12,657 research outputs found
Coupled logistic maps and non-linear differential equations
We study the continuum space-time limit of a periodic one dimensional array
of deterministic logistic maps coupled diffusively. First, we analyse this
system in connection with a stochastic one dimensional Kardar-Parisi-Zhang
(KPZ) equation for confined surface fluctuations. We compare the large-scale
and long-time behaviour of space-time correlations in both systems. The dynamic
structure factor of the coupled map lattice (CML) of logistic units in its deep
chaotic regime and the usual d=1 KPZ equation have a similar temporal stretched
exponential relaxation. Conversely, the spatial scaling and, in particular, the
size dependence are very different due to the intrinsic confinement of the
fluctuations in the CML. We discuss the range of values of the non-linear
parameter in the logistic map elements and the elastic coefficient coupling
neighbours on the ring for which the connection with the KPZ-like equation
holds. In the same spirit, we derive a continuum partial differential equation
governing the evolution of the Lyapunov vector and we confirm that its
space-time behaviour becomes the one of KPZ. Finally, we briefly discuss the
interpretation of the continuum limit of the CML as a
Fisher-Kolmogorov-Petrovsky-Piscounov (FKPP) non-linear diffusion equation with
an additional KPZ non-linearity and the possibility of developing travelling
wave configurations.Comment: 23 page
Stability of stochastic impulsive differential equations: integrating the cyber and the physical of stochastic systems
According to Newton's second law of motion, we humans describe a dynamical
system with a differential equation, which is naturally discretized into a
difference equation whenever a computer is used. The differential equation is
the physical model in human brains and the difference equation the cyber model
in computers for the dynamical system. The physical model refers to the
dynamical system itself (particularly, a human-designed system) in the physical
world and the cyber model symbolises it in the cyber counterpart. This paper
formulates a hybrid model with impulsive differential equations for the
dynamical system, which integrates its physical model in real world/human
brains and its cyber counterpart in computers. The presented results establish
a theoretic foundation for the scientific study of control and communication in
the animal/human and the machine (Norbert Wiener) in the era of rise of the
machines as well as a systems science for cyber-physical systems (CPS)
A detectability criterion and data assimilation for non-linear differential equations
In this paper we propose a new sequential data assimilation method for
non-linear ordinary differential equations with compact state space. The method
is designed so that the Lyapunov exponents of the corresponding estimation
error dynamics are negative, i.e. the estimation error decays exponentially
fast. The latter is shown to be the case for generic regular flow maps if and
only if the observation matrix H satisfies detectability conditions: the rank
of H must be at least as great as the number of nonnegative Lyapunov exponents
of the underlying attractor. Numerical experiments illustrate the exponential
convergence of the method and the sharpness of the theory for the case of
Lorenz96 and Burgers equations with incomplete and noisy observations
Pulses in the Zero-Spacing Limit of the GOY Model
We study the propagation of localised disturbances in a turbulent, but
momentarily quiescent and unforced shell model (an approximation of the
Navier-Stokes equations on a set of exponentially spaced momentum shells).
These disturbances represent bursts of turbulence travelling down the inertial
range, which is thought to be responsible for the intermittency observed in
turbulence. Starting from the GOY shell model, we go to the limit where the
distance between succeeding shells approaches zero (``the zero spacing limit'')
and helicity conservation is retained. We obtain a discrete field theory which
is numerically shown to have pulse solutions travelling with constant speed and
with unchanged form. We give numerical evidence that the model might even be
exactly integrable, although the continuum limit seems to be singular and the
pulses show an unusual super exponential decay to zero as when , where is the {\em
golden mean}. For finite momentum shell spacing, we argue that the pulses
should accelerate, moving to infinity in a finite time. Finally we show that
the maximal Lyapunov exponent of the GOY model approaches zero in this limit.Comment: 27 pages, submitted for publicatio
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