Strange Nonchaotic Attractors Research Articles

Strange nonchaotic attractors and bifurcation of a four-wheel-steering vehicle system with driver steering control

In this paper, a dynamical model is developed for the four-wheel-steering(4WS)vehicles, with nonlinear lateral tyre forces and quasi-periodic disturbances from the steering mechanism and quasi-periodic pavement external deformation. Initially, the stability and Hopf bifurcation of the 4WS vehicle system without disturbance are analyzed employing the central manifold theory and projection method. The effects of parameters on the stability and the types of Hopf bifurcation for the vehicle system are also discussed. It is shown that both Hopf bifurcation and degenerate Hopf bifurcation occur in the vehicle system. The critical speed decreases with increased driver’s perceptual time delay and distance from the vehicle’s center of gravity to the front axles, while it increases with the control strategy coefficient. However, the increase of the frontal visibility distance of the driver leads to an initial increase of the critical speed, decreasing afterwards. Subsequently, the strange nonchaotic dynamical behaviour of the disturbed 4WS system is investigated. Several routes and mechanisms for the birth of strange nonchaotic attractors (SNAs) are identified, including the torus doubling bifurcation, the torus fractalization, and the loss of transverse stability of a torus. Finally, the nonchaotic property is verified through the calculation of the maximum Lyapunov exponent, and the strange property is characterized using power spectra and rational approximation.

Birth of Strange Non-chaotic Attractors in Fractional-Order Systems

In this study, we present the construction of a quasiperiodically driven fractional-order Chua system and investigate its dynamical behavior by varying the fractional order parameters q1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q_1$$\\end{document} and q2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q_2$$\\end{document} using various characterization techniques. Our observations suggest that selecting appropriate combinations of fractional orders allows us to observe bifurcation phenomena for different values of the forcing amplitude. Through the use of Poincaré sections, we visualize the formation of tori and their doubling bifurcation route to strange nonchaotic attractor. Additionally, we employ numerical and statistical tools, such as singular continuous spectra, separation of nearby trajectories, and fast Fourier transform, to verify the occurrence and understand the mechanism behind the birth of strange nonchaotic attractor. We differentiate torus, strange nonchaos, and chaos with the help of 0–1 test and recurrence analysis. Our results provide valuable insights into the dynamical behavior of quasiperiodically driven fractional-order systems, with potential implications for various fields of science and engineering.

Режимы обобщенной синхронизации грубых генераторов широкополосного сигнала при наличии частотных помех в канале связи

Possibility of the onset of generalized synchronization regimes and these applications for communication in the unidirectionally coupled wide-band generators in a presence of a frequency distortion in a communication channel is studied. There are considered three options of these generators: 1) with a robust hyperbolic chaotic attractor; 2) with a non-robust non-hyperbolic chaotic attractor; 3) with a robust strange non-chaotic attractor. Results are provided for three significant cases of frequency distortion: 1) removals in the high-frequency range; 2) removals in the low-frequency range; 3) frequency removals in the neighborhood of the main carrier frequency. Advantages of the hyperbolic and the strange non-chaotic signals for data transmission are demonstrated.

Strange nonchaotic attractors in a class of quasiperiodically forced piecewise smooth systems

Strange nonchaotic attractors in a class of quasiperiodically forced piecewise smooth systems

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.

On the probability of positive finite-time Lyapunov exponents on strange nonchaotic attractors

On the probability of positive finite-time Lyapunov exponents on strange nonchaotic attractors

Quantifying strange property of attractors in quasiperiodically forced systems

Quasiperiodically forced systems are an important class of dynamical systems exhibiting quasiperiodic, strange nonchaotic, and chaotic attractors. A major concern is the identification of the parameter range in which each one of the attractors is present. In this work, based on the phase sensitivity proposed by Pikovsky and Feudel, we define a measure to quantitatively distinguish quasiperiodic attractors, strange nonchaotic attractors, and chaotic attractors. Particularly, we can determine the boundary points of these three attractors in parameter space. The reliability of this measure is verified in smooth and non-smooth systems.

