87 research outputs found
Can we trust in numerical computations of chaotic solutions of dynamical systems ?
From Laser Dynamics to Topology of Chaos, (celebrating the 70th birthday of Prof Robert Gilmore, Rouen, June 28-30, 2011), Ed. Ch. Letellier.Since the famous paper of E. Lorenz in 1963 numerical computations using computers play a central role in order to display and analyze solutions of nonlinear dynamical systems. By these means new structures have been emphasized like hyperbolic and/or strange attractors. However theoretical proofs of their existence are very di¢ cult and limited to very special linear cases. Computer aided proofs are also complex and require special interval arithmetic analysis. Nevertheless, numerous researchers in several fields linked to chaotic dynamical systems are confident in the numerical solutions they found using popular software and publish without checking carefully the reliability of their results. In the simple case of discrete dynamical systems (e.g. Hénon map) there are concerns about the nature of what a computer find out : long unstable pseudo-orbits or strange attractors? The shadowing property and its generalizations which ensure that pseudo-orbits of a homeomorphism can be traceable by actual orbits even if rounding errors are not inevitable are not of great help in order to validate the numerical results. Continuous dynamical systems (e.g. Chua, Lorenz, Rössler) are even more difficult to handle in this scope and researchers have to be very cautious to back up theory with numerical computations. We present a survey of the topic based on these, only few, but well studied models
Coexistence of bounded and unbounded geometry for area-preserving maps
The geometry of the period doubling Cantor sets of strongly dissipative
infinitely renormalizable H\'enon-like maps has been shown to be unbounded by
M. Lyubich, M. Martens and A. de Carvalho, although the measure of unbounded
"spots" in the Cantor set has been demonstrated to be zero.
We show that an even more extreme situation takes places for infinitely
renormalizable area-preserving H\'enon-like maps: bounded and unbounded
geometries coexist with both phenomena occuring on subsets of positive measure
in the Cantor sets
Topological-numerical analysis of a two-dimensional discrete neuron model
We conduct computer-assisted analysis of the two-dimensional model of a
neuron introduced by Chialvo in 1995 (Chaos, Solitons & Fractals 5, 461-479).
We apply the method for rigorous analysis of global dynamics based on a
set-oriented topological approach, introduced by Arai et al. in 2009 (SIAM J.
Appl. Dyn. Syst. 8, 757-789) and improved and expanded afterwards.
Additionally, we introduce a new algorithm to analyze the return times inside a
chain recurrent set. Based on this analysis, together with the information on
the size of the chain recurrent set, we develop a new method that allows one to
determine subsets of parameters for which chaotic dynamics may appear. This
approach can be applied to a variety of dynamical systems, and we discuss some
of its practical aspects. The data and the software described in the paper are
available at http://www.pawelpilarczyk.com/neuron/
Planar Radial Weakly-Dissipative Diffeomorphisms
We study the effect of a small dissipative radial perturbation acting on a one parameter family of area preserving diffeomorphisms. This is a specific type of dissipative perturbation. The interest is on the global effect of the dissipation on a fixed domain around an elliptic fixed/periodic point of the family, rather than on the effects around a single resonance. We describe the local/global bifurcations observed in the transition from the conservative to a weakly dissipative case: the location of the resonant islands, the changes in the domains of attraction of the foci inside these islands, how the resonances disappear, etc. The possible -limits are determined in each case. This topological description gives rise to three different dynamical regimes according to the size of dissipative perturbation. Moreover, we determine the conservative limit of the probability of capture in a generic resonance from the interpolating flow approximation, hence assuming no homoclinics in the resonance. As a paradigm of weakly dissipative radial maps, we use a dissipative version of the Hénon map
A deductive-nomological model for mathematical scientific explanation
I propose a deductive-nomological model for mathematical scientific explanation. In this regard, I modify Hempel’s deductive-nomological model and test it against some of the following recent paradigmatic examples of the mathematical explanation of empirical facts: the seven bridges of Königsberg, the North American synchronized cicadas, and Hénon-Heiles Hamiltonian systems. I argue that mathematical scientific explanations that invoke laws of nature are qualitative explanations, and ordinary scientific explanations that employ mathematics are quantitative explanations. I analyse the repercussions of this deductivenomological model on causal explanations.info:eu-repo/semantics/publishedVersio
Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation
We present an algorithm for the rigorous integration of Delay Differential
Equations (DDEs) of the form . As an application, we
give a computer assisted proof of the existence of two attracting periodic
orbits (before and after the first period-doubling bifurcation) in the
Mackey-Glass equation
- …