Strange Nonchaotic Research Articles

Strange Nonchaotic Attractors in a Quasiperiodically Excited Slender Rigid Rocking Block with Two Frequencies

Strange Nonchaotic Attractors in a Quasiperiodically Excited Slender Rigid Rocking Block with Two Frequencies

Emergence of Hidden Strange Nonchaotic Attractors in a Rational Memristive Map

To exemplify the existence of hidden strange nonchaotic attractors (HSNAs) and transition mechanism, we consider a rational memristive map with additional force. We find that the four-torus bifurcates into the eight-torus through torus doubling as a function of the control parameter. Following that, the formation of strange nonchaotic attractors occurs when increasing the control parameter. As a result, it is clear that the suggested rational maps reach hidden chaotic attractors via HSNAs through the route of torus doubling to torus breakdown. The obtained dynamical transitions are validated further using bifurcation analysis and the largest Lyapunov exponents. In particular, the obtained HSNAs are confirmed through distinct statistical measures including 0–1 test and singular continuous spectrum.

Route to extreme events in a parametrically driven position-dependent nonlinear oscillator.

We explore the dynamics of a damped and driven Mathews-Lakshmanan oscillator type model with position-dependent mass term and report two distinct bifurcation routes to the advent of sudden, intermittent large-amplitude chaotic oscillations in the system. We characterize these infrequent and recurrent large oscillations as extreme events (EE) when they are significantly greater than the pre-defined threshold height. In the first bifurcation route, the system exhibits a bifurcation from quasiperiodic (QP) attractor to chaotic attractor via strange non-chaotic (SNA) attractor as a function of damping parameter. In the second route, the chaotic attractor in the form of EE has emerged directly from the QP attractor. Hence, to the best of our knowledge, this is the first study to report the birth of EE from these two distinct bifurcation routes. We also discuss that EE are emerged due to the sudden expansion of the chaotic attractor via interior crisis in the system. Regions of different dynamical states are distinguished using the Lyapunov exponent spectrum. Further, SNA and QP dynamics are determined using the singular spectrum analysis and 0-1 test. The region of EE is characterized using the threshold height.

TUNING OF IMC CONTROLLER FOR SYSTEMS WITH TIME-DELAY USING AN IMPROVED STRANGE NONCHAOTIC OPTIMIZATION APPROACH

The internal model control (IMC) has received extensive attention due to strong robustness, while it is challenging to tune the parameters of the controller. So far, strange nonchaotic optimization algorithm, based on grazing bifurcation, has been proposed to tune the parameters of the IMC controller in some recent progress, which processes a strong ability to escape from local optimization. However, the nonuniformity of the strange nonchaotic sequence distribution can result in insufficient optimization capability. In this paper, in order to obtain an excellent optimization effect, an improved strange nonchaotic optimization algorithm (ISNOA) based on grazing bifurcation is proposed to tune the parameters of the IMC controller. First, the boundary of a strange nonchaotic sequence is redefined by analyzing the strange nonchaotic sequence distribution. Furthermore, normalization function is applied to determine the optimization space. Consequently, the proposed ISNOA has better ability to escape from the local minima, higher stability and finer performance than the general SNOA, which is more suitable to tune the parameters of the IMC controller. Simulation results illustrate the effectiveness of ISNOA. In addition, for mismatch model, numerical simulations demonstrate the good performance of the control system based on ISNOA, meanwhile, the ISNOA is also demonstrated to be superior to particle swarm optimization (PSO).

Existence and Generation Mechanisms of Strange Nonchaotic Attractors in Axially Accelerating Beam Systems

The axially accelerating beam systems are widely used in engineering fields, which have abundant dynamical phenomena, though there are still very few results on the complex dynamics in such systems under quasiperiodic excitations. In this work, the Strange Nonchaotic Attractors (SNAs) in the quasiperiodic forced axially accelerating beam systems are studied. Several generation mechanisms (type-I intermittency, fractalization, torus collision) determining the transition routes from quasiperiodic attractors (torus attractors) to SNAs are established. The strange properties are verified by phase portraits, rational approximations, power spectra and so on, while the nonchaotic properties are checked by Lyapunov exponents. These results reveal new dynamical phenomena of the axially accelerating beam systems.

Strange Nonchaotic Attractors in Memristor-Based Shimizu–Morioka Oscillator

The existence of strange nonchaotic attractors (SNAs) is examined using a memristor-based Shimizu–Morioka oscillator. In particular, we analyze the birth of SNAs by varying one of the frequency of external quasi-periodic forces. The occurrence and its mechanism are further analyzed through rational approximation theory. Following that, we also perform different nonlinear dynamical characterizations, including Poincaré surface sections, singular continuous spectrum, separation of nearby trajectories, 0–1 test, as well as finite-time Lyapunov exponents for more precise validation. Finally, the experimental evidence for the occurrence of SNA is inspected through a real-time analog circuit in the laboratory.

