221 research outputs found
Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory
We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motivated by a low-order model for magnetic activity in a stellar dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations, where the saddle-node bifurcation is assumed to have global reinjection of trajectories. The model can produce chaotic behaviour within each of a pair of invariant subspaces, and also it can show attractors that are stuck-on to both of the invariant subspaces. We investigate the detailed intermittent dynamics for such an attractor, investigating the effect of breaking the symmetry between the two Hopf bifurcations, and observing that it can appear via blowout bifurcations from the invariant subspaces.
We give a simple Markov chain model for the two-state intermittent dynamics that reproduces the time spent close to the invariant subspaces and the switching between the different possible invariant subspaces; this clarifies the observation that the proportion of time spent near the different subspaces depends on the average residence time and also on the probabilities of switching between the possible subspaces
Compressible magnetoconvection in three dimensions: pattern formation in a strongly stratified layer
The interaction between magnetic fields and convection is interesting both because of its astrophysical importance and because the nonlinear Lorentz force leads to an especially rich variety of behaviour. We present several sets of computational results for magnetoconvection in a square box, with periodic lateral boundary conditions, that show transitions from steady convection with an ordered planform through a regime with intermittent bursts to complicated spatiotemporal behaviour. The constraints imposed by the square lattice are relaxed as the aspect ratio is increased. In wide boxes we find a new regime, in which regions with strong fields are separated from regions with vigorous convection. We show also how considerations of symmetry and associated group theory can be used to explain the nature of these transitions and the sequence in which the relevant bifurcations occur
Bistability in the Complex Ginzburg-Landau Equation with Drift
Properties of the complex Ginzburg-Landau equation with drift are studied focusing on the Benjamin-Feir stable regime. On a finite interval with Neumann boundary conditions the equation exhibits bistability between a spatially uniform time-periodic state and a variety of nonuniform states with complex time dependence. The origin of this behavior is identified and contrasted with the bistable behavior present with periodic boundary conditions and no drift
Two-state intermittency near a symmetric interaction of saddle-node and Hopf bifurcations: a case study from dynamo theory
Copyright © 2004 Elsevier. NOTICE: This is the author’s version of a work accepted for publication by Elsevier. Changes resulting from the publishing process, including peer review, editing, corrections, structural formatting and other quality control mechanisms, may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Physica D, Vol 194, Issues 1-2, 2004, DOI:10.1016/j.physd.2004.02.002We consider a model of a Hopf bifurcation interacting as a codimension 2 bifurcation with a saddle-node on a limit cycle, motivated by a low-order model for magnetic activity in a stellar
dynamo. This model consists of coupled interactions between a saddle-node and two Hopf bifurcations,
where the saddle-node bifurcation is assumed to have global reinjection of trajectories.
The model can produce chaotic behaviour within each of a pair of invariant subspaces, and also
it can show attractors that are stuck-on to both of the invariant subspaces. We investigate the
detailed intermittent dynamics for such an attractor, investigating the effect of breaking the
symmetry between the two Hopf bifurcations, and observing that it can appear via blowout
bifurcations from the invariant subspaces.