Infinite strange non-chaotic attractors in a non-autonomous jerk system

Infinite strange non-chaotic attractors in a non-autonomous jerk system

Border-collision bifurcation route to strange nonchaotic attractors in the piecewise linear normal form map

A new route to strange nonchaotic attractors (SNAs) is investigated in a quasiperiodically driven nonsmooth map. It is shown that the smooth quasiperiodic torus becomes nonsmooth (continuous and non-differentiable) due to the border-collision bifurcation of the torus. As the coefficients change, the nonsmooth torus gets gradually fractal, forming strange nonchaotic attractors. It is termed the border-collision bifurcation route to SNAs. A novel feature of this route is that a large number of SNAs are formed, and the parameter area of SNAs makes up about 40 % of the given parameter regions. These SNAs are identified by the largest Lyapunov exponents and the phase sensitivity exponents. They are also characterized by the distribution of finite-time Lyapunov exponents. Unlike other types of SNAs, the distribution of finite-time Lyapunov exponents has its maximum at a relatively small negative Lyapunov exponent, which contributes largely to lead to the abundance of SNAs.

Generation mechanisms of strange nonchaotic attractors and multistable dynamics in a class of nonlinear economic systems

In this paper, we study strange nonchaotic attractors (SNAs) and multistable dynamics in a class of nonlinear economic systems. For quasiperiodically forced case, the generation and evolution mechanisms of SNAs are discussed. The fractal, Heagy–Hammel, torus doubling, and intermittency routes to SNAs are identified. The Lyapunov exponent, phase-sensitive function and power spectrum are used to characterize the dynamical and geometrical properties of SNAs. Moreover, when multistable phenomenon occur in the system, the boundaries of the basin of attraction may become intertwined, which leads to the economic unpredictability in the long run.

Hidden strange nonchaotic dynamics in a non-autonomous model

We report the existence of a hidden strange nonchaotic attractor (SNA) in the system devoid of equilibrium. To identify the hidden SNAs, we have added a two frequency sinusoidal driving. The resultant model exhibits smooth basin of attraction with extended bounded dynamics. Further, we found two prominent routes namely, gradual fractalization and Heagy–Hammel for the emergence of hidden SNAs, for two different parametric regimes. The contemplated hidden SNA is characterized through Lyapunov exponents, spectral measures and phase sensitivity exponent. Besides, we incorporated the recurrence quantification analysis to investigate the onset of hidden SNAs in the chosen model. We also provide experimental evidence for the observation through electronic analog simulation which agrees with numerical simulations. We propose this approach as a generic method to witness hidden SNAs in any model without equilibria.

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.

Continuous irregular dynamics with multiple neutral trajectories permit species coexistence in competitive communities

The colonization model formulates competition among propagules for habitable sites to colonize, which serves as a mechanism enabling coexistence of multiple species. This model traditionally assumes that encounters between propagules and sites occur as mass action events, under which species distribution can eventually reach an equilibrium state with multiple species in a constant environment. To investigate the effects of encounter mode on species diversity, we analyzed community dynamics in the colonization model by varying encounter processes. The analysis indicated that equilibrium is approximately neutrally-stable under perfect ratio-dependent encounter, resulting in temporally continuous variation of species’ frequencies with irregular trajectories even under a constant environment. Although the trajectories significantly depend on initial conditions, they are considered to be “strange nonchaotic attractors” (SNAs) rather than chaos from the asymptotic growth rates of displacement. In addition, trajectories with different initial conditions remain different through time, indicating that the system involves an infinite number of SNAs. This analysis presents a novel mechanism for transient dynamics under competition.

