Related papers: Spatial discretizations of generic dynamical syste…
This paper concerns the link between the dynamical behaviour of a dynamical system and the dynamical behaviour of its numerical simulations. Here, we model numerical truncation as a spatial discretization of the system. Some previous works…
We study spatial discretizations of dynamical systems: is it possible to recover some dynamical features of a system from numerical simulations? Here, we tackle this issue for the simplest algorithm possible: we compute long segments of…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
In this article, we study the behaviour of discrete one-dimensional dynamical systems associated to functions on finite sets. We formalise the global orbit pattern formed by all the periodic orbits (gop) as the ordered set of periods when…
We apply the ``consistent discretization'' approach to general relativity leaving the spatial slices continuous. The resulting theory is free of the diffeomorphism and Hamiltonian constraints, but one can impose the diffeomorphism…
Differential flatness serves as a powerful tool for controlling continuous time nonlinear systems in problems such as motion planning and trajectory tracking. A similar notion, called difference flatness, exists for discrete-time systems.…
This paper deals with the dynamics of discretizations of isometries of $\mathbf R^n$, and more precisely the density of the successive images of $\mathbf Z^n$ by the discretizations of a generic sequence of isometries. We show that this…
Gauge invariance in discrete dynamical systems and its connection with quantization are considered. For a complete description of gauge symmetries of a system we construct explicitly a class of groups unifying in a natural way the space and…
We investigate the convergence towards periodic orbits in discrete dynamical systems. We examine the probability that a randomly chosen point converges to a particular neighborhood of a periodic orbit in a fixed number of iterations, and we…
Certain intriguing consequences of the discreteness of time on the time evolution of dynamical systems are discussed. In the discrete-time classical mechanics proposed here, there is an {\it arrow of time} that follows from the fact that…
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…
We study the behaviour of discrete dynamical systems generated by a continuous map $f$ of a compact real interval into itself where at randomly chosen times a function different from $f$ - so called impulse function is applied. We show that…
We study invariant sets and measures generated by iterated function systems defined on countable discrete spaces that are uniform grids of a finite dimension. The discrete spaces of this type can be considered as models of spaces in which…
The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion…
The paper proposes an algorithm for a discretization (sampled-time implementation) of a homogeneous control preserving the finite-time and nearly fixed-time stability property of the original (sampling-free) system. The sampling period is…
A dynamical system with discrete time is studied by means of algebraic geometry. The system admits a reduction that is interpreted as a classical field theory in 2+1-dimensional wholly discrete space-time. The integrals of motion of a…
Several approaches to the dynamics of loop quantum gravity involve discretizing the equations of motion. The resulting discrete theories are known to be problematic since the first class algebra of constraints of the continuum theory…
Real-world dynamical systems with retardation effects are described in general not by a single, precisely defined time delay, but by a range of delay times. An exact mapping onto a set of $N+1$ ordinary differential equations exists when…
I analyse a generalised Burger's equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the…
We study classical Hamiltonian systems in which the intrinsic proper time evolution parameter is related through a probability distribution to the physical time, which is assumed to be discrete. In this way, a physical clock with discrete…