STRANGE NONCHAOTIC ATTRACTORS IN A PERIODICALLY FORCED PIECEWISE LINEAR SYSTEM WITH NOISE

The study of strange nonchaotic attractors (SNAs) has been mainly restricted to quasiperiodically forced systems. At present, SNAs have also been uncovered in several periodically forced smooth systems with noise. In this work, we consider a periodically forced nonsmooth system and find that SNAs are created by a small amount of noise. SNAs can be generated in different periodic windows with weak noise perturbation. If the parameter is varied further from the chaotic range, a larger noise intensity is required to induce SNAs. Besides, noise-induced SNAs can be generated by the periodic attractors near the boundary crisis. In addition, with the increasing noise intensity, the intermittency between SNAs and periodic attractors can be induced by transient chaos. The characteristics of SNAs are analyzed by the Lyapunov exponent, power spectrum, singular continuous spectrum, spectral distribution functions, and finite time Lyapunov exponent.

Dynamical Analysis of Biological Signals with the 0–1 Test: A Case Study of the PhotoPlethysmoGraphic (PPG) Signal

The 0–1 test distinguishes between regular and chaotic dynamics for a deterministic system using a time series as a starting point without appealing to any state space reconstruction method. A modification of the 0–1 test allows for the determination of a more comprehensive range of signal dynamic behaviors, particularly in the field of biological signals. We report the results of applying the test and study with more details the PhotoPlethysmoGraphic (PPG) signal behavior from different healthy young subjects, although its use is extensible to other biological signals. While mainly used for heart rate and blood oxygen saturation monitoring, the PPG signal contains extensive physiological dynamics information. We show that the PPG signal, on a healthy young individual, is predominantly quasi-periodic on small timescales (short span of time concerning the dominant frequency). However, on large timescales, PPG signals yield an aperiodic behavior that can be firmly chaotic or a prior transition via an SNA (Strange Nonchaotic Attractor). The results are based on the behavior of well-known time series that are random, chaotic, aperiodic, periodic, and quasi-periodic.

Strange Nonchaotic Attractors From a Family of Quasiperiodically Forced Piecewise Linear Maps

In this paper, a family of quasiperiodically forced piecewise linear maps is considered. It is proved that there exists a unique strange nonchaotic attractor for some set of parameter values. It is the graph of an upper semi-continuous function, which is invariant, discontinuous almost everywhere and attracts almost all orbits. Moreover, both Lyapunov exponents on the attractor is nonpositive. Finally, to demonstrate and validate our theoretical results, numerical simulations are presented to exhibit the corresponding phase portrait and Lyapunov exponents portrait.

Coexistence of Strange Nonchaotic Attractors in a Quasiperiodically Forced Dynamical Map

In this paper, we investigate coexisting strange nonchaotic attractors (SNAs) in a quasiperiodically forced system. We also describe the basins of attraction for coexisting attractors and identify the mechanism for the creation of coexisting attractors. We find three types of routes to coexisting SNAs, including intermittent route, Heagy–Hammel route and fractalization route. The mechanisms for the creation of coexisting SNAs are investigated by the interruption of coexisting torus-doubling bifurcations. We characterize SNAs by the largest Lyapunov exponents, phase sensitivity exponents and power spectrum. Besides, the SNAs with extremely fractal basins exhibit sensitive dependence on the initial condition for some particular parameters.

Observation of Strange Nonchaotic Dynamics in the Frame of State-Controlled Cellular Neural Network-Based Oscillator

Abstract In this paper, the strange nonchaotic dynamics of a quasi-periodically driven state-controlled cellular neural network (SC-CNN) based on a simple chaotic circuit is investigated using hardware experiments and numerical simulations. We report here two different routes to strange nonchaotic attractors (SNAs) taken by this SC-CNN based circuit system. These routes were confirmed using rational approximation (RA) theory, finite time Lyapunov exponents, spectrum of the largest Lyapunov exponents and their variance, and phase sensitivity exponent. It is observed that the results from both computer simulations as well as laboratory experiments have spectacular resemblance.

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.

Investigation of Chaotic and Strange Nonchaotic Phenomena in Nonautonomous Wien-Bridge Oscillator with Diode Nonlinearity

We have studied the chaotic and strange nonchaotic phenomena of a simple quasiperiodically forced Wien bridge oscillator circuit with diode as the only nonlinearity in this electronic oscillator system responsible for various nonlinear behaviors. Both the experimental results and the numerical simulation results for their confirmation are provided to show the bifurcation process. Various measures used for the numerical confirmation of SNA are power spectrum, maximal Lyapunov exponent, path of translational variables, mean square displacement, projection of poincaré section, log-log plot, and autocorrelation function. Based upon the numerical results, the birth of SNAs has been identified in the band merging route, intermittency route, and blowout bifurcation route. In addition, the birth of SNAs has been analyzed with peculiar mechanism, namely, “0-1 Test” employing the one state dynamical variable.

