dynamics can be understood by studying the flow of an associated differential equation,
Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2013
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Discrete-time differential systems
In this paper we formulate a coherent discrete-time signals and systems theory taking derivative concepts as basis. Two derivatives – nabla (forward) and delta (backward) – are defined and generalized to fractional orders, obtaining two formulations that are discrete versions of the well-known Grünwald–Letnikov derivatives. The eigenfunctions of such derivatives are the so-called nabla and delta exponentials. With these exponentials two generalized discrete-time Laplace transforms are deduced and their properties studied. These transforms are back compatible with the current Laplace and Z transforms. They are used to study the discrete-time linear systems defined by differential equations. These equations although discrete mimic the usual continuous-time equations that are uniformly approximated when the sampling interval becomes small. Impulse response and transfer function notions are introduced and obtained. The Fourier transform and the frequency response are also considered. This implies a unified mathematical framework that allows us to approximate the classic continuous-time case when the sampling rate is high or obtain the current discrete-time case based on difference equation.
Discrete Dynamical Systems: A Brief Survey
Journal of the Institute of Engineering
Dynamical system is a mathematical formalization for any fixed rule that is described in time dependent fashion. The time can be measured by either of the number systems - integers, real numbers, complex numbers. A discrete dynamical system is a dynamical system whose state evolves over a state space in discrete time steps according to a fixed rule. This brief survey paper is concerned with the part of the work done by José Sousa Ramos [2] and some of his research students. We present the general theory of discrete dynamical systems and present results from applications to geometry, graph theory and synchronization. Journal of the Institute of Engineering, 2018, 14(1): 35-51
A Discrete Dynamical System and Its Applications
Pesquisa Operacional
The main goal of this manuscript is to introduce a discrete dynamical system defined by symmetric matrices and a real parameter. By construction, we rediscovery the Power Iteration Method from the Projected Gradient Method. Convergence of the discrete dynamical system solution is established. Finally, we consider two applications, the first one consists in find a solution of non linear equation problem and the other one consists in verifies the optimality conditions when we solve quadratic optimization problems over linear equality constraints.
Differential representations of driftless discrete-time dynamics
Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), 1998
The paper deals with nonlinear driftless discrete-time dynamics. I t is shown that such dynamics can always be represented by an exponential form which corresponds to the solution of a suitable differential equation. This enables us t o characterize the associated Lie group structure and to further understand invariance, accessibility and passivity properties. 2 Nonlinear driftless dynamics Let the driftless discrete-time dynamics be where x belongs to M , an analytic n-dimensional manifold, F : M x I R -+ M , is an analytic function of both arguments satisfying F ( . , 0) = 0. For U in a neighbourhood U, of 0, let the expansion of Id + F ( ., U ) be Id + F ( . , u ) = Id + '11 22 1 0-7803-4394-8198 $1 0.00 0 1998 IEEE 4620 Authorized licensed use limited to: Universita degli Studi di Roma La Sapienza. Downloaded on April 23,2010 at 13:38:47 UTC from IEEE Xplore. Restrictions apply.