Strange Nonchaotic Attractors Research Articles

The non-smooth pitchfork bifurcation: a renormalization analysis

We give a renormalization group analysis of a system exhibiting a non-smooth pitchfork bifurcation to a strange non-chaotic attractor. For parameter choices satisfying two specified conditions, self-similar behaviour of the attractor on and near the bifurcation curve can be observed, which corresponds to a periodic orbit of an underlying renormalization operator. We examine the scaling properties for various parameter choices including the so-called pitchfork critical point. Finally, we study the autocorrelation function for the system and show that it is equivalent to that present in symmetric barrier billiards.

De-synchronization and chaos in two inductively coupled Van der Pol auto-generators

In this article, we consider a system of autonomous inductively coupled Van der Pol generators. For two coupled generators, we establish the presence of metastable chaos, a strange non-chaotic attractor, and several stable limiting cycles. Areas of parametric dependence of different modes of synchronization are obtained.

Dimensions of structurally stable and critical strange non-chaotic attractors

We provide evidence that the box-counting dimension of a structurally stable strange non-chaotic attractor (SNA) of pinched skew product type is equal to 2 by showing that it has non-negligible area. The argument presented is made more accurate in the study of a piecewise linear SNA. Furthermore we provide evidence that the fractal dimension of a critical SNA is not equal to 2, but in fact lies between 1 and 2. We numerically calculate the box-counting dimension for several critical SNAs, providing further evidence to support this conjecture.

Analytical Study of an MLC Circuit with Quasiperiodic Forcing

An Murali-Lakshmanan-Chua circuit is the simplest second order non-autonomous nonlinear circuit exhibiting chaos. The strange nonchaotic attractor behaviour of the circuit when subjected to quasiperiodic forcing is studied analytically. An explicit analytical solution for a system exhibiting strange nonchaotic attractor behaviour in its dynamics is presented here for the first time, with phase portraits showing the Heagy-Hammel and fractalization routes to chaos.

Hyperchaos and bifurcations in a driven Van der Pol–Duffing oscillator circuit

We investigate the dynamics of a driven Van der Pol–Duffing oscillator circuit and show the existence of higher-dimensional chaotic orbits (or hyperchaos), transient chaos, strange-nonchaotic attractors, as well as quasiperiodic orbits born from Hopf bifurcating orbits. By computing all the Lyapunov exponent spectra, scanning a wide range of the driving frequency and driving amplitude parameter space, we explore in two-parameter space the regimes of different dynamical behaviours.

Anomalous diffusion generated by quasiperiodically forced maps with strange nonchaotic attractors

Anomalous diffusion generated by quasiperiodically forced maps with strange nonchaotic attractors

Bifurcations and Chaos of Duffing–van der Pol Equation with Nonsymmetric Nonlinear Restoring and Two External Forcing Terms

Bifurcations and chaos of Duffing–van der Pol equation with nonsymmetric nonlinear restoring and two external forcing terms are investigated. The threshold values of the existence of chaotic motion are obtained under periodic perturbation. By the second-order averaging method, we prove the criteria of the existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 1, 2, 3, 5, and cannot prove the criterion of existence of chaos in an averaged system under quasi-periodic perturbation for ω2 = nω1 + εσ, n = 4, 6, 7, …, where σ is not rational to ω1, but can show the occurrence of chaos in the original system by numerical simulation. Numerical simulation including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincaré maps, not only show the consistence with the theoretical analysis but also exhibit more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations lead to interleaving occurrence of chaotic behaviors and quasi-periodic orbits, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos occurring more than once, chaos suddenly disappearing to periodic orbits, strange nonchaotic attractor, nonattracting chaotic set and nice chaotic attractors.

A numerical study of the local K spectrum of strange nonchaotic attractors

Quasiperiodically-driven dynamical systems generally show transitions from a quaisperiodic attractor to a strange nonchaotic attractor (SNA), and then to a chaotic attractor. Discriminating the SNA from the choatic or the quasiperiodic attractor is still not easy because the region of the SNA is narrow and the dynamics of the SNA looks similar to that of other attractors. In this paper, the local K spectrum of the 0–1 test for chaos is introduced to investigate numerically transitions in quasiperiodically-driven dynamical systems. The local K spectrum of the SNA shows clearly distinct characteristics as compared to that of other attractors. This indicates that the local K spectrum is useful for discriminating the SNA from among other attractors of nonlinear dynamical systems.

