12,453 research outputs found
Two-Particle Circular Billiards Versus Randomly Perturbed One-Particle Circular Billiards
We study a two-particle circular billiard containing two finite-size circular
particles that collide elastically with the billiard boundary and with each
other. Such a two-particle circular billiard provides a clean example of an
"intermittent" system. This billiard system behaves chaotically, but the time
scale on which chaos manifests can become arbitrarily long as the sizes of the
confined particles become smaller. The finite-time dynamics of this system
depends on the relative frequencies of (chaotic) particle-particle collisions
versus (integrable) particle-boundary collisions, and investigating these
dynamics is computationally intensive because of the long time scales involved.
To help improve understanding of such two-particle dynamics, we compare the
results of diagnostics used to measure chaotic dynamics for a two-particle
circular billiard with those computed for two types of one-particle circular
billiards in which a confined particle undergoes random perturbations.
Importantly, such one-particle approximations are much less computationally
demanding than the original two-particle system, and we expect them to yield
reasonable estimates of the extent of chaotic behavior in the two-particle
system when the sizes of confined particles are small. Our computations of
recurrence-rate coefficients, finite-time Lyapunov exponents, and
autocorrelation coefficients support this hypothesis and suggest that studying
randomly perturbed one-particle billiards has the potential to yield insights
into the aggregate properties of two-particle billiards, which are difficult to
investigate directly without enormous computation times (especially when the
sizes of the confined particles are small).Comment: 9 pages, 7 figures (some with multiple parts); published in Chao
Formal Definitions of Unbounded Evolution and Innovation Reveal Universal Mechanisms for Open-Ended Evolution in Dynamical Systems
Open-ended evolution (OEE) is relevant to a variety of biological, artificial
and technological systems, but has been challenging to reproduce in silico.
Most theoretical efforts focus on key aspects of open-ended evolution as it
appears in biology. We recast the problem as a more general one in dynamical
systems theory, providing simple criteria for open-ended evolution based on two
hallmark features: unbounded evolution and innovation. We define unbounded
evolution as patterns that are non-repeating within the expected Poincare
recurrence time of an equivalent isolated system, and innovation as
trajectories not observed in isolated systems. As a case study, we implement
novel variants of cellular automata (CA) in which the update rules are allowed
to vary with time in three alternative ways. Each is capable of generating
conditions for open-ended evolution, but vary in their ability to do so. We
find that state-dependent dynamics, widely regarded as a hallmark of life,
statistically out-performs other candidate mechanisms, and is the only
mechanism to produce open-ended evolution in a scalable manner, essential to
the notion of ongoing evolution. This analysis suggests a new framework for
unifying mechanisms for generating OEE with features distinctive to life and
its artifacts, with broad applicability to biological and artificial systems.Comment: Main document: 17 pages, Supplement: 21 pages Presented at OEE2: The
Second Workshop on Open-Ended Evolution, 15th International Conference on the
Synthesis and Simulation of Living Systems (ALIFE XV), Canc\'un, Mexico, 4-8
July 2016 (http://www.tim-taylor.com/oee2/
A novel Recurrence-Transience transition and Tracy-Widom growth in a cellular automaton with quenched noise
We study the growing patterns formed by a deterministic cellular automaton,
the rotor-router model, in the presence of quenched noise. By the detailed
study of two cases, we show that: (a) the boundary of the pattern displays KPZ
fluctuations with a Tracy-Widom distribution, (b) as one increases the amount
of randomness, the rotor-router path undergoes a transition from a recurrent to
a transient walk. This transition is analysed here for the first time, and it
is shown that it falls in the 3D Anisotropic Directed Percolation universality
class.Comment: 6 pages + 8 pages SI, updated version with some correction
Strong stochastic stability for non-uniformly expanding maps
We consider random perturbations of discrete-time dynamical systems. We give
sufficient conditions for the stochastic stability of certain classes of maps,
in a strong sense. This improves the main result in J. F. Alves, V. Araujo,
Random perturbations of non-uniformly expanding maps, Asterisque 286 (2003),
25--62, where it was proved the convergence of the stationary measures of the
random process to the SRB measure of the initial system in the weak* topology.
Here, under slightly weaker assumptions on the random perturbations, we obtain
a stronger version of stochastic stability: convergence of the densities of the
stationary measures to the density of the SRB measure of the unperturbed system
in the L1-norm. As an application of our results we obtain strong stochastic
stability for two classes of non-uniformly expanding maps. The first one is an
open class of local diffeomorphisms introduced in J. F. Alves, C. Bonatti, M.
Viana, SRB measures for partially hyperbolic systems whose central direction is
mostly expanding, Invent. Math. 140 (2000), 351--398, and the second one the
class of Viana maps.Comment: 43 page
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