32,207 research outputs found
Recurrence Plots 25 years later -- gaining confidence in dynamical transitions
Recurrence plot based time series analysis is widely used to study changes
and transitions in the dynamics of a system or temporal deviations from its
overall dynamical regime. However, most studies do not discuss the significance
of the detected variations in the recurrence quantification measures. In this
letter we propose a novel method to add a confidence measure to the recurrence
quantification analysis. We show how this approach can be used to study
significant changes in dynamical systems due to a change in control parameters,
chaos-order as well as chaos-chaos transitions. Finally we study and discuss
climate transitions by analysing a marine proxy record for past sea surface
temperature. This paper is dedicated to the 25th anniversary of the
introduction of recurrence plots
Recurrence-based time series analysis by means of complex network methods
Complex networks are an important paradigm of modern complex systems sciences
which allows quantitatively assessing the structural properties of systems
composed of different interacting entities. During the last years, intensive
efforts have been spent on applying network-based concepts also for the
analysis of dynamically relevant higher-order statistical properties of time
series. Notably, many corresponding approaches are closely related with the
concept of recurrence in phase space. In this paper, we review recent
methodological advances in time series analysis based on complex networks, with
a special emphasis on methods founded on recurrence plots. The potentials and
limitations of the individual methods are discussed and illustrated for
paradigmatic examples of dynamical systems as well as for real-world time
series. Complex network measures are shown to provide information about
structural features of dynamical systems that are complementary to those
characterized by other methods of time series analysis and, hence,
substantially enrich the knowledge gathered from other existing (linear as well
as nonlinear) approaches.Comment: To be published in International Journal of Bifurcation and Chaos
(2011
Systematic Derivation of Amplitude Equations and Normal Forms for Dynamical Systems
We present a systematic approach to deriving normal forms and related
amplitude equations for flows and discrete dynamics on the center manifold of a
dynamical system at local bifurcations and unfoldings of these. We derive a
general, explicit recurrence relation that completely determines the amplitude
equation and the associated transformation from amplitudes to physical space.
At any order, the relation provides explicit expressions for all the
nonvanishing coefficients of the amplitude equation together with
straightforward linear equations for the coefficients of the transformation.
The recurrence relation therefore provides all the machinery needed to solve a
given physical problem in physical terms through an amplitude equation. The new
result applies to any local bifurcation of a flow or map for which all the
critical eigenvalues are semisimple i.e. have Riesz index unity). The method is
an efficient and rigorous alternative to more intuitive approaches in terms of
multiple time scales. We illustrate the use of the method by deriving amplitude
equations and associated transformations for the most common simple
bifurcations in flows and iterated maps. The results are expressed in tables in
a form that can be immediately applied to specific problems.Comment: 40 pages, 1 figure, 4 tables. Submitted to Chaos. Please address any
correspondence by email to [email protected]
Regular and Chaotic Motion in General Relativity: The Case of a Massive Magnetic Dipole
Circular motion of particles, dust grains and fluids in the vicinity of
compact objects has been investigated as a model for accretion of gaseous and
dusty environment. Here we further discuss, within the framework of general
relativity, figures of equilibrium of matter under the influence of combined
gravitational and large-scale magnetic fields, assuming that the accreted
material acquires a small electric charge due to interplay of plasma processes
and photoionization. In particular, we employ an exact solution describing the
massive magnetic dipole and we identify the regions of stable motion. We also
investigate situations when the particle dynamics exhibits the onset of chaos.
In order to characterize the measure of chaoticness we employ techniques of
Poincar\'e surfaces of section and of recurrence plots.Comment: 11 pages, 6 figures, published in the proceedings of the conference
"Relativity and Gravitation: 100 Years after Einstein in Prague" (25. - 29.
6. 2012, Prague
Geometric and dynamic perspectives on phase-coherent and noncoherent chaos
Statistically distinguishing between phase-coherent and noncoherent chaotic
dynamics from time series is a contemporary problem in nonlinear sciences. In
this work, we propose different measures based on recurrence properties of
recorded trajectories, which characterize the underlying systems from both
geometric and dynamic viewpoints. The potentials of the individual measures for
discriminating phase-coherent and noncoherent chaotic oscillations are
discussed. A detailed numerical analysis is performed for the chaotic R\"ossler
system, which displays both types of chaos as one control parameter is varied,
and the Mackey-Glass system as an example of a time-delay system with
noncoherent chaos. Our results demonstrate that especially geometric measures
from recurrence network analysis are well suited for tracing transitions
between spiral- and screw-type chaos, a common route from phase-coherent to
noncoherent chaos also found in other nonlinear oscillators. A detailed
explanation of the observed behavior in terms of attractor geometry is given.Comment: 12 pages, 13 figure
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