12,036 research outputs found
Coherent transport structures in magnetized plasmas II: Numerical results
In a pair of linked articles (called Article I and II respectively) we apply
the concept of Lagrangian Coherent Structures borrowed from the study of
Dynamical Systems to magnetic field configurations in order to separate regions
where field lines have different kind of behavior. In the present article,
article II, by means of a numerical procedure we investigate the Lagrangian
Coherent Structures in the case of a two-dimensional magnetic configuration
with two island chains that are generated by magnetic reconnection and evolve
nonlinearly in time. The comparison with previous results, obtained by assuming
a fixed magnetic field configuration, allows us to explore the dependence of
transport barriers on the particle velocity
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
Hyperbolic Covariant Coherent Structures in two dimensional flows
A new method to describe hyperbolic patterns in two dimensional flows is
proposed. The method is based on the Covariant Lyapunov Vectors (CLVs), which
have the properties to be covariant with the dynamics, and thus being mapped by
the tangent linear operator into another CLVs basis, they are norm independent,
invariant under time reversal and can be not orthonormal. CLVs can thus give a
more detailed information on the expansion and contraction directions of the
flow than the Lyapunov Vector bases, that are instead always orthogonal. We
suggest a definition of Hyperbolic Covariant Coherent Structures (HCCSs), that
can be defined on the scalar field representing the angle between the CLVs.
HCCSs can be defined for every time instant and could be useful to understand
the long term behaviour of particle tracers.
We consider three examples: a simple autonomous Hamiltonian system, as well
as the non-autonomous "double gyre" and Bickley jet, to see how well the angle
is able to describe particular patterns and barriers. We compare the results
from the HCCSs with other coherent patterns defined on finite time by the
Finite Time Lyapunov Exponents (FTLEs), to see how the behaviour of these
structures change asymptotically
Sensitivity to the initial state of interacting ultracold bosons in disordered lattices
We study the dynamics of a nonlinear one-dimensional disordered system
obtained by coupling the Anderson model with the Gross-Pitaevskii equation. An
analytical model provides us with a single quantity globally characterizing the
localization of the system. This quantity obeys a scaling law with respect to
the width of the initial state, which can be used to characterize the dynamics
independently of the initial state.Comment: 10 pages, 12 figures, revtex4, submited to PR
Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection
We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this ‘cycling chaos’ manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets. Such behaviour is robust to perturbations that preserve the symmetry of the system; we examine bifurcations of this state.
We discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace. This differs from the standard scenario for the blowout bifurcation in that in our case, the blowout is neither subcritical nor supercritical. The robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors.
The model we consider is a ninth-order truncated ordinary differential equation (ODE) model of three-dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient. Symmetries of the model lead to the existence of invariant subspaces for the dynamics; in particular there are invariant subspaces that correspond to regimes of two-dimensional flows, with variation in the vertical but only one of the two horizontal directions. Stable two-dimensional chaotic flow can go unstable to three-dimensional flow via the cross-roll instability. We show how the bifurcations mentioned above can be located by examination of various transverse Liapunov exponents. We also consider a reduction of the ODE to a map and demonstrate that the same behaviour can be found in the corresponding map. This allows us to describe and predict a number of observed transitions in these models. The dynamics we describe is new but nonetheless robust, and so should occur in other applications
Horizontal Visibility graphs generated by type-I intermittency
The type-I intermittency route to (or out of) chaos is investigated within
the Horizontal Visibility graph theory. For that purpose, we address the
trajectories generated by unimodal maps close to an inverse tangent bifurcation
and construct, according to the Horizontal Visibility algorithm, their
associated graphs. We show how the alternation of laminar episodes and chaotic
bursts has a fingerprint in the resulting graph structure. Accordingly, we
derive a phenomenological theory that predicts quantitative values of several
network parameters. In particular, we predict that the characteristic power law
scaling of the mean length of laminar trend sizes is fully inherited in the
variance of the graph degree distribution, in good agreement with the numerics.
We also report numerical evidence on how the characteristic power-law scaling
of the Lyapunov exponent as a function of the distance to the tangent
bifurcation is inherited in the graph by an analogous scaling of the block
entropy over the degree distribution. Furthermore, we are able to recast the
full set of HV graphs generated by intermittent dynamics into a renormalization
group framework, where the fixed points of its graph-theoretical RG flow
account for the different types of dynamics. We also establish that the
nontrivial fixed point of this flow coincides with the tangency condition and
that the corresponding invariant graph exhibit extremal entropic properties.Comment: 8 figure
Non-perturbative features of driven scattering systems
We investigate the scattering properties of one-dimensional, periodically and
non-periodically forced oscillators. The pattern of singularities of the
scattering function, in the periodic case, shows a characteristic hierarchical
structure where the number Nc of zeros of the solutions plays the role of an
order parameter marking the level of the observed self-similar structure. The
behavior is understood both in terms of the return map and of the intersections
pattern of the invariant manifolds of the outermost fixed points. In the
non-periodic case the scattering function does not provide a complete
development of the hierarchical structure. The singularities pattern of the
outgoing energy as a function of the driver amplitude is connected to the
arrangement of gaps in the fundamental regions. The survival probability
distribution of temporarily bound orbits is shown to decay asymptotically as a
power law. The "stickiness" of regular regions of phase space, given by KAM
surfaces and remnant of KAM curves, is responsible for this observation
Dynamics of Unperturbed and Noisy Generalized Boolean Networks
For years, we have been building models of gene regulatory networks, where
recent advances in molecular biology shed some light on new structural and
dynamical properties of such highly complex systems. In this work, we propose a
novel timing of updates in Random and Scale-Free Boolean Networks, inspired by
recent findings in molecular biology. This update sequence is neither fully
synchronous nor asynchronous, but rather takes into account the sequence in
which genes affect each other. We have used both Kauffman's original model and
Aldana's extension, which takes into account the structural properties about
known parts of actual GRNs, where the degree distribution is right-skewed and
long-tailed. The computer simulations of the dynamics of the new model compare
favorably to the original ones and show biologically plausible results both in
terms of attractors number and length. We have complemented this study with a
complete analysis of our systems' stability under transient perturbations,
which is one of biological networks defining attribute. Results are
encouraging, as our model shows comparable and usually even better behavior
than preceding ones without loosing Boolean networks attractive simplicity.Comment: 29 pages, publishe
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