2,060 research outputs found
Noise, Bifurcations, and Modeling of Interacting Particle Systems
We consider the stochastic patterns of a system of communicating, or coupled,
self-propelled particles in the presence of noise and communication time delay.
For sufficiently large environmental noise, there exists a transition between a
translating state and a rotating state with stationary center of mass. Time
delayed communication creates a bifurcation pattern dependent on the coupling
amplitude between particles. Using a mean field model in the large number
limit, we show how the complete bifurcation unfolds in the presence of
communication delay and coupling amplitude. Relative to the center of mass, the
patterns can then be described as transitions between translation, rotation
about a stationary point, or a rotating swarm, where the center of mass
undergoes a Hopf bifurcation from steady state to a limit cycle. Examples of
some of the stochastic patterns will be given for large numbers of particles
Dynamics of the Desai-Zwanzig model in multiwell and random energy landscapes
We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {\bf 19}1-24 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multi-well potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional and a continuation algorithm for the stationary solutions
Canonical active Brownian motion
Active Brownian motion is the complex motion of active Brownian particles.
They are active in the sense that they can transform their internal energy into
energy of motion and thus create complex motion patterns. Theories of active
Brownian motion so far imposed couplings between the internal energy and the
kinetic energy of the system. We investigate how this idea can be naturally
taken further to include also couplings to the potential energy, which finally
leads to a general theory of canonical dissipative systems. Explicit analytical
and numerical studies are done for the motion of one particle in harmonic
external potentials. Apart from stationary solutions, we study non-equilibrium
dynamics and show the existence of various bifurcation phenomena.Comment: 11 pages, 6 figures, a few remarks and references adde
Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions
In this manuscript we analyze the collective behavior of mean-field limits of
large-scale, spatially extended stochastic neuronal networks with delays.
Rigorously, the asymptotic regime of such systems is characterized by a very
intricate stochastic delayed integro-differential McKean-Vlasov equation that
remain impenetrable, leaving the stochastic collective dynamics of such
networks poorly understood. In order to study these macroscopic dynamics, we
analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics
and sigmoidal interactions. In that case, we prove that the solution of the
mean-field equation is Gaussian, hence characterized by its two first moments,
and that these two quantities satisfy a set of coupled delayed
integro-differential equations. These equations are similar to usual neural
field equations, and incorporate noise levels as a parameter, allowing analysis
of noise-induced transitions. We identify through bifurcation analysis several
qualitative transitions due to noise in the mean-field limit. In particular,
stabilization of spatially homogeneous solutions, synchronized oscillations,
bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from
static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow
further exploring the role of noise in the nervous system.Comment: Updated to the latest version published, and clarified the dependence
in space of Brownian motion
Mean Field Limits for Interacting Diffusions in a Two-Scale Potential
In this paper we study the combined mean field and homogenization limits for
a system of weakly interacting diffusions moving in a two-scale, locally
periodic confining potential, of the form considered
in~\cite{DuncanPavliotis2016}. We show that, although the mean field and
homogenization limits commute for finite times, they do not, in general,
commute in the long time limit. In particular, the bifurcation diagrams for the
stationary states can be different depending on the order with which we take
the two limits. Furthermore, we construct the bifurcation diagram for the
stationary McKean-Vlasov equation in a two-scale potential, before passing to
the homogenization limit, and we analyze the effect of the multiple local
minima in the confining potential on the number and the stability of stationary
solutions
Particle interactions mediated by dynamical networks: assessment of macroscopic descriptions
We provide a numerical study of the macroscopic model of [3] derived from an
agent-based model for a system of particles interacting through a dynamical
network of links. Assuming that the network remodelling process is very fast,
the macroscopic model takes the form of a single aggregation diffusion equation
for the density of particles. The theoretical study of the macroscopic model
gives precise criteria for the phase transitions of the steady states, and in
the 1-dimensional case, we show numerically that the stationary solutions of
the microscopic model undergo the same phase transitions and bifurcation types
as the macroscopic model. In the 2-dimensional case, we show that the numerical
simulations of the macroscopic model are in excellent agreement with the
predicted theoretical values. This study provides a partial validation of the
formal derivation of the macroscopic model from a microscopic formulation and
shows that the former is a consistent approximation of an underlying particle
dynamics, making it a powerful tool for the modelling of dynamical networks at
a large scale
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