Renormalisation of correlations in a barrier billiard: Quadratic irrational trajectories

We present an analysis of autocorrelation functions in symmetric barrier billiards using a renormalisation approach for quadratic irrational trajectories. Depending on the nature of the barrier, this leads to either self-similar or chaotic behaviour. In the self-similar case we give an analysis of the half barrier and present a detailed calculation of the locations, asymptotic heights and signs of the main peaks in the autocorrelation function. Then we consider arbitrary barriers, illustrating that typically these give rise to chaotic correlations of the autocorrelation function which we further represent by showing the invariant sets associated with these correlations. Our main ingredient here is a functional recurrence which has been previously derived and used in work on the Harper equation, strange non-chaotic attractors and a quasi-periodically forced two-level system.

Numerical simulation of torus breakdown to chaos in an atmospheric-pressure dielectric barrier discharge

Understanding the routes to chaos occurring in atmospheric-pressure dielectric barrier discharge systems by changing controlling parameters is very important to predict and control the dynamical behaviors. In this paper, a route of a quasiperiodic torus to chaos via the strange nonchaotic attractor is observed in an atmospheric-pressure dielectric barrier discharge driven by triangle-wave voltage. By increasing the driving frequency, the discharge system first bifurcates to a quasiperiodic torus from a stable single periodic state, and then torus and phase-locking periodic state appear and disappear alternately. In the meantime, the torus becomes increasingly wrinkling and stretching, and gradually approaches a fractal structure with the nonpositive largest Lyapunov exponent, i.e., a strange nonchaotic attractor. After that, the discharge system enters into chaotic state. If the driving frequency is further increased, another well known route of period-doubling bifurcation to chaos is also observed.

Torus-doubling bifurcations and strange nonchaotic attractors in a vibro-impact system

Whether strange nonchaotic attractors (SNAs) can occur typically in dynamical systems other than quasiperiodically driven systems has been an open problem. Here we show that the SNAs can be induced by the interruption of torus-doubling bifurcation in a periodically driven vibro-impact system. It is found that the creation of SNAs occurs due to the collision of doubled torus with some unstable periodic orbits. Based on the Poincaré map technique, two types of codimension-3 bifurcations are represented and typical torus-doubling bifurcations can occur in the neighborhood of these bifurcation points. We use the singular continuous Fourier spectrum and its scaling to verify the strangeness (nondifferentiability) of the attractors. The implicitly transcendental map is used to calculate the largest Lyapunov exponent by QR-based algorithm, which ensures that the underlying dynamics (with nonpositive Lyapunov exponents) is nonchaotic. The associated mechanism is described for the creation of SNAs through detecting unstable periodic orbits from Poincaré map series.

Applicability of 0-1 test for strange nonchaotic attractors

We show that the recently introduced 0-1 test can successfully distinguish between strange nonchaotic attractors (SNAs) and periodic/quasiperiodic/chaotic attractors, by suitably choosing the arbitrary parameter associated with the translation variables in terms of the golden mean number which avoids resonance with the quasiperiodic force. We further characterize the transition from quasiperiodic to chaotic motion via SNAs in terms of the 0-1 test. We demonstrate that the test helps to detect different dynamical transitions to SNAs from quasiperiodic attractor or the transitions from SNAs to chaos. We illustrate the performance of the 0-1 test in detecting transitions to SNAs in quasiperiodically forced logistic map, cubic map, and Duffing oscillator.

Strange nonchaotic attractors with Wada basins

We demonstrate strange nonchaotic attractors with Wada basins (SNAsWB), and verify the abundance of SNAsWB in a quasiperiodically forced Holmes map. We identify the routes to the creation of the SNAsWB in a two-parameter space. The SNAsWB are characterized by the maximal Lyapunov exponent, by the estimation of the phase sensitivity exponent, and by the singular-continuous spectra. We observe that the SNAs’ basins are totally Wada basins in a large range of parameters. The topological structures of the SNAs’ Wada basins are distinguished by the basin cell method. We investigate the underlying mechanism for the abundance of SNAsWB, which is responsible for different types of basin cell in the absence of forcing. This suggests that the SNAs cannot be predicted reliably for specific initial conditions. These SNAsWB can thus be expected to occur more commonly in dynamical systems.

