380 research outputs found
Analysis of chaotic motion and its shape dependence in a generalized piecewise linear map
We analyse the chaotic motion and its shape dependence in a piecewise linear
map using Fujisaka's characteristic function method. The map is a
generalization of the one introduced by R. Artuso. Exact expressions for
diffusion coefficient are obtained giving previously obtained results as
special cases. Fluctuation spectrum relating to probability density function is
obtained in a parametric form. We also give limiting forms of the above
quantities. Dependence of diffusion coefficient and probability density
function on the shape of the map is examined.Comment: 4 pages,4 figure
Synchronization of Coupled Systems with Spatiotemporal Chaos
We argue that the synchronization transition of stochastically coupled
cellular automata, discovered recently by L.G. Morelli {\it et al.} (Phys. Rev.
{\bf 58 E}, R8 (1998)), is generically in the directed percolation universality
class. In particular, this holds numerically for the specific example studied
by these authors, in contrast to their claim. For real-valued systems with
spatiotemporal chaos such as coupled map lattices, we claim that the
synchronization transition is generically in the universality class of the
Kardar-Parisi-Zhang equation with a nonlinear growth limiting term.Comment: 4 pages, including 3 figures; submitted to Phys. Rev.
Phase synchronization in time-delay systems
Though the notion of phase synchronization has been well studied in chaotic
dynamical systems without delay, it has not been realized yet in chaotic
time-delay systems exhibiting non-phase coherent hyperchaotic attractors. In
this article we report the first identification of phase synchronization in
coupled time-delay systems exhibiting hyperchaotic attractor. We show that
there is a transition from non-synchronized behavior to phase and then to
generalized synchronization as a function of coupling strength. These
transitions are characterized by recurrence quantification analysis, by phase
differences based on a new transformation of the attractors and also by the
changes in the Lyapunov exponents. We have found these transitions in coupled
piece-wise linear and in Mackey-Glass time-delay systems.Comment: 4 pages, 3 Figures (To appear in Physical Review E Rapid
Communication
Studying Attractor Symmetries by Means of Cross Correlation Sums
We use the cross correlation sum introduced recently by H. Kantz to study
symmetry properties of chaotic attractors. In particular, we apply it to a
system of six coupled nonlinear oscillators which was shown by Kroon et al. to
have attractors with several different symmetries, and compare our results with
those obtained by ``detectives" in the sense of Golubitsky et al.Comment: LaTeX file, 16 pages and 16 postscript figures; tarred, gzipped and
uuencoded; submitted to 'Nonlinearity
Fundamental scaling laws of on-off intermittency in a stochastically driven dissipative pattern forming system
Noise driven electroconvection in sandwich cells of nematic liquid crystals
exhibits on-off intermittent behaviour at the onset of the instability. We
study laser scattering of convection rolls to characterize the wavelengths and
the trajectories of the stochastic amplitudes of the intermittent structures.
The pattern wavelengths and the statistics of these trajectories are in
quantitative agreement with simulations of the linearized electrohydrodynamic
equations. The fundamental distribution law for the durations
of laminar phases as well as the power law of the amplitude distribution
of intermittent bursts are confirmed in the experiments. Power spectral
densities of the experimental and numerically simulated trajectories are
discussed.Comment: 20 pages and 17 figure
Infinities of stable periodic orbits in systems of coupled oscillators
We consider the dynamical behavior of coupled oscillators with robust heteroclinic cycles between saddles that may be periodic or chaotic. We differentiate attracting cycles into types that we call phase resetting and free running depending on whether the cycle approaches a given saddle along one or many trajectories. At loss of stability of attracting cycling, we show in a phase-resetting example the existence of an infinite family of stable periodic orbits that accumulate on the cycling, whereas for a free-running example loss of stability of the cycling gives rise to a single quasiperiodic or chaotic attractor
Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory
We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motivated by a low-order model for magnetic activity in a stellar dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations, where the saddle-node bifurcation is assumed to have global reinjection of trajectories. The model can produce chaotic behaviour within each of a pair of invariant subspaces, and also it can show attractors that are stuck-on to both of the invariant subspaces. We investigate the detailed intermittent dynamics for such an attractor, investigating the effect of breaking the symmetry between the two Hopf bifurcations, and observing that it can appear via blowout bifurcations from the invariant subspaces.
We give a simple Markov chain model for the two-state intermittent dynamics that reproduces the time spent close to the invariant subspaces and the switching between the different possible invariant subspaces; this clarifies the observation that the proportion of time spent near the different subspaces depends on the average residence time and also on the probabilities of switching between the possible subspaces
Asymptotic power law of moments in a random multiplicative process with weak additive noise
It is well known that a random multiplicative process with weak additive
noise generates a power-law probability distribution. It has recently been
recognized that this process exhibits another type of power law: the moment of
the stochastic variable scales as a function of the additive noise strength. We
clarify the mechanism for this power-law behavior of moments by treating a
simple Langevin-type model both approximately and exactly, and argue this
mechanism is universal. We also discuss the relevance of our findings to noisy
on-off intermittency and to singular spatio-temporal chaos recently observed in
systems of non-locally coupled elements.Comment: 11 pages, 9 figures, submitted to Phys. Rev.
Absence of First-order Transition and Tri-critical Point in the Dynamic Phase Diagram of a Spatially Extended Bistable System in an Oscillating Field
It has been well established that spatially extended, bistable systems that
are driven by an oscillating field exhibit a nonequilibrium dynamic phase
transition (DPT). The DPT occurs when the field frequency is on the order of
the inverse of an intrinsic lifetime associated with the transitions between
the two stable states in a static field of the same magnitude as the amplitude
of the oscillating field. The DPT is continuous and belongs to the same
universality class as the equilibrium phase transition of the Ising model in
zero field [G. Korniss et al., Phys. Rev. E 63, 016120 (2001); H. Fujisaka et
al., Phys. Rev. E 63, 036109 (2001)]. However, it has previously been claimed
that the DPT becomes discontinuous at temperatures below a tricritical point
[M. Acharyya, Phys. Rev. E 59, 218 (1999)]. This claim was based on
observations in dynamic Monte Carlo simulations of a multipeaked probability
density for the dynamic order parameter and negative values of the fourth-order
cumulant ratio. Both phenomena can be characteristic of discontinuous phase
transitions. Here we use classical nucleation theory for the decay of
metastable phases, together with data from large-scale dynamic Monte Carlo
simulations of a two-dimensional kinetic Ising ferromagnet, to show that these
observations in this case are merely finite-size effects. For sufficiently
small systems and low temperatures, the continuous DPT is replaced, not by a
discontinuous phase transition, but by a crossover to stochastic resonance. In
the infinite-system limit the stochastic-resonance regime vanishes, and the
continuous DPT should persist for all nonzero temperatures
Chaos and Synchronized Chaos in an Earthquake Model
We show that chaos is present in the symmetric two-block Burridge-Knopoff
model for earthquakes. This is in contrast with previous numerical studies, but
in agreement with experimental results. In this system, we have found a rich
dynamical behavior with an unusual route to chaos. In the three-block system,
we see the appearance of synchronized chaos, showing that this concept can have
potential applications in the field of seismology.Comment: To appear in Physical Review Letters (13 pages, 6 figures
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