19,803 research outputs found
The influence of noise on scalings for in-out intermittency
We study the effects of noise on a recently discovered form of intermittency,
referred to as in-out intermittency. This type of intermittency, which reduces
to on-off in systems with a skew product structure, has been found in the
dynamics of maps, ODE and PDE simulations that have symmetries. It shows itself
in the form of trajectories that spend a long time near a symmetric state
interspersed with short bursts away from symmetry. In contrast to on-off
intermittency, there are clearly distinct mechanisms of approach towards and
away from the symmetric state, and this needs to be taken into account in order
to properly model the long time statistics. We do this by using a
diffusion-type equation with delay integral boundary condition. This model is
validated by considering the statistics of a two-dimensional map with and
without the addition of noise.Comment: Submitted to Physical Review E, also available at
http://www.eurico.web.co
Low frequency noise controls on-off intermittency of bifurcating systems
A bifurcating system subject to multiplicative noise can display on-off
intermittency. Using a canonical example, we investigate the extreme
sensitivity of the intermittent behavior to the nature of the noise. Through a
perturbative expansion and numerical studies of the probability density
function of the unstable mode, we show that intermittency is controlled by the
ratio between the departure from onset and the value of the noise spectrum at
zero frequency. Reducing the noise spectrum at zero frequency shrinks the
intermittency regime drastically. This effect also modifies the distribution of
the duration that the system spends in the off phase. Mechanisms and
applications to more complex bifurcating systems are discussed
On-off intermittency over an extended range of control parameter
We propose a simple phenomenological model exhibiting on-off intermittency
over an extended range of control parameter. We find that the distribution of
the 'off' periods has as a power-law tail with an exponent varying continuously
between -1 and -2, at odds with standard on-off intermittency which occurs at a
specific value of the control parameter, and leads to the exponent -3/2. This
non-trivial behavior results from the competition between a strong slowing down
of the dynamics at small values of the observable, and a systematic drift
toward large values.Comment: 4 pages, 3 figure
Transverse instability for non-normal parameters
We consider the behaviour of attractors near invariant subspaces on varying a
parameter that does not preserve the dynamics in the invariant subspace but is
otherwise generic, in a smooth dynamical system. We refer to such a parameter
as ``non-normal''. If there is chaos in the invariant subspace that is not
structurally stable, this has the effect of ``blurring out'' blowout
bifurcations over a range of parameter values that we show can have positive
measure in parameter space.
Associated with such blowout bifurcations are bifurcations to attractors
displaying a new type of intermittency that is phenomenologically similar to
on-off intermittency, but where the intersection of the attractor by the
invariant subspace is larger than a minimal attractor. The presence of distinct
repelling and attracting invariant sets leads us to refer to this as ``in-out''
intermittency. Such behaviour cannot appear in systems where the transverse
dynamics is a skew product over the system on the invariant subspace.
We characterise in-out intermittency in terms of its structure in phase space
and in terms of invariants of the dynamics obtained from a Markov model of the
attractor. This model predicts a scaling of the length of laminar phases that
is similar to that for on-off intermittency but which has some differences.Comment: 15 figures, submitted to Nonlinearity, the full paper available at
http://www.maths.qmw.ac.uk/~eo
Effects of the low frequencies of noise on On-Off intermittency
A bifurcating system subject to multiplicative noise can exhibit on-off
intermittency close to the instability threshold. For a canonical system, we
discuss the dependence of this intermittency on the Power Spectrum Density
(PSD) of the noise. Our study is based on the calculation of the Probability
Density Function (PDF) of the unstable variable. We derive analytical results
for some particular types of noises and interpret them in the framework of
on-off intermittency. Besides, we perform a cumulant expansion for a random
noise with arbitrary power spectrum density and show that the intermittent
regime is controlled by the ratio between the departure from the threshold and
the value of the PSD of the noise at zero frequency. Our results are in
agreement with numerical simulations performed with two types of random
perturbations: colored Gaussian noise and deterministic fluctuations of a
chaotic variable. Extensions of this study to another, more complex, system are
presented and the underlying mechanisms are discussed.Comment: 13pages, 13 figure
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