Nonchaotic Attractors Research Articles

Dynamical analysis of the Rulkov model with quasiperiodic forcing

This paper investigates the dynamical behavior of the Rulkov neuron model under quasiperiodic forcing, focusing on the emergence and mechanisms of strange nonchaotic attractors (SNAs). SNAs are characterized by fractal geometry without sensitive dependence on initial conditions, and their formation mechanisms, such as torus-doubling, fractal, bubble, Heagy–Hammel, and multi-intermittency routes, are explored in detail. The study employs Lyapunov exponents, singular continuous spectrum, and phase sensitivity to detect and characterize the strange and nonchaotic properties of attractors. Numerical simulations reveal the transition dynamics between strange nonchaotic, quasiperiodic, and chaotic states, emphasizing the role of transient dynamics in understanding the complex behavior of neural systems under multifrequency stimuli. The findings provide deeper insights into the intricate dynamical processes in neuron models, potentially aiding the development of advanced models for neural response prediction in varying environments.

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.

Birth of Catastrophe and Strange Attractors through Generalized Hopf Bifurcations in Covid-19 Transmission Mathematical Model

Coronavirus can be transmitted through the things that people carry or the things where it sticks to after being spread by the sufferer. Instead, various preventive measures have been carried out. We create a new mathematical model that represents Coronavirus that exists in non-living objects, susceptible, and infected subpopulations interaction by considering the Coronavirus transmission through non-living objects caused by susceptible and infected subpopulations along with its prevention to characterize the dynamics of Coronavirus transmission in the population under those conditions. One disease-free and two infection equilibrium points along with their local stability and coexistence are identified. Global stability of the disease-free equilibria and basic reproduction number are also investigated. Changes in susceptible-Coronavirus interaction rate generate Fold and Hopf bifurcations which represent the emergence of a cycle and the collision of two infection equilibrium points respectively. Catastrophe generated by the collision between an attractor and a repeller is found around a Generalized Hopf bifurcation point by changing susceptible-Coronavirus interaction rate and increasing rate of Coronavirus originating from infected subpopulation. It represents a momentary unpredictable dynamics as the effect of Coronavirus addition and infection. Non-chaotic strange attractors that represent complex but still predictable dynamics are also triggered by Generalized Hopf bifurcation when the susceptible-Coronavirus interaction rate and one of the following parameters, i.e. increasing rate of Coronavirus originating from infected subpopulation or infected subpopulation recovery rate vary.

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

Double grazing bifurcation route in a quasiperiodically driven piecewise linear oscillator.

Considering a piecewise linear oscillator with quasiperiodic excitation, we uncover the route of double grazing bifurcation of quasiperiodic torus to strange nonchaotic attractors (i.e., SNAs). The maximum displacement for double grazing bifurcation of the quasiperiodic torus can be obtained analytically. After double grazing of quasiperiodic orbits, the smooth quasiperiodic torus wrinkles increasingly with the continuous change of the parameter. Subsequently, the whole quasiperiodic torus loses the smoothness by becoming everywhere non-differentiable, which indicates the birth of SNAs. The Lyapunov exponent is adopted to verify the nonchaotic property of the SNA. The strange property of SNAs can be characterized by the phase sensitivity, the power spectrum, the singular continuous spectrum, and the fractal structure. Our detailed analysis shows that the SNAs induced by double grazing may exist in a short parameter interval between 1 T quasiperiodic orbit and 2 T quasiperiodic orbit or between 1 T quasiperiodic orbit and 4 T quasiperiodic orbit or between 1 T quasiperiodic orbit and chaotic motion. Noteworthy, SNAs may also exist in a large parameter interval after double grazing, which does not lead to any quasiperiodic or chaotic orbits.

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.

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.

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.

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.

Coexistence of chaotic and non-chaotic attractors in a three-species slow–fast system

The dynamical complexity of conceptual few-species systems has long been attracting considerable attention. In particular, significant attention has been paid to the three trophic level systems to demonstrate that even a baseline prey/predator/top-predator system can exhibit complex dynamics. However, previous studies left many open questions. In this paper, we consider a generalization of the Hastings–Powell model that include intraspecific competition among the predators and top predators. To enhance the ecological relevance of the model, we incorporate different timescales for different species. Specifically, we aim to study how the dynamics of the system is affected in the presence of multiple timescales along the trophic levels. We employ the geometric singular perturbation approach to analyze the change in species abundance over three different timescales, slow, fast, and intermediate. We analytically determine the entry and exit points from the vicinity of the critical manifold; the corresponding values act as critical thresholds for a sudden, fast transition in population density. We show that the properties of this generalized form of the Hastings–Powell model are dynamically more rich than it was observed in previous studies; particularly, the system can exhibit bi-stability or tri-stability in certain parameter intervals. The existence of a homoclinic orbit in a limiting subsystem (prey–predator) indicates period-doubling cascades to chaos in the full system (prey–predator–top predator). We further investigate the impact of the intraspecific competition of predators and top predators on the system’s dynamics. Strong intraspecific competition among the predators and top predators shows periodic coexistence of all species. In contrast, weak competition can drive the system to exhibit tri-stability, which includes the coexistence of chaotic and periodic (of different periods) attractors.

Rich complex dynamics in new fractional-order hyperchaotic systems using a modified Caputo operator based on the extended Gamma function

In this work, rich complex dynamics are observed in new fractional-order hyperchaotic systems when employing a modified Caputo-operator based on the extended Gamma function. Thus, the new parameter of the extended Gamma function provides higher degree of freedom that plays a vital role in the appearance (or disappearance) of chaotic attractors. Two numerical examples are provided to validate the proposed method. The numerical examples show that various chaotic, hyperchaotic and non-chaotic attractors are obtained when varying the new parameter of the modified Caputo-operator.In conclusion, this work shows that the modified Caputo-operator possesses higher degree of freedom than the classic one. This interesting phenomena result in obtaining rich complex dynamics in the systems that involve this new operator including phase transition from hyperchaotic to chaotic states and also phase transition from chaotic to non-chaotic states. Moreover, we show in the second numerical example that chaotic attractors exist only with the modified Caputo-operator using a specific choice of the system’s parameters, while the system’s counterpart does not show chaotic behaviors when using the classic Caputo-operator. Thus, the fractional systems with the modified Caputo-operator are shown to exhibit stronger chaotic dynamics than their counterparts with the old (classic) Caputo-operator.