22,327 research outputs found
Applications of Dynamical Systems in Engineering
This paper presents the current possible applications of Dynamical Systems in
Engineering. The applications of chaos, fractals have proven to be an exciting
and fruitful endeavor. These applications are highly diverse ranging over such
fields as Electrical, Electronics and Computer Engineering. Dynamical Systems
theory describes general patterns found in the solution of systems of nonlinear
equations. The theory focuses upon those equations representing the change of
processes in time. This paper offers the issue of applying dynamical systems
methods to a wider circle of Engineering problems. There are three components
to our approach: ongoing and possible applications of Fractals, Chaos Theory
and Dynamical Systems. Some basic and useful computer simulation of Dynamical
System related problems have been shown also.Comment: 32 Page
Koopman analysis of the long-term evolution in a turbulent convection cell
We analyse the long-time evolution of the three-dimensional flow in a closed
cubic turbulent Rayleigh-B\'{e}nard convection cell via a Koopman eigenfunction
analysis. A data-driven basis derived from diffusion kernels known in machine
learning is employed here to represent a regularized generator of the unitary
Koopman group in the sense of a Galerkin approximation. The resulting Koopman
eigenfunctions can be grouped into subsets in accordance with the discrete
symmetries in a cubic box. In particular, a projection of the velocity field
onto the first group of eigenfunctions reveals the four stable large-scale
circulation (LSC) states in the convection cell. We recapture the preferential
circulation rolls in diagonal corners and the short-term switching through roll
states parallel to the side faces which have also been seen in other
simulations and experiments. The diagonal macroscopic flow states can last as
long as a thousand convective free-fall time units. In addition, we find that
specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced
oscillatory fluctuations for particular stable diagonal states of the LSC. The
corresponding velocity field structures, such as corner vortices and swirls in
the midplane, are also discussed via spatiotemporal reconstructions.Comment: 32 pages, 9 figures, article in press at Journal of Fluid Mechanic
Discrete wave turbulence of rotational capillary water waves
We study the discrete wave turbulent regime of capillary water waves with
constant non-zero vorticity. The explicit Hamiltonian formulation and the
corresponding coupling coefficient are obtained. We also present the
construction and investigation of resonance clustering. Some physical
implications of the obtained results are discussed.Comment: 13 pages, 10 figure
Numerical simulations of nonlinear modes in mica: past, present and future
We review research on the role of nonlinear coherent phenomena (e.g breathers
and kinks) in the formation linear decorations in mica crystal. The work is
based on a new model for the motion of the mica hexagonal K layer, which allows
displacement of the atoms from the unit cell. With a simple piece-wise
polynomial inter-particle potential, we verify the existence of localized
long-lived breathers in an idealized lattice at 0K. Moreover, our model allows
us to observe long-lived localized kinks. We study the interactions of such
localized modes along a lattice direction, and in addition demonstrate fully
two dimensional scattering of such pulses for the first time. For large
interatomic forces we observe a spreading horseshoe-shaped wave, a type of
shock wave but with a breather profile
A system of ODEs for a Perturbation of a Minimal Mass Soliton
We study soliton solutions to a nonlinear Schrodinger equation with a
saturated nonlinearity. Such nonlinearities are known to possess minimal mass
soliton solutions. We consider a small perturbation of a minimal mass soliton,
and identify a system of ODEs similar to those from Comech and Pelinovsky
(2003), which model the behavior of the perturbation for short times. We then
provide numerical evidence that under this system of ODEs there are two
possible dynamical outcomes, which is in accord with the conclusions of
Pelinovsky, Afanasjev, and Kivshar (1996). For initial data which supports a
soliton structure, a generic initial perturbation oscillates around the stable
family of solitons. For initial data which is expected to disperse, the finite
dimensional dynamics follow the unstable portion of the soliton curve.Comment: Minor edit
A probabilistic decomposition-synthesis method for the quantification of rare events due to internal instabilities
We consider the problem of probabilistic quantification of dynamical systems
that have heavy-tailed characteristics. These heavy-tailed features are
associated with rare transient responses due to the occurrence of internal
instabilities. Here we develop a computational method, a probabilistic
decomposition-synthesis technique, that takes into account the nature of
internal instabilities to inexpensively determine the non-Gaussian probability
density function for any arbitrary quantity of interest. Our approach relies on
the decomposition of the statistics into a `non-extreme core', typically
Gaussian, and a heavy-tailed component. This decomposition is in full
correspondence with a partition of the phase space into a `stable' region where
we have no internal instabilities, and a region where non-linear instabilities
lead to rare transitions with high probability. We quantify the statistics in
the stable region using a Gaussian approximation approach, while the
