50 research outputs found
The optimal sink and the best source in a Markov chain
It is well known that the distributions of hitting times in Markov chains are
quite irregular, unless the limit as time tends to infinity is considered. We
show that nevertheless for a typical finite irreducible Markov chain and for
nondegenerate initial distributions the tails of the distributions of the
hitting times for the states of a Markov chain can be ordered, i.e., they do
not overlap after a certain finite moment of time.
If one considers instead each state of a Markov chain as a source rather than
a sink then again the states can generically be ordered according to their
efficiency. The mechanisms underlying these two orderings are essentially
different though.Comment: 12 pages, 1 figur
On the influence of noise on chaos in nearly Hamiltonian systems
The simultaneous influence of small damping and white noise on Hamiltonian
systems with chaotic motion is studied on the model of periodically kicked
rotor. In the region of parameters where damping alone turns the motion into
regular, the level of noise that can restore the chaos is studied. This
restoration is created by two mechanisms: by fluctuation induced transfer of
the phase trajectory to domains of local instability, that can be described by
the averaging of the local instability index, and by destabilization of motion
within the islands of stability by fluctuation induced parametric modulation of
the stability matrix, that can be described by the methods developed in the
theory of Anderson localization in one-dimensional systems.Comment: 10 pages REVTEX, 9 figures EP
Delayed Self-Synchronization in Homoclinic Chaos
The chaotic spike train of a homoclinic dynamical system is self-synchronized
by re-inserting a small fraction of the delayed output. Due to the sensitive
nature of the homoclinic chaos to external perturbations, stabilization of very
long periodic orbits is possible. On these orbits, the dynamics appears chaotic
over a finite time, but then it repeats with a recurrence time that is slightly
longer than the delay time. The effect, called delayed self-synchronization
(DSS), displays analogies with neurodynamic events which occur in the build-up
of long term memories.Comment: Submitted to Phys. Rev. Lett., 13 pages, 7 figure
Universal neural field computation
Turing machines and G\"odel numbers are important pillars of the theory of
computation. Thus, any computational architecture needs to show how it could
relate to Turing machines and how stable implementations of Turing computation
are possible. In this chapter, we implement universal Turing computation in a
neural field environment. To this end, we employ the canonical symbologram
representation of a Turing machine obtained from a G\"odel encoding of its
symbolic repertoire and generalized shifts. The resulting nonlinear dynamical
automaton (NDA) is a piecewise affine-linear map acting on the unit square that
is partitioned into rectangular domains. Instead of looking at point dynamics
in phase space, we then consider functional dynamics of probability
distributions functions (p.d.f.s) over phase space. This is generally described
by a Frobenius-Perron integral transformation that can be regarded as a neural
field equation over the unit square as feature space of a dynamic field theory
(DFT). Solving the Frobenius-Perron equation yields that uniform p.d.f.s with
rectangular support are mapped onto uniform p.d.f.s with rectangular support,
again. We call the resulting representation \emph{dynamic field automaton}.Comment: 21 pages; 6 figures. arXiv admin note: text overlap with
arXiv:1204.546
Generalized Phase Synchronization in unidirectionally coupled chaotic oscillators
We investigate phase synchronization between two identical or detuned
response oscillators coupled to a slightly different drive oscillator. Our
result is that phase synchronization can occur between response oscillators
when they are driven by correlated (but not identical) inputs from the drive
oscillator. We call this phenomenon Generalized Phase Synchronization (GPS) and
clarify its characteristics using Lyapunov exponents and phase difference
plots.Comment: 4 pages, 5 figure
Frozen spatial chaos induced by boundaries
We show that rather simple but non-trivial boundary conditions could induce
the appearance of spatial chaos (that is stationary, stable, but spatially
disordered configurations) in extended dynamical systems with very simple
dynamics. We exemplify the phenomenon with a nonlinear reaction-diffusion
equation in a two-dimensional undulated domain. Concepts from the theory of
dynamical systems, and a transverse-single-mode approximation are used to
describe the spatially chaotic structures.Comment: 9 pages, 6 figures, submitted for publication; for related work visit
http://www.imedea.uib.es/~victo
On the recurrence and robust properties of Lorenz'63 model
Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its
dynamical and statistical behavior considering the evolution of the associated
Casimir functions. We study the invariant density and other recurrence features
of a Markov expanding Lorenz-like map of the interval arising in the analysis
of the predictability of the extreme values reached by particular physical
observables evolving in time under the Lorenz'63 dynamics with the classical
set of parameters. Moreover, we prove the statistical stability of such an
invariant measure. This will allow us to further characterize the SRB measure
of the system.Comment: 44 pages, 7 figures, revised version accepted for pubblicatio
Partial symmetry breaking and heteroclinic tangencies
We study some global aspects of the bifurcation of an equivariant family of
volume-contracting vector fields on the three-dimensional sphere. When part of
the symmetry is broken, the vector fields exhibit Bykov cycles. Close to the
symmetry, we investigate the mechanism of the emergence of heteroclinic
tangencies coexisting with transverse connections. We find persistent suspended
horseshoes accompanied by attracting periodic trajectories with long periods