1,686 research outputs found
Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion
In this tutorial, we discuss self-excited and hidden attractors for systems
of differential equations. We considered the example of a Lorenz-like system
derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to
demonstrate the analysis of self-excited and hidden attractors and their
characteristics. We applied the fishing principle to demonstrate the existence
of a homoclinic orbit, proved the dissipativity and completeness of the system,
and found absorbing and positively invariant sets. We have shown that this
system has a self-excited attractor and a hidden attractor for certain
parameters. The upper estimates of the Lyapunov dimension of self-excited and
hidden attractors were obtained analytically.Comment: submitted to EP
Hidden attractors in fundamental problems and engineering models
Recently a concept of self-excited and hidden attractors was suggested: an
attractor is called a self-excited attractor if its basin of attraction
overlaps with neighborhood of an equilibrium, otherwise it is called a hidden
attractor. For example, hidden attractors are attractors in systems with no
equilibria or with only one stable equilibrium (a special case of
multistability and coexistence of attractors). While coexisting self-excited
attractors can be found using the standard computational procedure, there is no
standard way of predicting the existence or coexistence of hidden attractors in
a system. In this plenary survey lecture the concept of self-excited and hidden
attractors is discussed, and various corresponding examples of self-excited and
hidden attractors are considered
Hysteresis in Adiabatic Dynamical Systems: an Introduction
We give a nontechnical description of the behaviour of dynamical systems
governed by two distinct time scales. We discuss in particular memory effects,
such as bifurcation delay and hysteresis, and comment the scaling behaviour of
hysteresis cycles. These properties are illustrated on a few simple examples.Comment: 28 pages, 10 ps figures, AMS-LaTeX. This is the introduction of my
Ph.D. dissertation, available at
http://dpwww.epfl.ch/instituts/ipt/berglund/these.htm
Predictability of catastrophic events: material rupture, earthquakes, turbulence, financial crashes and human birth
We propose that catastrophic events are "outliers" with statistically
different properties than the rest of the population and result from mechanisms
involving amplifying critical cascades. Applications and the potential for
prediction are discussed in relation to the rupture of composite materials,
great earthquakes, turbulence and abrupt changes of weather regimes, financial
crashes and human parturition (birth).Comment: Latex document of 22 pages including 6 ps figures, in press in PNA
Characteristics of in-out intermittency in delay-coupled FitzHugh-Nagumo oscillators
We analyze a pair of delay-coupled FitzHugh-Nagumo oscillators exhibiting
in-out intermittency as a part of the generating mechanism of extreme events.
We study in detail the characteristics of in-out intermittency and identify the
invariant subsets involved --- a saddle fixed point and a saddle periodic orbit
--- neither of which are chaotic as in the previously reported cases of in-out
intermittency. Based on the analysis of a periodic attractor possessing in-out
dynamics, we can characterize the approach to the invariant synchronization
manifold and the spiralling out to the saddle periodic orbit with subsequent
ejection from the manifold. Due to the striking similarities, this analysis of
in-out dynamics explains also in-out intermittency.Comment: 15 pages, 6 figure
Contributions of plasma physics to chaos and nonlinear dynamics
This topical review focusses on the contributions of plasma physics to chaos
and nonlinear dynamics bringing new methods which are or can be used in other
scientific domains. It starts with the development of the theory of Hamiltonian
chaos, and then deals with order or quasi order, for instance adiabatic and
soliton theories. It ends with a shorter account of dissipative and high
dimensional Hamiltonian dynamics, and of quantum chaos. Most of these
contributions are a spin-off of the research on thermonuclear fusion by
magnetic confinement, which started in the fifties. Their presentation is both
exhaustive and compact. [15 April 2016
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