Comparative analysis of the secure communication schemes based on the generators of hyperbolic strange attractor and strange nonchaotic attractor

The purpose of this work is to analyse qualitative features of the information transmission process via several communication schemes based on the synchronization of transmitter and receiver, both being complex signal generators. For this purpose generators of the hyperbolic chaos and generators with the strange nonchaotic attractor are employed. Evaluation of advantages and disadvantages of such schemes is made comparing themselves with each other as well as with schemes based on the nonhyperbolic chaotic generators. Methods. The power spectra and the distributions of the largest finite-time Lyapunov exponent are used to confirm the complexity of the dynamics of the generators in use and to verify the wide-bandness, robustness and stochasticity of their signals. Confidentiality of the informational signal transmission is achieved using its nonlinear mixing to the dynamics of the transmitter. The special phase mixing is used since the model generators employed for the research demonstrate nontrivial dynamics for the angular variable — oscillations phase shift. The digital image is used as an information for transmission. Visual control during the transmission process allows to carry out the qualitative analysis of the success of the signal coding and its detecting by the receiver. Results. Successful transmission and decoding of information for all schemes under investigation are demonstrated for the case of identical transmitter and receiver. Parameter detuning of these generators leads to difficulties in separation of the informational signal from the chaotic/complex carrier due to loss of the full synchronization. For the nonhyperbolic chaos detuning of the parameter responsible for the amplitude of the signal leads to the bad quality of the detection while frequency detuning makes detection absolutely impossible. Schemes with the hyperbolic chaos and strange nonchaotic dynamics appear to demonstrate much better results. The information detection is much better in this case because of the robustness of the generalized synchronization. Conclusion. Robust chaotic and complex nonchaotic generators appear to have significant advantages for communication systems comparing to the chaotic generators of nonhyperbolic type.

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.

Birth of strange nonchaotic attractors in a piecewise linear oscillator.

Nonsmooth systems are widely encountered in engineering fields. They have abundant dynamical phenomena, including some results on the complex dynamics in such systems under quasiperiodically forced excitations. In this work, we consider a quasiperiodically forced piecewise linear oscillator and show that strange nonchaotic attractors (SNAs) do exist in such nonsmooth systems. The generation and evolution mechanisms of SNAs are discussed. The torus-doubling, fractal, bubbling, and intermittency routes to SNAs are identified. The strange properties of SNAs are characterized with the aid of the phase sensitivity function, singular continuous spectrum, rational frequency approximation, and the path of the partial Fourier sum of state variables in a complex plane. The nonchaotic properties of SNAs are verified by the methods of maximum Lyapunov exponent and power spectrum.

Manifestation of strange nonchaotic attractors in extended systems: a study through out-of-time-ordered correlators

We study the spatial spread of out-of-time-ordered correlators (OTOCs) in coupled map lattices (CMLs) of quasiperiodically forced nonlinear maps. We use instantaneous speed (IS) and finite-time Lyapunov exponents (FTLEs) to investigate the role of strange non-chaotic attractors (SNAs) on the spatial spread of the OTOC. We find that these CMLs exhibit a characteristic on and off type of spread of the OTOC for SNA. Further, we provide a broad spectrum of the various dynamical regimes in a two-parameter phase diagram using IS and FTLEs. We substantiate our results by confirming the presence of SNA using established tools and measures, namely the distribution of finite-time Lyapunov exponents, phase sensitivity, spectrum of partial Fourier sums, and $0-1$ test.

Strange nonchaotic dynamics in a discrete FitzHugh-Nagumo neuron model with sigmoidal recovery variable.

We report the appearance of strange nonchaotic attractors in a discrete FitzHugh-Nagumo neuron model with discontinuous resetting. The well-known strange nonchaotic attractors appear in quasiperiodically forced continuous-time dynamical systems as well as in a discrete map with a small intensity of noise. Interestingly, we show that a discrete FitzHugh-Nagumo neuron model with a sigmoidal recovery variable and discontinuous resetting generates strange nonchaotic attractors without external force. These strange nonchaotic attractors occur as intermittency behavior (locally unstable behavior in laminar flow) in the periodic dynamics. We use various characterization techniques to validate the existence of strange nonchaotic attractors in the considered system.

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.