Dimensions of strange nonchaotic attractors (original) (raw)
Related papers
Is the dimension of chaotic attractors invariant under coordinate changes?
Journal of Statistical Physics, 1984
Several different dimensionlike quantities, which have been suggested as being relevant to the study of chaotic attractors, are examined. In particular, we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points. It is found that some are and some are not. It is suggested that the word "dimension" be reversed only for those quantities have this invariance property.
D S ] 1 3 A pr 2 00 6 How chaotic are strange nonchaotic attractors ?
2014
We show that the classic examples of quasi-periodically forced maps with strange nonchaotic attractors described by Grebogi et al and Herman in the mid-1980s have some chaotic properties. More precisely, we show that these systems exhibit sensitive dependence on initial conditions, both on the whole phase space and restricted to the attractor. The results also remain valid in more general classes of quasiperiodically forced systems. Further, we include an elementary proof of a classic result by Glasner and Weiss on sensitive dependence, and we clarify the structure of the attractor in an example with two-dimensional fibers also introduced by Grebogi et al.
Dimension of strange attractors in four-dimensional maps
Physics Letters A, 1990
The method of Tsang for calculating the Hausdorifdimension ofa strange attractor in two-dimensional maps is generalized to a four-dimensional map for the double rotor system. The computed dimension compares well with previous work, and is much faster computationally.
Fractal Properties of Robust Strange Nonchaotic Attractors
Physical Review Letters, 2001
We consider the existence of robust strange nonchaotic attractors (SNA's) in a simple class of quasiperiodically forced systems. Rigorous results are presented demonstrating that the resulting attractors are strange in the sense that their box-counting dimension is N +1 while their information dimension is N. We also show how these properties are manifested in numerical experiments.
2011
The correlation dimension of the fractal attractor of the general two-dimensional iterated quadratic map is obtained by using the cell-count algorithm. However, the cell-size necessary to cover all points of the attractor object is determined. Also, an explicit criterion for the selection of the cell size in cell-count algorithm is presented. When the log-log graph is plotted by Matlab program, the result will be an approximate polynomial of degree one (straight line) that fits the data in a least squares method. The correlation dimension of the fractal attractor object is the slope of this straight line. Mathematics Subject Classification: 28A80
The necessity for a time local dimension in systems with time-varying attractors
Physica A: Statistical Mechanics and its Applications, 1997
We show that a simple non-linear system of ordinary differential equations may possess a time varying attractor dimension. This indicates that it is infeasible to characterize EEG and MEG time series with a single time global dimension. We suggest another measure for the description of non-stationary attractors.
An estimate on the fractal dimension of attractors of gradient-like dynamical systems
Nonlinear Analysis: Theory, Methods & Applications, 2012
This paper is dedicated to estimate the fractal dimension of exponential global attractors of some generalized gradient-like semigroups in a general Banach space in terms of the maximum of the dimension of the local unstable manifolds of the isolated invariant sets, Lipschitz properties of the semigroup and the rate of exponential attraction. We also generalize this result for some special evolution processes, introducing a concept of Morse decomposition with pullback attractivity. Under suitable assumptions, if (A, A * ) is an attractor-repeller pair for the attractor A of a semigroup {T (t) : t ≥ 0}, then the fractal dimension of A can be estimated in terms of the fractal dimension of the local unstable manifold of A * , the fractal dimension of A, the Lipschitz properties of the semigroup and the rate of the exponential attraction. The ingredients of the proof are the notion of generalized gradient-like semigroups and their regular attractors, Morse decomposition and a fine analysis of the structure of the attractors. As we said previously, we generalize this result for some evolution processes using the same basic ideas.
Topological dimension of singular-hyperbolic attractors
2003
An {\em attractor} is a transitive set of a flow to which all positive orbit close to it converges. An attractor is {\em singular-hyperbolic} if it has singularities (all hyperbolic) and is partially hyperbolic with volume expanding central direction \cite{MPP}. The geometric Lorenz attractor \cite{GW} is an example of a singular-hyperbolic attractor with topological dimension geq2\geq 2geq2. We shall prove that {\em all} singular-hyperbolic attractors on compact 3-manifolds have topological dimension geq2\geq 2geq2. The proof uses the methods in \cite{MP}.