Dynamics between order and chaos in conceptual models of glacial cycles

The dynamics of glacial cycles is studied in terms of the dynamical systems theory. We explore the dependence of the climate state on the phase of astronomical forcing by examining five conceptual models of glacial cycles proposed in the literature. The models can be expressed as quasiperiodically forced dynamical systems. It is shown that four of them exhibit a strange nonchaotic attractor (SNA), which is an intermediate regime between quasiperiodicity and chaos. Then, the dependence of the climate state on the phase of astronomical forcing is not given by smooth relations, but constitutes a geometrically strange set. Our result suggests that the dynamics of SNA is a candidate for the dynamics of glacial cycles, in addition to the well-known dynamics of quasiperiodicity and chaos.

Experimental distinction between chaotic and strange nonchaotic attractors on the basis of consistency

We experimentally study strange nonchaotic attractors (SNAs) and chaotic attractors by using a nonlinear integrated circuit driven by a quasiperiodic input signal. An SNA is a geometrically strange attractor for which typical orbits have nonpositive Lyapunov exponents. It is a difficult problem to distinguish between SNAs and chaotic attractors experimentally. If a system has an SNA as a unique attractor, the system produces an identical response to a repeated quasiperiodic signal, regardless of the initial conditions, after a certain transient time. Such reproducibility of response outputs is called consistency. On the other hand, if the attractor is chaotic, the consistency is low owing to the sensitive dependence on initial conditions. In this paper, we analyze the experimental data for distinguishing between SNAs and chaotic attractors on the basis of the consistency.

Dimensions of Attractors in Pinched Skew Products

We study dimensions of strange non-chaotic attractors and their associated physical measures in so-called pinched skew products, introduced by Grebogi and his coworkers in 1984. Our main results are that the Hausdorff dimension, the pointwise dimension and the information dimension are all equal to one, although the box-counting dimension is known to be two. The assertion concerning the pointwise dimension is deduced from the stronger result that the physical measure is rectifiable. Our findings confirm a conjecture by Ding, Grebogi and Ott from 1989.

Nature of weak generalized synchronization in chaotically driven maps

Weak generalized synchrony in a drive-response system occurs when the response dynamics is a unique but nondifferentiable function of the drive, in a manner that is similar to the formation of strange nonchaotic attractors in quasiperiodically driven dynamical systems. We consider a chaotically driven monotone map and examine the geometry of the limit set formed in the regime of weak generalized synchronization. The fractal dimension of the set of zeros is studied both analytically and numerically. We further examine the stable and unstable sets formed and measure the regularity of the coupling function. The stability index as well as the dimension spectrum of the equilibrium measure can be computed analytically.

Wada basins of strange nonchaotic attractors in a quasiperiodically forced system

Abstract Whether Wada basins of strange nonchaotic attractors (SNAs) can exist has been an open problem. Here we verify the existence of Wada basin for SNAs in a quasiperiodically forced Duffing map. We show that the SNAsʼ basins are full Wada for a set of parameters of positive measure. We identify two types of SNAsʼ Wada basins by the basin cell method. It suggests that SNAs cannot be predicted reliably for the specific initial conditions.

Quasiperiodically driven maps in the low-dissipation limit

(Received 24 September 2012; revised manuscript received 18 February 2013; published 11 March 2013)We study the quasiperiodically driven Henon and Standard maps in the weak dissipative limit. In the absence of´forcing, there are a large number of coexisting periodic attractors. Although chaotic attractors can also be found,these typically have vanishingly small basins of attraction. Quasiperiodic forcing reduces the multistability in thesystem, and as the bifurcation parameter is varied, strange nonchaotic attractors (SNAs) are created. The attractorbasin for SNAs appears to be the largest among those of all coexisting attractors at such a transition.DOI: 10.1103/PhysRevE.87.034901 PACS number(s): 05

Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chuaʼs oscillator

Abstract We have constructed a quasiperiodically forced Chuaʼs circuit and report the occurrence of strange nonchaos in it. The birth of SNA is not through the well known mechanisms reported in literature, but through a novel route which we christen as “Formation and merging of bubbles route”. This is so named because the Poincare strands undergo a series of splittings and mergings in certain regions, giving bubbles like appearance. These bubbles coalesce as the control parameter is varied, resulting in the birth of SNA. This has been confirmed numerically using various statistical measures.