69 research outputs found
Potentials and Limits to Basin Stability Estimation
Acknowledgments The authors gratefully acknowledge the support of BMBF, CoNDyNet, FK. 03SF0472A.Peer reviewedPublisher PD
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
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe
Strange attractors for the generalized Lozi-like family
We generalize the Lozi-like family introduced in Misiurewicz and \v{S}timac
work from 2017. The generalized Lozi-like family encompasses in particular
certain Lozi-like maps, orientation preserving or reversing Lozi maps or large
parameter regions of 2-dimensional border collision normal forms. We prove that
it possesses a strange attractor, arising as a homoclinic class
Can we trust in numerical computations of chaotic solutions of dynamical systems ?
From Laser Dynamics to Topology of Chaos, (celebrating the 70th birthday of Prof Robert Gilmore, Rouen, June 28-30, 2011), Ed. Ch. Letellier.Since the famous paper of E. Lorenz in 1963 numerical computations using computers play a central role in order to display and analyze solutions of nonlinear dynamical systems. By these means new structures have been emphasized like hyperbolic and/or strange attractors. However theoretical proofs of their existence are very di¢ cult and limited to very special linear cases. Computer aided proofs are also complex and require special interval arithmetic analysis. Nevertheless, numerous researchers in several fields linked to chaotic dynamical systems are confident in the numerical solutions they found using popular software and publish without checking carefully the reliability of their results. In the simple case of discrete dynamical systems (e.g. Hénon map) there are concerns about the nature of what a computer find out : long unstable pseudo-orbits or strange attractors? The shadowing property and its generalizations which ensure that pseudo-orbits of a homeomorphism can be traceable by actual orbits even if rounding errors are not inevitable are not of great help in order to validate the numerical results. Continuous dynamical systems (e.g. Chua, Lorenz, Rössler) are even more difficult to handle in this scope and researchers have to be very cautious to back up theory with numerical computations. We present a survey of the topic based on these, only few, but well studied models
Discrete-Time Chaotic-Map Truly Random Number Generators: Design, Implementation, and Variability Analysis of the Zigzag Map
In this paper, we introduce a novel discrete chaotic map named zigzag map
that demonstrates excellent chaotic behaviors and can be utilized in Truly
Random Number Generators (TRNGs). We comprehensively investigate the map and
explore its critical chaotic characteristics and parameters. We further present
two circuit implementations for the zigzag map based on the switched current
technique as well as the current-mode affine interpolation of the breakpoints.
In practice, implementation variations can deteriorate the quality of the
output sequence as a result of variation of the chaotic map parameters. In
order to quantify the impact of variations on the map performance, we model the
variations using a combination of theoretical analysis and Monte-Carlo
simulations on the circuits. We demonstrate that even in the presence of the
map variations, a TRNG based on the zigzag map passes all of the NIST 800-22
statistical randomness tests using simple post processing of the output data.Comment: To appear in Analog Integrated Circuits and Signal Processing (ALOG
Analysis and synthesis of self-synchronizing chaotic systems
Includes bibliographical references (p. 225-228).Supported by the U.S. Air Force Office of Scientific Research. AFOSR-91-0034-C Supported by the U.S. Navy Office of Naval Research. N00014-91-C-0125 N00014-93-1-0686 Supported by Lockheed Sanders, Inc.Kevin M. Cuomo
Dynamics meets Morphology: towards Dymorph Computation
In this dissertation, approaches are presented for both technically using and investigating biological principles with oscillators in the context of electrical engineering, in particular neuromorphic engineering. Thereby, dynamics as well as morphology as important neuronal principles were explicitly selected, which shape the information processing in the human brain and distinguish it from other technical systems. The aspects and principles selected here are adaptation during the encoding of stimuli, the comparatively low signal transmission speed, the continuous formation and elimination of connections, and highly complex, partly chaotic, dynamics. The selection of these phenomena and properties has led to the development of a sensory unit that is capable of encoding mechanical stress into a series of voltage pulses by the use of a MOSFET augmented by AlScN. The circuit is based on a leaky integrate and fire neuron model and features an adaptation of the pulse frequency. Furthermore, the slow signal transmission speed of biological systems was the motivation for the investigation of a temporal delay in the feedback of the output pulses of a relaxation oscillator. In this system stable pulse patterns which form due to so-called jittering bifurcations could be observed. In particular, switching between different stable pulse patterns was possible to induce. In the further course of the work, the first steps towards time-varying coupling of dynamic systems are investigated. It was shown that in a system consisting of dimethyl sulfoxid and zinc acetate, oscillators can be used to force the formation of filaments. The resulting filaments then lead to a change in the dynamics of the oscillators. Finally, it is shown that in a system with chaotic dynamics, the extension of it with a memristive device can lead to a transient stabilisation of the dynamics, a behaviour that can be identified as a repeated pass of Hopf bifurcations
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