Strange Nonchaotic Bursting in a Quasiperiodically-forced Hindmarsh-Rose Neuron

We study the transition from a silent state to a bursting state by varying the dc stimulus in the Hindmarsh-Rose neuron under quasiperiodic stimulation. For this quasiperiodically-forced case, a new type of strange nonchaotic (SN) bursting state is found to occur between the silent state and the chaotic bursting state. This is in contrast to the periodically-forced case where the silent state transforms directly to a chaotic bursting state. Using a rational approximation to the quasiperiodic forcing, we investigate the mechanism for the appearance of such an SN bursting state. Thus, a smooth torus (corresponding to a silent state) is found to transform to an SN bursting attractor through a phase-dependent subcritical period-doubling bifurcation. These SN bursting states, together with chaotic bursting states, are characterized in terms of the interburst interval, the bursting length, and the number of spikes in each burst. Both bursting states are found to be aperiodic complex ones. Consequently, aperiodic complex burstings may result from two dynamically different states with strange geometry (one is chaotic and the other is nonchaotic). Thus, in addition to chaotic burstings, SN burstings may become a dynamical origin for the complex physiological rhythms that are ubiquitous in organisms.

Strange nonchaotic oscillations in the quasiperiodically forced Hodgkin–Huxley neuron

We numerically study dynamical behaviors of the quasiperiodically forced Hodgkin–Huxley neuron and compare the dynamical responses with those for the case of periodic stimulus. In the periodically forced case, a transition from a periodic to a chaotic oscillation was found to occur via period doublings in previous numerical and experimental works. We investigate the effect of the quasiperiodic forcing on this period-doubling route to chaotic oscillation. In contrast to the case of periodic forcing, a new type of strange nonchaotic (SN) oscillating states (that are geometrically strange but have no positive Lyapunov exponents) is found to exist between the regular and chaotic oscillating states as intermediate ones. Their strange fractal geometry leads to aperiodic ‘complex’ spikings. Various dynamical routes to SN oscillations are identified, as in the quasiperiodically forced logistic map. These SN spikings are expected to be observed in experiments of the quasiperiodically forced squid giant axon.

Strange Nonchaotic Responses of the Quasiperiodically Forced Morris-Lecar Neuron

We study dynamical responses of the self-oscillating Morris-Lecar (ML) neuron under quasiperiodic stimulation. For the case of periodic stimulation on the self-oscillating ML neuron, a transition from a periodic to a chaotic oscillation occurs through period doublings. We investigate the effect of the quasiperiodic forcing on this period-doubling route to chaotic oscillation. In contrast to the periodically-forced case, a new type of strange nonchaotic (SN) oscillating states (that are geometrically strange but have no positive Lyapunov exponents) is thus found to appear between the regular and chaotic oscillating states. Strange fractal geometry of these SN oscillating states, which is characterized in terms of the phase sensitivity exponent and the distribution of local finite-time Lyapunov exponent, leads to aperiodic “complex” spikings. Diverse routes to SN oscillations are found, as in the quasiperiodically forced logistic map.

On the definition of strange nonchaotic attractor

We discuss the notion of Strange Nonchaotic Attractor widely considered in the literature, and formulate a precise definition of this notion that includes the well known models proposed by Grebogi et al [1] and Keller [2]; as well as a wide range of other similar examples proposed in the literature. We will pay special attention to the properties of the Lyapunov exponents and to the validity of the dierent methods of checking the nonchaoticity condition.

SNA’s in the Quasi-Periodic Quadratic Family

We rigorously show that there can exist Strange Nonchaotic Attractors (SNA) in the quasi-periodically forced quadratic (or logistic) map $$(\theta,x)\mapsto(\theta+\omega,c(\theta)x(1-x))$$ for certain choices of \({c:\mathbb{T} \to [3/2,4]}\) and Diophantine ω.

Strange Nonchaotic Attractor in a Nonautonomous Oscillatory System with Nonlinear Capacity

The mathematical model of a nonautonomous dynamic system with nonlinear capacity and four-dimensional phase space is numerically investigated. The behavior of the maximum characteristic Lyapunov parameter and attractor capacity, information, and correlation dimensions attendant to variations of the external force frequency is investigated. The existence of the strange nonchaotic attractor that retains the roughness property for a wide range of variations of the system parameters is established in the phase space. Results of field experiments that confirm the correctness of numerical calculations are presented.

Intermittent Strange Nonchaotic Attractors in Quasiperiodically Forced Systems

Intermittent strange nonchaotic attractors (SNAs) appear typically in quasiperiodically forced systems. As a basic model, we consider the quasiperiodically forced Henon map and investigate the mechanism for the intermittent transition to SNAs. Using rational approximations to the quasiperiodic forcing, it is shown that dynamical transition to an intermittent SNA occurs via a phase-dependent saddle-node bifurcation when a smooth torus collides with a new kind of invariant “ring-shaped” unstable set.