Strange Nonchaotic Attractors Research Articles

New periodic-chaotic attractors in slow-fast Duffing system with periodic parametric excitation

A new type of responses called as periodic-chaotic motion is found by numerical simulations in a Duffing oscillator with a slowly periodically parametric excitation. The periodic-chaotic motion is an attractor, and simultaneously possesses the feature of periodic and chaotic oscillations, which is a new addition to the rich nonlinear motions of the Duffing system including equlibria, periodic responses, quasi-periodic oscillations and chaos. In the current slow-fast Duffing system, we find three new attractors in the form of periodic-chaotic motions. These are called the fixed-point chaotic attractor, the fixed-point strange nonchaotic attractor, and the critical behavior with the maximum Lyapunov exponent fluctuating around zero. The system periodically switches between one attractor with a fixed single-well potential and the other with time-varying two-well potentials in every period of excitation. This behavior is apparently the mechanism to generate the periodic-chaotic motion.

Strange nonchaotic attractors in a nonsmooth dynamical system

Strange nonchaotic attractors (SNAs) have fractal geometric structure, but are nonchaotic in the dynamical sense. Since Grebogi et al. discovered SNA in 1984, it has become one of the important topics in nonlinear dynamics. Now, the study of SNAs has been mainly confined to smooth dynamics with quasiperiodic excitation or random excitation. In this paper, we consider a class of single-degree-of-freedom gear dynamical system with quasiperiodic forcing. We show that the gear transmission system can be modeled as a three-dimensional piecewise linear system, which belongs to a typical class of nonsmooth system. We then show that SNAs do exist in such nonsmooth dynamical system with quasiperiodic force. The dynamical behavior of the nonsmooth system is analyzed as a parameter is varied. The dynamics is analyzed through phase diagrams and bifurcation diagrams, Lyapunov exponents, singular continuous power spectrum, phase sensitivity of time series and rational approximations.

Mechanisms of strange nonchaotic attractors in a nonsmooth system with border-collision bifurcations

It is not very clear to understand genesis and mechanisms for the creation of strange nonchaotic attractors (SNAs) due to the nonsmooth bifurcations in the nonsmooth systems. A quasiperiodically forced piecewise Logistic system is shown to exhibit many types of routes to the creation of SNAs. We point out that the truncation of border-collision torus-doubling bifurcation can lead to different types of SNAs. We identify and describe the Heagy–Hammel routes, fractalization route and intermittent routes after the two coexisting tori collide at the border and the doubled torus is interrupted in this system. It has been shown that there exist two critical tongue-type regions in the parameter space, where the different mechanisms for the birth of SNAs are investigated. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents. Different types of SNAs are also characterized by the singular-continuous spectrum, Fourier transform, rational approximations, distribution of finite-time Lyapunov exponents and recurrence analysis.

Электронное устройство, реализующее странный нехаотический аттрактор Ханта–Отта

Электронное устройство, реализующее странный нехаотический аттрактор Ханта–Отта

Strange nonchaotic attractors in oscillators sharing nonlinearity

We identified the plausibility of the existence of strange nonchaotic attractors (SNAs) in a model of two sinusoidal driven LCR dissipative oscillators sharing a common piecewise nonlinearity. The detected strange nonchaos has been assayed through numerical studies encompassing phase space analysis, Poincaré cross sections, Lyapunov exponents, etc., The strange nonchaotic nature of attractors was examined by recurrence quantification measure. Phase synchronization features of the attractors in the SNA regime have also been included. Experimental evidences have been furnished by means of a real-time electronic circuit to reinforce the observed results. It is expected that the outcomes of the present study could facilitate the understanding and the observation of SNAs in nonlinearly coupled networks, including small-world, scale-free networks and neuronal networks.

Strange nonchaotic and chaotic attractors in a self-excited thermoacoustic oscillator subjected to external periodic forcing

We experimentally investigate the synchronization dynamics of a self-excited thermoacoustic system forced beyond its phase-locked state. The system consists of a laminar premixed flame in a tube combustor subjected to periodic acoustic forcing. On increasing the forcing amplitude above that required for phase locking, we find that the system can transition out of phase locking and into chaos, which is consistent with the Afraimovich-Shilnikov theorem for the breakdown of a phase-locked torus. However, we also find some unexpected behavior, most notably the emergence of a strange nonchaotic attractor (SNA) before the onset of chaos. We verify the existence of the SNA and chaotic attractor by examining the correlation dimension, the autocorrelation function, the power-law scaling in the Fourier amplitude spectrum, the permutation entropy in a pseudoperiodic surrogate test, and the permutation spectrum. In summary, this study explores the SNA and chaotic dynamics of a thermoacoustic system forced beyond its phase-locked state, opening up new pathways for the development of alternative strategies to control self-excited thermoacoustic oscillations in combustion devices such as gas turbines and rocket engines.

