101,448 research outputs found
Approximate dynamical systems on interval
AbstractLet I be a an open real interval. We show that if a function H:I×R→I satisfies the inequality |H(H(x0,s),t)−H(x0,s+t)|≤δfor s,t∈R with a δ≥0 and an x0∈I such that the function H(x0,⋅) is a continuous surjection of R onto I, then there exists a dynamical system F on I such that |H(x,t)−F(x,t)|≤9δfor x∈I,t∈R
Universal functions and exactly solvable chaotic systems
A universal differential equation is a nontrivial differential equation the
solutions of which approximate to arbitrary accuracy any continuous function on
any interval of the real line. On the other hand, there has been much interest
in exactly solvable chaotic maps. An important problem is to generalize these
results to continuous systems.
Theoretical analysis would allow us to prove theorems about these systems and
predict new phenomena. In the present paper we discuss the concept of universal
functions and their relevance to the theory of universal differential
equations. We present a connection between universal functions and solutions to
chaotic systems. We will show the statistical independence between and
(when is not equal to zero) and is a solution to
some chaotic systems. We will construct universal functions that behave as
delta-correlated noise. We will construct universal dynamical systems with
truly noisy solutions. We will discuss physically realizable dynamical systems
with universal-like properties.Comment: 12 Pages, 9 figures. Proceedings 1st Meeting IST-IM
Convergence Time Towards Periodic Orbits in Discrete Dynamical Systems
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 use linearized equations to examine the evolution near that
neighborhood. The underlying idea is that points of stable periodic orbit are
associated with intervals. We state and prove a theorem that details what
regions of phase space are mapped into these intervals (once they are known)
and how many iterations are required to get there. We also construct algorithms
that allow our theoretical results to be implemented successfully in practice.Comment: 17 pages; 7 figure
Reachability in Biochemical Dynamical Systems by Quantitative Discrete Approximation (extended abstract)
In this paper, a novel computational technique for finite discrete
approximation of continuous dynamical systems suitable for a significant class
of biochemical dynamical systems is introduced. The method is parameterized in
order to affect the imposed level of approximation provided that with
increasing parameter value the approximation converges to the original
continuous system. By employing this approximation technique, we present
algorithms solving the reachability problem for biochemical dynamical systems.
The presented method and algorithms are evaluated on several exemplary
biological models and on a real case study.Comment: In Proceedings CompMod 2011, arXiv:1109.104
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