Chaos, Solitons & Fractals, 1998
In this paper we present an overview of a classification of alternative mathematical schemes which determine possible impositions of noise on dynamic systems either of a continuous or discrete formulation in time; see [1]. When a noise interferes with the evolution of a dynamic system it is called a dynamic noise. Such a dynamic noise may take the form of an additive or multiplicative expression which illustrate the kind of parameters by which noise may enter into the equations of a dynamic system. We also consider the case of an output noise, i.e. a noise which does not influence the evolution of a dynamic system. The output noise is again divided into additive and multiplicative forms, depending on how it is introduced into the formulation of the system. We present some numerical investigations concerning the influence of noise on i) the correlation dimension, ii) the largest Lyapunov exponent and iii) the geometric structure of the Henon and Lorenz attractors. We also recall, a method of constructing models for effecting predictions of time series with a finite correlation dimension as obtained in [1].
Related topics
Cited by
Cluster-weighted modeling: Estimation of the Lyapunov spectrum in driven systems
Physical Review E, 2005
Cluster-weighted modeling based techniques are shown to be accurate, efficient, and robust in application to the problem of computing the Lyapunov spectrum from time-series data. We develop a method that is appropriate for application to driven nonlinear dynamical systems and show, in particular, that it is possible to estimate both global and local Lyapunov exponents through this technique. For dynamics on strange nonchaotic attractors, the present approach correctly determines a largest Lyapunov exponent that is negative.
Rotation numbers for quasi-periodically forced monotone circle maps
Dynamical Systems, 2002
Rotation numbers have played a central role in the study of (unforced) monotone circle maps. In such a case it is possible to obtain a priori bounds of the form » ¡ 1=n µ …1=n †…y n ¡ y 0 † µ » ‡ 1=n, where …1=n †…y n ¡ y 0 † is an estimate of the rotation number obtained from an orbit of length n with initial condition y 0 , and » is the true rotation number. This allows rotation numbers to be computed reliably and e ciently. Although Herman has proved that quasi-periodically forced circle maps also possess a well-de®ned rotation number, independent of initial condition, the analogous bound does not appear to hold. In particular, two of the authors have recently given numerical evidence that there exist quasi-periodically forced circle maps for which y n ¡ y 0 ¡ »n is not bounded. This renders the estimation of rotation numbers for quasi-periodically forced circle maps much more problematical. In this paper, a new characterization of the rotation number is derived for quasiperiodically forced circle maps based upon integrating iterates of an arbitrary smooth curve. This satis®es analogous bounds to above and permits us to develop improved numerical techniques for computing the rotation number. Additionally, the boundedness of y n ¡ y 0 ¡ »n is considered. It is shown that if this quantity is bounded (both above and below) for one orbit, then it is bounded for all orbits. Conversely, if for any orbit y n ¡ y 0 ¡ »n is unbounded either above or below, then there is a residual set of orbits for which y n ¡ y 0 ¡ »n is unbounded both above and below. In proving these results a min±max characterization of the rotation number is also presented. The performance of an algorithm based on this is evaluated, and on the whole it is found to be inferior to the integral based method.
Physical Review Letters, 2015
The unprecedented light curves of the Kepler space telescope document how the brightness of some stars pulsates at primary and secondary frequencies whose ratios are near the golden mean, the most irrational number. A nonlinear dynamical system driven by an irrational ratio of frequencies generically exhibits a strange but nonchaotic attractor. For Kepler 's "golden" stars, we present evidence of the first observation of strange nonchaotic dynamics in nature outside the laboratory. This discovery could aid the classification and detailed modeling of variable stars. PACS numbers: 05.45.Tp, 05.45.Df, 97.10.Sj, 95.75.Wx
The birth of strange nonchaotic attractors
Physica D: Nonlinear Phenomena, 1994
A mechanism is described for the development of strange nonchaotic attractors from two-frequency quasiperiodic attractors in quasiperiodically driven maps. This mechanism is intimately tied to the phenomenon of torus-doubling. The transition to strange nonchaotic behavior occurs when a period-doubled torus collides with its unstable parent torus. As the collision is approached, the period-doubled torus becomes extremely wrinkled, ultimately becoming fractal at the collision. The Lyapunov exponent remains negative through the collision. These collisions are shown to be a new type of attractor merging crisis; the new feature is the possibility of nonchaotic attractors taking part in the crisis. This mechanism is illustrated via numerical and analytical studies of a quasiperiodically driven logistic map.