Investigation on Unconventional Synthesis of Astroinformatic Data Classifier Powered by Irregular Dynamics

This paper discusses the mutual combination of the unconventional algorithm (in this case evolutionary algorithms), deterministic chaos and modeling on real data from astrophysics. Analytical programming with selected evolutionary algorithm is used to synthesize suitable models. The main attention in this paper is focused on various chaotic generators, which are used instead of classical pseudo-random number generators. Chaotic generators are used in conjunction with a special case - a generator based on a strange non-chaotic attractor. The performance of all chaotic and non-chaotic based generators is then mutually compared at the end.

Applicability of strange nonchaotic Wien-bridge oscillators for secure communication

In this paper, experimental realisation of synchronisation of strange nonchaotic attractor in unidirectionally coupled Wien-bridge oscillator is presented. Preliminary work of chaotic and strange nonchaotic dynamics with diode as the nonlinear element is published in J. Nonlinear Dyn. 2015, 1 (2015). In this work, two Wien-bridge oscillators are coupled at the aperiodic dynamics for application in secure communication. The dynamics of the coupled system is explored by experimental and numerical studies. Employing the simple communication masking method with aperiodic dynamics offers some advantages rather than using chaotic attractors, i.e., the range of control parameter value is smaller than the range for which the system exhibits chaotic behaviour. The performance of the system is evaluated and it is found that experimental result agrees with numerical results.

Strange non-chaotic attractors in a state controlled-cellular neural network-based quasiperiodically forced MLC circuit

In this paper, we report the dynamical transitions to strange non-chaotic attractors in a quasiperiodically forced state controlled-cellular neural network (SC-CNN)-based MLC circuit via two different mechanisms, namely the Heagy–Hammel route and the gradual fractalisation route. These transitions were observed through numerical simulations and hardware experiments and confirmed using statistical tools, such as maximal Lyapunov exponent spectrum and its variance and singular continuous spectral analysis. We find that there is a remarkable agreement of the results from both numerical simulations as well as from hardware experiments.

Strange nonchaotic dynamics of parametrically enhanced MLC circuit

We report experimental and numerical studies of a strange nonchaotic attractor (SNA) using a parametrically perturbed Murali–Lakshmanan–Chua (MLC) circuit. One of the energy storage elements in the circuit, i.e., the capacitor, is perturbed parametrically with a sinusoidal excitation. Under this condition, the circuit exhibits both chaotic and strange nonchaotic behavior for suitably adjusted capacitor values. This constitutes the novelty of this work. The other point to note in this circuit is that one of the two alternating-current (AC) stimulations (which are usually connected in series to observe SNA) is obtained by varying the capacitance with a perturbation consisting of a periodic stimulating force. The experimental results are confirmed by numerical study, in particular calculations of the correlation coefficient, path of translation variables, modified mean square displacement, singular continuous spectrum, power spectrum, and autocorrelation coefficient, to confirm the existence of an SNA in the dynamics of the modified MLC circuit.

Strange nonchaotic attractors for computation.

We investigate the response of quasiperiodically driven nonlinear systems exhibiting strange nonchaotic attractors (SNAs) to deterministic input signals. We show that if one uses two square waves in an aperiodic manner as input to a quasiperiodically driven double-well Duffing oscillator system, the response of the system can produce logical output controlled by such a forcing. Changing the threshold or biasing of the system changes the output to another logic operation and memory latch. The interplay of nonlinearity and quasiperiodic forcing yields logical behavior, and the emergent outcome of such a system is a logic gate. It is further shown that the logical behaviors persist even for an experimental noise floor. Thus the SNA turns out to be an efficient tool for computation.

Strange non-chaotic attractors in spin valve systems

The current-driven magnetization dynamics in spin torque oscillators is investigated because of its high potential for high-frequency applications. The system consists of a pinned layer with a fixed magnetization and a single-domain free layer, such that it is governed by the Landau-Lifshitz-Gilbert-Slonczewski equation. In particular, we study the effect of a time-dependent quasi-periodic current. We numerically characterize the dynamical behavior of the system by monitoring the Lyapunov exponents, and by calculating the Fourier spectra. We find a rather complicated landscape of sometimes closely intermingled chaotic and non-chaotic areas in the parameters space. Finally, we compute a phase diagram of the existence of the strange non-chaotic attractors.

Multi-affinity and Phase Sensitivity of Strange Nonchaotic Attractors

We study the multi-affinity and phase sensitivity of strange nonchaotic attractors (SNAs) in quasiperiodically driven systems and show that the q-dependent scaling exponent η(q), which determines the scaling of the qth correlation function of the height differences of the graph of SNAs, exhibits phase-transition-like behaviors. Furthermore, we derive hyperscaling relations, which connect two kinds of exponents characterizing the phase sensitivity and multi-affinity. We point out that these hyperscaling laws seem to be universal in SNAs.

