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14
2001] “Construction of classes of circuitindependent chaotic oscillatorsusing passiveonly nonlinear devices
 IEEE Trans. Circuits Syst.I
"... Abstract—Two generic classes of chaotic oscillators comprising four different configurations are constructed. The proposed structures are based on the simplest possible abstract models of generic secondorder RC sinusoidal oscillators that satisfy the basic condition for oscillation and the frequenc ..."
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Cited by 7 (0 self)
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Abstract—Two generic classes of chaotic oscillators comprising four different configurations are constructed. The proposed structures are based on the simplest possible abstract models of generic secondorder RC sinusoidal oscillators that satisfy the basic condition for oscillation and the frequency of oscillation formulas. By linking these sinusoidal oscillator engines to simple passive firstorder or secondorder nonlinear composites, chaos is generated and the evolution of the twodimensional sinusoidal oscillator dynamics into a higher dimensional state space is clearly recognized. We further discuss three architectures into which autonomous chaotic oscillators can be decomposed. Based on one of these architectures we classify a large number of the available chaotic oscillators and propose a novel reconstruction of the classical Chua’s circuit. The wellknown Lorenz system of equations is also studied and a simplified model with equivalent dynamics, but containing no multipliers, is introduced.
From template analysis to generating partitions I: Periodic orbits, knots and symbolic encodings
 Physica D
"... We present a detailed algorithm to construct symbolic encodings for chaotic attractors of threedimensional flows. It is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by Lefranc et al. [Phys. Rev. Lett. 73 (1994) 1364]. For ea ..."
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Cited by 3 (0 self)
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We present a detailed algorithm to construct symbolic encodings for chaotic attractors of threedimensional flows. It is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by Lefranc et al. [Phys. Rev. Lett. 73 (1994) 1364]. For each orbit, the symbolic names that are consistent with its knottheoretic invariants and with the topological structure of the attractor are first obtained using template analysis. This information and the locations of the periodic orbits in the section plane are then used to construct a generating partition by means of triangulations. We provide numerical evidence of the validity of this method by applying it successfully to sets of more than 1500 periodic orbits extracted from numerical simulations, and obtain partitions whose border is localized with a precision of 0.01%. A distinctive advantage of this approach is that the solution is progressively refined using higherperiod orbits, which makes it robust to noise, and suitable for analyzing experimental time series. Furthermore, the resulting encodings are by construction consistent in the corresponding limits with those rigorously known for both onedimensional and hyperbolic maps. © 2000 Elsevier
IDENTIFICATION OF COMPLEX PROCESSES BASED ON ANALYSIS OF PHASE SPACE STRUCTURES
"... Abstract. The problem of investigation of temporal and/or spatial behavior of highly nonlinear or complex natural systems has long been of fundamental scientific interest. At the same time it is presently well understood that identification of dynamics of processes in complex natural systems, throug ..."
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Abstract. The problem of investigation of temporal and/or spatial behavior of highly nonlinear or complex natural systems has long been of fundamental scientific interest. At the same time it is presently well understood that identification of dynamics of processes in complex natural systems, through their qualitative description and quantitative evaluation, is far from a purely academic question and has an essential practical importance. This is quite understandable as systems with complex dynamics abound in nature and examples can be found in very different areas such as medicine and biology (rhythms, physiological cycles, epidemics), atmosphere (climate and weather change), geophysics (tides, earthquakes, volcanoes, magnetic field variations), economy (financial markets behavior, exchange rates), engineering (friction, fracturing), communication (electronic networks, internet packet dynamics) etc. The past two decades of research on qualitative and especially quantitative investigations of dynamics of real processes of different origin brought significant progress in the understanding of behavior of natural processes. At the same time serious drawbacks have also been revealed. This is why exhaustive investigation of
Some considerations on Poincaré maps for chaotic flows
"... . The standard way to reduce a flow to a timediscrete dynamical system is by the technique of the Poincare surface of section. We discuss the relationship between Poincare maps obtained from di#erent surfaces of section and give some considerations for practical application for this method. PACS ..."
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. The standard way to reduce a flow to a timediscrete dynamical system is by the technique of the Poincare surface of section. We discuss the relationship between Poincare maps obtained from di#erent surfaces of section and give some considerations for practical application for this method. PACS numbers: 05.45.Ac, 02.30.Hq Short title: LETTER TO THE EDITOR December 10, 1999 + Present address: MaxPlanckInstitut fur Stromungsforschung, Bunsenstrae 10, D37073 Gottingen, Germany # email: wolfram@chaos.gwdg.de 2 1. Introduction One century ago Poincare introduced the concept of timediscrete dynamical systems in his study of twodimensional autonomous di#erential equations [1]. Such construction of Poincare maps has become one of the very basic tools in nonlinear dynamics and is nowadays contained in every textbook on nonlinear dynamics (c.f. e.g. [2]). The method is to a good deal at the heart of e.g. data analysis [3], control in nonlinear systems [4], and numerical ...
Abstract
, 2003
"... Transition from chaotic to quasiperiodic phase in modified Lorenz model is analyzed by performing the contact transformation such that the trajectory in R 3 is projected on R 2. The relative torsion number and the characteristics of the template are measured using the eigenvector of the Jacobian in ..."
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Transition from chaotic to quasiperiodic phase in modified Lorenz model is analyzed by performing the contact transformation such that the trajectory in R 3 is projected on R 2. The relative torsion number and the characteristics of the template are measured using the eigenvector of the Jacobian instead of vectors on moving frame along the closed trajectory. Application to the circulation of a fluid in a convection loop and oscillation of the electric field in singlemode laser system are performed. The time series of the eigenvalues of the Jacobian and the scatter plot of the trajectory in the transformed coordinate plane X − Z in the former and X  − Z  in the latter, allow to visualize characteristic pattern change at the transition from quasiperiodic to chaotic. In the case of single mode laser, we observe the correlation between the critical movement of the eigenvalues of the Jacobian in the complex plane and intermittency.
PERIODIC ATTRACTORS OF RANDOM TRUNCATOR MAPS
, 2006
"... Abstract. This paper introduces the truncator map as a dynamical system on the space of configurations of an interacting particle system. We represent the symbolic dynamics generated by this system as a noncommutative algebra and classify its periodic orbits using properties of endomorphisms of the ..."
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Abstract. This paper introduces the truncator map as a dynamical system on the space of configurations of an interacting particle system. We represent the symbolic dynamics generated by this system as a noncommutative algebra and classify its periodic orbits using properties of endomorphisms of the resulting algebraic structure. A stochastic model is constructed on these endomorphisms, which leads to the classification of the distribution of periodic orbits for random truncator maps. This framework is applied to investigate the periodic transitions of Bornholdt’s spin market model. 1. Model Description Let Ω = [−1, 1] N for some positive dimension N and consider a set {Sj} M j=1 of mutually exclusive and exhaustive subsets of Ω. A typical example will be the generalized quadrants, i.e. Sj = x ∈ Ω
unknown title
, 1999
"... From template analysis to generating partitions I: Periodic orbits, knots and symbolic encodings ..."
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From template analysis to generating partitions I: Periodic orbits, knots and symbolic encodings