Physical Review E, 1997
Elimination of chaotic behavior in the harmonically driven van der Pol oscillator by means of a comparatively weak additional forcing was studied through geometrical resonance analysis. We considered commensurate and incommensurate cases together with the effect of the phase difference between the forcings. The analysis provided parameter-space regions for regularization that were corroborated by numerical experiments, including instances with clearly large chaos-inducing forcing. A reinterpretation of a classical result, due to Cartwright and Littlewood ͓J. London Math. Soc. 20, 180 ͑1945͔͒, was also derived by means of geometrical resonance analysis. ͓S1063-651X͑97͒06008-X͔ PACS number͑s͒: 05.45.ϩb
Universal scaling of rotation intervals for quasi-periodically forced circle maps
Dynamical Systems, 2012
We introduce a simplifying assumption which makes it possible to approximate the rotation number of an invertible quasi-periodically forced circle map by an integral in the limit of large forcing. We use this to describe universal scaling laws for the width of the non-trivial rotation interval of non-invertible quasi-periodically forced circle maps in this limit, and compare the results with numerical simulations. Dedicated to the memory of Jaroslav Stark 1. Introduction Noninvertible circle maps have been used to model the breakdown of invariant tori in differential equations [5, 9], and there are many interesting results about the bifurcation structure for these maps [9, 10]. Whilst invertible maps of the circle have a unique rotation number measuring the average speed of orbits around the circle, noninvertible circle maps have a rotation interval: different orbits can lead to different rotation rates, but the set of all rotation rates is a closed interval, which may be a point [1, 9, 11]. If the differential equation being modelled is quasi-periodically forced then it is
On the route to strangeness without chaos in the quasiperiodically forced van der Pol oscillator
Chaos, Solitons & Fractals, 1996
4bstract-The non-chaotic strange attractor of a non-autonomous dissipative dynamical system with a two-frequency-quasiperiodic forcing is studied. Following results from T. Kapitaniak [Chaos Solituns & Fractals 1, 67 (1991)], numerical observations are refined and the route from torus to strange non-chaotic attractor is described in detail. Possible misinterpretations of experimental observations are discussed.
Fractal Properties of Robust Strange Nonchaotic Attractors
Physical Review Letters, 2001
We consider the existence of robust strange nonchaotic attractors (SNA's) in a simple class of quasiperiodically forced systems. Rigorous results are presented demonstrating that the resulting attractors are strange in the sense that their box-counting dimension is N +1 while their information dimension is N. We also show how these properties are manifested in numerical experiments.
On the Amplitude of External Perturbation and Chaos via Devil's Staircase — Stability of Attractors
International Journal of Bifurcation and Chaos, 2015
We designed a chaotic memristive circuit proposed by Chua and Muthuswamy, and analyzed the behavior of the voltage of the capacitor, electric current in the inductor and the voltage of the memristor by adding an external sinusoidal oscillation of a type γω cos ωt to [Formula: see text], when [Formula: see text] is expressed by y(t)/C, and studied Devil's staircase route to chaos. We compared the frequency of the driving oscillation fs and the frequency of the response fd in the window and assigned W = fs/fd to each window. When capacitor C = 1.0, we observe stable attractors of Farey sequences [Formula: see text], which can be interpreted as hidden attractors, while when C = 1.2, we observe disturbed attractors. Frequency dependent stability of chaotic circuits due to octonions is discussed.
Related papers
Unstable periodic orbits and the dimensions of multifractal chaotic attractors
Physical review. A, 1988
The probability measure generated by typical chaotic orbits of a dynamical system can have an arbitrarily Sne-scaled interwoven structure of points with different singularity scalings. Recent work has characterized such measures via a spectrum of fractal dimension values. In this paper we pursue the idea that the infinite number of unstable periodic orbits embedded in the support of the measure provides the key to an understanding of the structure of the subsets with different singularity scalings. In particular, a formulation relating the spectrum of dimensions to unstable periodic orbits is presented for hyperbolic maps of arbitrary dimensionality. Both chaotic attractors and chaotic repeBers are considered.
Unstable dimension variability: A source of nonhyperbolicity in chaotic systems
International Symposium on Physical Design, 1996
The hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set. A familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set. Here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible (respectively noninvertible) maps of dimension 3 (respectively 2) and higher. In particular, we investigate a situation (first considered by Abraham and Smale in 1970 for different purposes) in which the dimension of the unstable (and stable) tangent spaces are not constant over the chaotic set; we call this unstable dimension variability. A simple two-dimensional map that displays behavior typical of this phenomenon is presented and analyzed.
How chaotic are strange non-chaotic attractors?
Nonlinearity, 2006
We show that the classic examples of quasi-periodically forced maps with strange nonchaotic attractors described by Grebogi et al and Herman in the mid-1980s have some chaotic properties. More precisely, we show that these systems exhibit sensitive dependence on initial conditions, both on the whole phase space and restricted to the attractor. The results also remain valid in more general classes of quasiperiodically forced systems. Further, we include an elementary proof of a classic result by Glasner and Weiss on sensitive dependence, and we clarify the structure of the attractor in an example with two-dimensional fibers also introduced by Grebogi et al.
