148 research outputs found

### Very long transients in globally coupled maps

Very long transients are found in the partially ordered phase of type II of
globally coupled logistic maps. The transients always lead the system in this
phase to a state with a few synchronous clusters. This transient behaviour is
not significantly influenced by the introduction of weak noises. However, such
noises generally favor cluster partitions with more stable periodic dynamics.Comment: 4 pages, 5 figures include

### On the genealogy of a population of biparental individuals

If one goes backward in time, the number of ancestors of an individual
doubles at each generation. This exponential growth very quickly exceeds the
population size, when this size is finite. As a consequence, the ancestors of a
given individual cannot be all different and most remote ancestors are repeated
many times in any genealogical tree. The statistical properties of these
repetitions in genealogical trees of individuals for a panmictic closed
population of constant size N can be calculated. We show that the distribution
of the repetitions of ancestors reaches a stationary shape after a small number
Gc ~ log N of generations in the past, that only about 80% of the ancestral
population belongs to the tree (due to coalescence of branches), and that two
trees for individuals in the same population become identical after Gc
generations have elapsed. Our analysis is easy to extend to the case of
exponentially growing population.Comment: 14 pages, 7 figures, to appear in the Journal of Theoretical Biolog

### Replica-symmetry breaking in dynamical glasses

Systems of globally coupled logistic maps (GCLM) can display complex
collective behaviour characterized by the formation of synchronous clusters. In
the dynamical clustering regime, such systems possess a large number of
coexisting attractors and might be viewed as dynamical glasses. Glass
properties of GCLM in the thermodynamical limit of large system sizes $N$ are
investigated. Replicas, representing orbits that start from various initial
conditions, are introduced and distributions of their overlaps are numerically
determined. We show that for fixed-field ensembles of initial conditions, as
used in previous numerical studies, all attractors of the system become
identical in the thermodynamical limit up to variations of order $1/\sqrt{N}$
because the initial value of the coupling field is characterized by vanishing
fluctuations, and thus replica symmetry is recovered for $N\to \infty$. In
contrast to this, when random-field ensembles of initial conditions are chosen,
replica symmetry remains broken in the thermodynamical limit.Comment: 19 pages, 18 figure

### Long transients and cluster size in globally coupled maps

We analyze the asymptotic states in the partially ordered phase of a system
of globally coupled logistic maps. We confirm that, regardless of initial
conditions, these states consist of a few clusters, and they properly belong in
the ordered phase of these systems. The transient times necessary to reach the
asymptotic states can be very long, especially very near the transition line
separating the ordered and the coherent phases. We find that, where two
clusters form, the distribution of their sizes corresponds to windows of
regular or narrow-band chaotic behavior in the bifurcation diagram of a system
of two degrees of freedom that describes the motion of two clusters, where the
size of one cluster acts as a bifurcation parameter.Comment: To appear in Europhysics Letter

### Time delay in the Kuramoto model with bimodal frequency distribution

We investigate the effects of a time-delayed all-to-all coupling scheme in a
large population of oscillators with natural frequencies following a bimodal
distribution. The regions of parameter space corresponding to synchronized and
incoherent solutions are obtained both numerically and analytically for
particular frequency distributions. In particular we find that bimodality
introduces a new time scale that results in a quasiperiodic disposition of the
regions of incoherence.Comment: 5 pages, 4 figure

### Chimera Ising Walls in Forced Nonlocally Coupled Oscillators

Nonlocally coupled oscillator systems can exhibit an exotic spatiotemporal
structure called chimera, where the system splits into two groups of
oscillators with sharp boundaries, one of which is phase-locked and the other
is phase-randomized. Two examples of the chimera states are known: the first
one appears in a ring of phase oscillators, and the second one is associated
with the two-dimensional rotating spiral waves. In this article, we report yet
another example of the chimera state that is associated with the so-called
Ising walls in one-dimensional spatially extended systems, which is exhibited
by a nonlocally coupled complex Ginzburg-Landau equation with external forcing.Comment: 7 pages, 5 figures, to appear in Phys. Rev.

### Coherence in scale-free networks of chaotic maps

We study fully synchronized states in scale-free networks of chaotic logistic
maps as a function of both dynamical and topological parameters. Three
different network topologies are considered: (i) random scale-free topology,
(ii) deterministic pseudo-fractal scale-free network, and (iii) Apollonian
network. For the random scale-free topology we find a coupling strength
threshold beyond which full synchronization is attained. This threshold scales
as $k^{-\mu}$, where $k$ is the outgoing connectivity and $\mu$ depends on the
local nonlinearity. For deterministic scale-free networks coherence is observed
only when the coupling strength is proportional to the neighbor connectivity.
We show that the transition to coherence is of first-order and study the role
of the most connected nodes in the collective dynamics of oscillators in
scale-free networks.Comment: 9 pages, 8 figure

### Propagation of small perturbations in synchronized oscillator networks

We study the propagation of a harmonic perturbation of small amplitude on a
network of coupled identical phase oscillators prepared in a state of full
synchronization. The perturbation is externally applied to a single oscillator,
and is transmitted to the other oscillators through coupling. Numerical results
and an approximate analytical treatment, valid for random and ordered networks,
show that the response of each oscillator is a rather well-defined function of
its distance from the oscillator where the external perturbation is applied.
For small distances, the system behaves as a dissipative linear medium: the
perturbation amplitude decreases exponentially with the distance, while
propagating at constant speed. We suggest that the pattern of interactions may
be deduced from measurements of the response of individual oscillators to
perturbations applied at different nodes of the network

### The Kuramoto model with distributed shear

We uncover a solvable generalization of the Kuramoto model in which shears
(or nonisochronicities) and natural frequencies are distributed and
statistically dependent. We show that the strength and sign of this dependence
greatly alter synchronization and yield qualitatively different phase diagrams.
The Ott-Antonsen ansatz allows us to obtain analytical results for a specific
family of joint distributions. We also derive, using linear stability analysis,
general formulae for the stability border of incoherence.Comment: 6 page

### Existence of hysteresis in the Kuramoto model with bimodal frequency distributions

We investigate the transition to synchronization in the Kuramoto model with
bimodal distributions of the natural frequencies. Previous studies have
concluded that the model exhibits a hysteretic phase transition if the bimodal
distribution is close to a unimodal one, due to the shallowness the central
dip. Here we show that proximity to the unimodal-bimodal border does not
necessarily imply hysteresis when the width, but not the depth, of the central
dip tends to zero. We draw this conclusion from a detailed study of the
Kuramoto model with a suitable family of bimodal distributions.Comment: 9 pages, 5 figures, to appear in Physical Review

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