We give a simple Markov chain model for the two-state intermittent dynamics that reproduces
the time spent close to the invariant subspaces and the switching between the different
possible invariant subspaces; this clarifes the observation that the proportion of time spent near
the different subspaces depends on the average residence time and also on the probabilities of
switching between the possible subspaces
Driven-Dissipative Dynamics of Atomic Ensembles in a Resonant Cavity II: Quasiperiodic Route to Chaos and Chaotic Synchronization
We analyze the origin and properties of the chaotic dynamics of two atomic
ensembles in a driven-dissipative experimental setup, where they are
collectively damped by a bad cavity mode and incoherently pumped by a Raman
laser. Starting from the mean-field equations, we explain the emergence of
chaos by way of quasiperiodicity -- presence of two or more incommensurate
frequencies. This is known as the Ruelle-Takens-Newhouse route to chaos. The
equations of motion have a -symmetry with respect to the
interchange of the two ensembles. However, some of the attractors of these
equations spontaneously break this symmetry. To understand the emergence and
subsequent properties of various attractors, we concurrently study the
mean-field trajectories, Poincar\'{e} sections, maximum and conditional
Lyapunov exponents, and power spectra. Using Floquet analysis, we show that
quasiperiodicity is born out of non -symmetric oscillations via
a supercritical Neimark-Sacker bifurcation. Changing the detuning between the
level spacings in the two ensembles and the repump rate results in the
synchronization of the two chaotic ensembles. In this regime, the chaotic
intensity fluctuations of the light radiated by the two ensembles are
identical. Identifying the synchronization manifold, we understand the origin
of synchronized chaos as a tangent bifurcation intermittency of the
-symmetric oscillations. At its birth, synchronized chaos is
unstable. The interaction of this attractor with other attractors causes on-off
intermittency until the synchronization manifold becomes sufficiently
attractive. We also show coexistence of different phases in small pockets near
the boundaries.Comment: 52 pages, 36 figures, published versio
Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I: General presentation and periodic solutions
We present experimental results on hydrothermal traveling-waves dynamics in
long and narrow 1D channels. The onset of primary traveling-wave patterns is
briefly presented for different fluid heights and for annular or bounded
channels, i.e., within periodic or non-periodic boundary conditions. For
periodic boundary conditions, by increasing the control parameter or changing
the discrete mean-wavenumber of the waves, we produce modulated waves patterns.
These patterns range from stable periodic phase-solutions, due to supercritical
Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes
and/or counter-propagating-waves competition, i.e., traveling sources and
sinks. The transition from non-linearly saturated Eckhaus modulations to
transient pattern-breaks by traveling holes and spatio-temporal defects is
documented. Our observations are presented in the framework of coupled complex
Ginzburg-Landau equations with additional fourth and fifth order terms which
account for the reflection symmetry breaking at high wave-amplitude far from
onset. The second part of this paper (nlin.PS/0208030) extends this study to
spatially non-periodic patterns observed in both annular and bounded channel.Comment: 45 pages, 21 figures (elsart.cls + AMS extensions). Accepted in
Physica D. See also companion paper "Nonlinear dynamics of waves and
modulated waves in 1D thermocapillary flows. II: Convective/absolute
transitions" (nlin.PS/0208030). A version with high resolution figures is
available on N.G. web pag
Effect of Noise on Excursions To and Back From Infinity
The effect of additive white noise on a model for bursting behavior in large
aspect-ratio binary fluid convection is considered. Such bursts are present in
systems with nearly square symmetry and are the result of heteroclinic cycles
involving infinite amplitude states created when the square symmetry is broken.
A combination of numerical results and analytical arguments show how even a
very small amount of noise can have a very large effect on the amplitudes of
successive bursts. Large enough noise can also affect the physical
manifestations of the bursts. Finally, it is shown that related bursts may
occur when white noise is added to the normal form equations for the Hopf
bifurcation with exact square symmetry.Comment: 17 pages, 9 figure
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
Whirling Hexagons and Defect Chaos in Hexagonal Non-Boussinesq Convection
We study hexagon patterns in non-Boussinesq convection of a thin rotating
layer of water. For realistic parameters and boundary conditions we identify
various linear instabilities of the pattern. We focus on the dynamics arising
from an oscillatory side-band instability that leads to a spatially disordered
chaotic state characterized by oscillating (whirling) hexagons. Using
triangulation we obtain the distribution functions for the number of pentagonal
and heptagonal convection cells. In contrast to the results found for defect
chaos in the complex Ginzburg-Landau equation and in inclined-layer convection,
the distribution functions can show deviations from a squared Poisson
distribution that suggest non-trivial correlations between the defects.Comment: 4 mpg-movies are available at
http://www.esam.northwestern.edu/~riecke/lit/lit.html submitted to New J.
Physic
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