1,290 research outputs found
The dynamics of correlated novelties
One new thing often leads to another. Such correlated novelties are a
familiar part of daily life. They are also thought to be fundamental to the
evolution of biological systems, human society, and technology. By opening new
possibilities, one novelty can pave the way for others in a process that
Kauffman has called "expanding the adjacent possible". The dynamics of
correlated novelties, however, have yet to be quantified empirically or modeled
mathematically. Here we propose a simple mathematical model that mimics the
process of exploring a physical, biological or conceptual space that enlarges
whenever a novelty occurs. The model, a generalization of Polya's urn, predicts
statistical laws for the rate at which novelties happen (analogous to Heaps'
law) and for the probability distribution on the space explored (analogous to
Zipf's law), as well as signatures of the hypothesized process by which one
novelty sets the stage for another. We test these predictions on four data sets
of human activity: the edit events of Wikipedia pages, the emergence of tags in
annotation systems, the sequence of words in texts, and listening to new songs
in online music catalogues. By quantifying the dynamics of correlated
novelties, our results provide a starting point for a deeper understanding of
the ever-expanding adjacent possible and its role in biological, linguistic,
cultural, and technological evolution
Scaling and singularities in the entrainment of globally-coupled oscillators
The onset of collective behavior in a population of globally coupled
oscillators with randomly distributed frequencies is studied for phase
dynamical models with arbitrary coupling. The population is described by a
Fokker-Planck equation for the distribution of phases which includes the
diffusive effect of noise in the oscillator frequencies. The bifurcation from
the phase-incoherent state is analyzed using amplitude equations for the
unstable modes with particular attention to the dependence of the nonlinearly
saturated mode on the linear growth rate . In general
we find where is the
diffusion coefficient and is the mode number of the unstable mode. The
unusual factor arises from a singularity in the cubic term of
the amplitude equation.Comment: 11 pages (Revtex); paper submitted to Phys. Rev. Let
Hydrodynamic synchronisation of non-linear oscillators at low Reynolds number
We introduce a generic model of weakly non-linear self-sustained oscillator
as a simplified tool to study synchronisation in a fluid at low Reynolds
number. By averaging over the fast degrees of freedom, we examine the effect of
hydrodynamic interactions on the slow dynamics of two oscillators and show that
they can lead to synchronisation. Furthermore, we find that synchronisation is
strongly enhanced when the oscillators are non-isochronous, which on the limit
cycle means the oscillations have an amplitude-dependent frequency.
Non-isochronity is determined by a nonlinear coupling being non-zero.
We find that its () sign determines if they synchronise in- or
anti-phase. We then study an infinite array of oscillators in the long
wavelength limit, in presence of noise. For , hydrodynamic
interactions can lead to a homogeneous synchronised state. Numerical
simulations for a finite number of oscillators confirm this and, when , show the propagation of waves, reminiscent of metachronal coordination.Comment: 4 pages, 2 figure
Multi-shocks in asymmetric simple exclusions processes: Insights from fixed-point analysis of the boundary-layers
The boundary-induced phase transitions in an asymmetric simple exclusion
process with inter-particle repulsion and bulk non-conservation are analyzed
through the fixed points of the boundary layers. This system is known to have
phases in which particle density profiles have different kinds of shocks. We
show how this boundary-layer fixed-point method allows us to gain physical
insights on the nature of the phases and also to obtain several quantitative
results on the density profiles especially on the nature of the boundary-layers
and shocks.Comment: 12 pages, 8 figure
Synchronization of globally coupled two-state stochastic oscillators with a state dependent refractory period
We present a model of identical coupled two-state stochastic units each of
which in isolation is governed by a fixed refractory period. The nonlinear
coupling between units directly affects the refractory period, which now
depends on the global state of the system and can therefore itself become time
dependent. At weak coupling the array settles into a quiescent stationary
state. Increasing coupling strength leads to a saddle node bifurcation, beyond
which the quiescent state coexists with a stable limit cycle of nonlinear
coherent oscillations. We explicitly determine the critical coupling constant
for this transition
Synchronization from Disordered Driving Forces in Arrays of Coupled Oscillators
The effects of disorder in external forces on the dynamical behavior of
coupled nonlinear oscillator networks are studied. When driven synchronously,
i.e., all driving forces have the same phase, the networks display chaotic
dynamics. We show that random phases in the driving forces result in regular,
periodic network behavior. Intermediate phase disorder can produce network
synchrony. Specifically, there is an optimal amount of phase disorder, which
can induce the highest level of synchrony. These results demonstrate that the
spatiotemporal structure of external influences can control chaos and lead to
synchronization in nonlinear systems.Comment: 4 pages, 4 figure
Spatial synchronization and extinction of species under external forcing
We study the interplay between synchronization and extinction of a species.
Using a general model we show that under a common external forcing, the species
with a quadratic saturation term in the population dynamics first undergoes
spatial synchronization and then extinction, thereby avoiding the rescue
effect. This is because the saturation term reduces the synchronization time
scale but not the extinction time scale. The effect can be observed even when
the external forcing acts only on some locations provided there is a
synchronizing term in the dynamics. Absence of the quadratic saturation term
can help the species to avoid extinction.Comment: 4 pages, 2 figure
Phase Diagram for the Winfree Model of Coupled Nonlinear Oscillators
In 1967 Winfree proposed a mean-field model for the spontaneous
synchronization of chorusing crickets, flashing fireflies, circadian pacemaker
cells, or other large populations of biological oscillators. Here we give the
first bifurcation analysis of the model, for a tractable special case. The
system displays rich collective dynamics as a function of the coupling strength
and the spread of natural frequencies. Besides incoherence, frequency locking,
and oscillator death, there exist novel hybrid solutions that combine two or
more of these states. We present the phase diagram and derive several of the
stability boundaries analytically.Comment: 4 pages, 4 figure
Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode
An amplitude equation for an unstable mode in a collisionless plasma is
derived from the dynamics on the two-dimensional unstable manifold of the
equilibrium. The mode amplitude decouples from the phase due to the
spatial homogeneity of the equilibrium, and the resulting one-dimensional
dynamics is analyzed using an expansion in . As the linear growth rate
vanishes, the expansion coefficients diverge; a rescaling
of the mode amplitude absorbs these
singularities and reveals that the mode electric field exhibits trapping
scaling as . The dynamics for
depends only on the phase where is the derivative of the dielectric as
.Comment: 11 pages (Latex/RevTex), 2 figures available in hard copy from the
Author ([email protected]); paper accepted by Physical Review
Letter
A neuronal network model of interictal and recurrent ictal activity
We propose a neuronal network model which undergoes a saddle-node bifurcation
on an invariant circle as the mechanism of the transition from the interictal
to the ictal (seizure) state. In the vicinity of this transition, the model
captures important dynamical features of both interictal and ictal states. We
study the nature of interictal spikes and early warnings of the transition
predicted by this model. We further demonstrate that recurrent seizures emerge
due to the interaction between two networks.Comment: 9 pages, 7 figure
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