Observation of chaotic and strange nonchaotic attractors in a simple multi-scroll system

The dynamics of third-order chaotic circuit with threshold controller as a nonlinear element is reported. The circuit and the preliminary result reported by Chithra et al. (J Comput Electron 16:883---889, 2017), are considered as a prototype by adding quasiperiodic force in the original circuit. This present work focuses attention on exhibiting all possible nonlinear dynamical phenomena (i.e., periodic motion, chaotic, strange nonchaotic and multi-scroll attractor) for a different set of circuit parametric values. The approach helps in proof-of-experimental concept to implement all possible complex dynamical behaviors from a single circuit. Further, we also place emphasis on numerical simulation and physical circuit realization on the breadboard. The existence of strange nonchaotic attractor is quantitatively confirmed by calculating $${ 0}{-}{} { 1}$$0-1 test and singular-continuous spectrum analysis. We then find this circuit has potential to produce multi-scroll strange nonchaotic attractor by increasing threshold values. The experimental results are identical to numerical simulations.

Power law asymptotics in the creation of strange attractors in the quasi-periodically forced quadratic family

Let Φ be a quasi-periodically forced quadratic map, where the rotation constant ω is a Diophantine irrational. A strange non-chaotic attractor (SNA) is an invariant (under Φ) attracting graph of a nowhere continuous measurable function ψ from the circle to .This paper investigates how a smooth attractor degenerates into a strange one, as a parameter approaches a critical value , and the asymptotics behind the bifurcation of the attractor from smooth to strange. In our model, the cause of the strange attractor is a so-called torus collision, whereby an attractor collides with a repeller.Our results show that the asymptotic minimum distance between the two colliding invariant curves decreases linearly in the parameter , as approaches the critical parameter value from below.Furthermore, we have been able to show that the asymptotic growth of the supremum of the derivative of the attracting graph is asymptotically bounded from both sides by a constant times the reciprocal of the square root of the minimum distance above.

Early warning signal for interior crises in excitable systems.

The ability to reliably predict critical transitions in dynamical systems is a long-standing goal of diverse scientific communities. Previous work focused on early warning signals related to local bifurcations (critical slowing down) and nonbifurcation-type transitions. We extend this toolbox and report on a characteristic scaling behavior (critical attractor growth) which is indicative of an impending global bifurcation, an interior crisis in excitable systems. We demonstrate our early warning signal in a conceptual climate model as well as in a model of coupled neurons known to exhibit extreme events. We observed critical attractor growth prior to interior crises of chaotic as well as strange-nonchaotic attractors. These observations promise to extend the classes of transitions that can be predicted via early warning signals.

Successive torus doubling and birth of strange non‐chaotic attractors in non‐linear electronic circuit

The authors experimentally demonstrate successive torus doubling and the birth of strange non-chaotic attractors (SNAs) in a quasi-periodically forced Chua's oscillator. So far it has been believed that the torus doubling occurs only a finite number of times. In this Letter, the authors report experimental observation of a certain kind of torus doubling called swollen shape bifurcation occurs an infinite number of times and leads to the birth of SNA. In the Poincare of the section, the torus undergoes a series of period doubling and after a particular value of control parameter, the infinitely period doubled torus in the localised regime termed as a chaotic band. While the Lyapunov exponent of the attractor has a negative value and confirms the attractor becomes strange and non-chaotic. Experimentally observed results are confirmed by a Poincare map, singular-continuous spectrum analysis and the finite-time Lyapunov exponent.

Dynamic Analysis of a Lü Model in Six Dimensions and Its Projections

Abstract In this article, extended complex Lü models (ECLMs) are proposed. They are obtained by substituting the real variables of the classical Lü model by complex variables. These projections, spanning from five dimensions (5D) and six dimensions (6D), are studied in their dynamics, which include phase spaces, calculations of eigenvalues and Lyapunov’s exponents, Poincaré maps, bifurcation diagrams, and related analyses. It is shown that in the case of a 5D extension, we have obtained chaotic trajectories; meanwhile the 6D extension shows quasiperiodic and hyperchaotic behaviors and it exhibits strange nonchaotic attractor (SNA) features.

On non-smooth pitchfork bifurcations in invertible quasi-periodically forced 1-D maps

In this note we revisit an example introduced by T. Jäger in which a Strange Non-chaotic Attractor seems to appear during a pitchfork bifurcation of invariant curves in a quasi-periodically forced 1-d map. In this example, it is remarkable that the map is invertible and, hence, the invariant curves are always reducible. In the first part of the paper we give a numerical description (based on a precise computation of invariant curves and Lyapunov exponents) of the phenomenon. The second part consists in a preliminary study of the phenomenon, in which we prove that an analytic self-symmetric invariant curve is persistent under perturbations.

A system governed by a set of nonautonomous differential equations with robust strange nonchaotic attractor of Hunt and Ott type

A physically realizable nonautonomous system of ring structure is considered, which manifests a robust strange nonchaotic attractor (SNA), similar to the attractor in the map on a torus proposed earlier by Hunt and Ott. Numerical simulation of the dynamics for the corresponding non-autonomous set of differential equations with quasi-periodic coefficients is provided. It is demonstrated that in terms of appropriately chosen phase variables the dynamics is consistent with the topology of the mapping of Hunt and Ott on the characteristic period. It has been shown that the occurrence of SNA agrees with the criterion of Pikovsky and Feudel. Also, the computations confirm that the Fourier spectrum in sustained SNA mode is of intermediate class between the continuous and discrete spectra (the singular continuous spectrum).