non-Gaussian distributions associated with the intermittently unstable regions
of the phase space are inexpensively computed through order-reduction methods
that take into account the strongly nonlinear character of the dynamics. The
probabilistic information in the two domains is analytically synthesized
through a total probability argument. The proposed approach allows for the
accurate quantification of non-Gaussian tails at more than 10 standard
deviations, at a fraction of the cost associated with the direct Monte-Carlo
simulations. We demonstrate the probabilistic decomposition-synthesis method
for rare events for two dynamical systems exhibiting extreme events: a
two-degree-of-freedom system of nonlinearly coupled oscillators, and in a
nonlinear envelope equation characterizing the propagation of unidirectional
water waves
Nonlinear physics of electrical wave propagation in the heart: a review
The beating of the heart is a synchronized contraction of muscle cells
(myocytes) that are triggered by a periodic sequence of electrical waves (action
potentials) originating in the sino-atrial node and propagating over the atria and
the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF)
or ventricular tachycardia (VT) are caused by disruptions and instabilities of these
electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent
wave patterns (AF,VF). Numerous simulation and experimental studies during the
last 20 years have addressed these topics. In this review we focus on the nonlinear
dynamics of wave propagation in the heart with an emphasis on the theory of pulses,
spirals and scroll waves and their instabilities in excitable media and their application
to cardiac modeling. After an introduction into electrophysiological models for action
potential propagation, the modeling and analysis of spatiotemporal alternans, spiral
and scroll meandering, spiral breakup and scroll wave instabilities like negative line
tension and sproing are reviewed in depth and discussed with emphasis on their impact
in cardiac arrhythmias.Peer ReviewedPreprin
An iterative action minimizing method for computing optimal paths in stochastic dynamical systems
We present a numerical method for computing optimal transition pathways and
transition rates in systems of stochastic differential equations (SDEs). In
particular, we compute the most probable transition path of stochastic
equations by minimizing the effective action in a corresponding deterministic
Hamiltonian system. The numerical method presented here involves using an
iterative scheme for solving a two-point boundary value problem for the
Hamiltonian system. We validate our method by applying it to both continuous
stochastic systems, such as nonlinear oscillators governed by the Duffing
equation, and finite discrete systems, such as epidemic problems, which are
governed by a set of master equations. Furthermore, we demonstrate that this
method is capable of dealing with stochastic systems of delay differential
equations.Comment: 20 pages, 8 figure
Efficiency characterization of a large neuronal network: a causal information approach
When inhibitory neurons constitute about 40% of neurons they could have an
important antinociceptive role, as they would easily regulate the level of
activity of other neurons. We consider a simple network of cortical spiking
neurons with axonal conduction delays and spike timing dependent plasticity,
representative of a cortical column or hypercolumn with large proportion of
inhibitory neurons. Each neuron fires following a Hodgkin-Huxley like dynamics
and it is interconnected randomly to other neurons. The network dynamics is
investigated estimating Bandt and Pompe probability distribution function
associated to the interspike intervals and taking different degrees of
inter-connectivity across neurons. More specifically we take into account the
fine temporal ``structures'' of the complex neuronal signals not just by using
the probability distributions associated to the inter spike intervals, but
instead considering much more subtle measures accounting for their causal
information: the Shannon permutation entropy, Fisher permutation information
and permutation statistical complexity. This allows us to investigate how the
information of the system might saturate to a finite value as the degree of
inter-connectivity across neurons grows, inferring the emergent dynamical
properties of the system.Comment: 26 pages, 3 Figures; Physica A, in pres
Finite-time Partitions for Lagrangian Structure Identification in Gulf Stream Eddy Transport
We develop a methodology to identify finite-time Lagrangian structures from
data and models using an extension of the Koopman operator-theoretic methods
developed for velocity fields with simple (periodic, quasi-periodic)
time-dependence. To achieve this, the notion of the Finite Time Ergodic (FiTER)
partition is developed and rigorously justified. In combination with a
clustering-based approach, the methodology enables identification of the
temporal evolution of Lagrangian structures in a classic, benchmark,
oceanographic transport problem, namely the cross-stream flux induced by the
interaction of a meso- scale Gulf Stream Ring eddy with the main jet. We focus
on a single mixing event driven by the interaction between an energetic cold
core ring (a cyclone), the strong jet, and a number of smaller scale cyclones
and anticyclones. The new methodology enab les reconstruction of Lagrangian
structures in three dimensions and analysis of their time-evolution
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