237,298 research outputs found
Variations on topological recurrence
Recurrence properties of systems and associated sets of integers that suffice
for recurrence are classical objects in topological dynamics. We describe
relations between recurrence in different sorts of systems, study ways to
formulate finite versions of recurrence, and describe connections to
combinatorial problems. In particular, we show that sets of Bohr recurrence
(meaning sets of recurrence for rotations) suffice for recurrence in
nilsystems. Additionally, we prove an extension of this property for multiple
recurrence in affine systems
Non-recurrence sets for weakly mixing linear dynamical systems
We study non-recurrence sets for weakly mixing dynamical systems by using
linear dynamical systems. These are systems consisting of a bounded linear
operator acting on a separable complex Banach space X, which becomes a
probability space when endowed with a non-degenerate Gaussian measure. We
generalize some recent results of Bergelson, del Junco, Lema\'nczyk and
Rosenblatt, and show in particular that sets \{n_k\} such that n_{k+1}/{n_k}
tends to infinity, or such that n_{k} divides n_{k+1} for each k, are
non-recurrence sets for weakly mixing linear dynamical systems. We also give
examples, for each r, of r-Bohr sets which are non-recurrence sets for some
weakly mixing systems
Bergelson's Theorem for weakly mixing C*-dynamical systems
We study a nonconventional ergodic average for asymptotically abelian weakly
mixing C*-dynamical systems, related to a second iteration of Khintchine's
recurrence theorem obtained by Bergelson in the measure theoretic case. A
noncommutative recurrence theorem for such systems is obtained as a corollary.Comment: 26 page
Phase Synchronization in Unidirectionally Coupled Ikeda Time-delay Systems
Phase synchronization in unidirectionally coupled Ikeda time-delay systems
exhibiting non-phase-coherent hyperchaotic attractors of complex topology with
highly interwoven trajectories is studied. It is shown that in this set of
coupled systems phase synchronization (PS) does exist in a range of the
coupling strength which is preceded by a transition regime (approximate PS) and
a nonsynchronous regime. However, exact generalized synchronization does not
seem to occur in the coupled Ikeda systems (for the range of parameters we have
studied) even for large coupling strength, in contrast to our earlier studies
in coupled piecewise-linear and Mackey-Glass systems
\cite{dvskml2006,dvskml2008}. The above transitions are characterized in terms
of recurrence based indices, namely generalized autocorrelation function
, correlation of probability of recurrence (CPR), joint probability of
recurrence (JPR) and similarity of probability of recurrence (SPR). The
existence of phase synchronization is also further confirmed by typical
transitions in the Lyapunov exponents of the coupled Ikeda time-delay systems
and also using the concept of localized sets.Comment: 10 pages, 7 figure
Recurrence spectrum in smooth dynamical systems
We prove that for conformal expanding maps the return time does have constant
multifractal spectrum. This is the counterpart of the result by Feng and Wu in
the symbolic setting
Fractional recurrence in discrete-time quantum walk
Quantum recurrence theorem holds for quantum systems with discrete energy
eigenvalues and fails to hold in general for systems with continuous energy. We
show that during quantum walk process dominated by interference of amplitude
corresponding to different paths fail to satisfy the complete quantum
recurrence theorem. Due to the revival of the fractional wave packet, a
fractional recurrence characterized using quantum P\'olya number can be seen.Comment: 10 pages, 11 figure : Accepted to appear in Central European Journal
of Physic
How to avoid potential pitfalls in recurrence plot based data analysis
Recurrence plots and recurrence quantification analysis have become popular
in the last two decades. Recurrence based methods have on the one hand a deep
foundation in the theory of dynamical systems and are on the other hand
powerful tools for the investigation of a variety of problems. The increasing
interest encompasses the growing risk of misuse and uncritical application of
these methods. Therefore, we point out potential problems and pitfalls related
to different aspects of the application of recurrence plots and recurrence
quantification analysis
- …