Chaotic attractors of two-dimensional invertible maps
Discrete Dynamics in Nature and Society, 1998
In this paper, we investigate the characteristics of quasihyperbolic attractors and quasiattractors in invertible dissipative maps of the plane. The criteria which allow one to diagnose the indicated types of attractors in numerical experiments are formulated.
Characterization of noise-induced strange nonchaotic attractors
Physical Review E, 2006
Strange nonchaotic attractors ͑SNAs͒ were previously thought to arise exclusively in quasiperiodic dynamical systems. A recent study has revealed, however, that such attractors can be induced by noise in nonquasiperiodic discrete-time maps or in periodically driven flows. In particular, in a periodic window of such a system where a periodic attractor coexists with a chaotic saddle ͑nonattracting chaotic invariant set͒, none of the Lyapunov exponents of the asymptotic attractor is positive. Small random noise is incapable of causing characteristic changes in the Lyapunov spectrum, but it can make the attractor geometrically strange by dynamically connecting the original periodic attractor with the chaotic saddle. Here we present a detailed study of noise-induced SNAs and the characterization of their properties. Numerical calculations reveal that the fractal dimensions of noise-induced SNAs typically assume fractional values, in contrast to SNAs in quasiperiodically driven systems whose dimensions are integers. An interesting finding is that the fluctuations of the finite-time Lyapunov exponents away from their asymptotic values obey an exponential distribution, the generality of which we are able to establish by a theoretical analysis using random matrices. We suggest a possible experimental test. We expect noise-induced SNAs to be general.
Influence of noise on statistical properties of nonhyperbolic attractors
Physical Review E, 2000
We analyze effects of bounded white and colored noise on nonhyperbolic chaotic attractors in twodimensional invertible maps. It is shown that first the nonhyperbolic nature is kept even in the presence of strong noise, but secondly already due to weak noise some properties of nonhyperbolic chaos can become similar to those of hyperbolic and almost hyperbolic chaos. We also estimate the stationary probability measure of noisy nonhyperbolic attractors. For this purpose two different methods for calculating the probability density are applied and the obtained results are compared in detail.
The birth of strange nonchaotic attractors
Physica D: Nonlinear Phenomena, 1994
A mechanism is described for the development of strange nonchaotic attractors from two-frequency quasiperiodic attractors in quasiperiodically driven maps. This mechanism is intimately tied to the phenomenon of torus-doubling. The transition to strange nonchaotic behavior occurs when a period-doubled torus collides with its unstable parent torus. As the collision is approached, the period-doubled torus becomes extremely wrinkled, ultimately becoming fractal at the collision. The Lyapunov exponent remains negative through the collision. These collisions are shown to be a new type of attractor merging crisis; the new feature is the possibility of nonchaotic attractors taking part in the crisis. This mechanism is illustrated via numerical and analytical studies of a quasiperiodically driven logistic map.
On determining the dimension of chaotic flows
Physica D: Nonlinear Phenomena, 1981
We describe a method for determining the approximate fractal dimension of an attractor. Our technique fits linear subspaces of appropriate dimension to sets of points on the attractor. The deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques. We show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time-varying quantity is available for analysis. These methods are applied to several dissipative dynamical systems.
Unusual Chaotic Attractors in Nonsmooth Dynamic Systems
International Journal of Bifurcation and Chaos, 2005
The present paper describes an unusual example of chaotic motion occurring in a nonsmooth mechanical system affected by dry friction. The mechanical system generates one-dimensional maps the orbits of which seem to exhibit sensitive dependence on initial conditions only in an extremely small set of their field of definition. The chaotic attractor is composed of zones characterized by very different rates of divergence of nearby orbits: in a large portion of the chaotic attractor the system motion follows a regular pattern whereas the more usual irregular motion affects only a small portion of the attractor. The irregular phase reintroduces the orbit in the regular zone and the sequence is repeated. The Lyapunov exponent of the map is computed to characterize the steady state motions and confirm their chaotic nature.
Physical Review E, 2002
We study local and global correlations between the naturally invariant measure of a chaotic one-dimensional map f and the conditionally invariant measure of the transiently chaotic map f H. The two maps differ only within a narrow interval H, while the two measures significantly differ within the images f l (H), where l is smaller than some critical number l c. We point out two different types of correlations. Typically, the critical number l c is small. The 2 value, which characterizes the global discrepancy between the two measures, typically obeys a power-law dependence on the width ⑀ of the interval H, with the exponent identical to the information dimension. If H is centered on an image of the critical point, then l c increases indefinitely with the decrease of ⑀, and the 2 value obeys a modulated power-law dependence on ⑀.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.