Quasi-periodic Attractors Research Articles

Scale-Free Chaos in the 2D Harmonically Confined Vicsek Model.

Animal motion and flocking are ubiquitous nonequilibrium phenomena that are often studied within active matter. In examples such as insect swarms, macroscopic quantities exhibit power laws with measurable critical exponents and ideas from phase transitions and statistical mechanics have been explored to explain them. The widely used Vicsek model with periodic boundary conditions has an ordering phase transition but the corresponding homogeneous ordered or disordered phases are different from observations of natural swarms. If a harmonic potential (instead of a periodic box) is used to confine particles, then the numerical simulations of the Vicsek model display periodic, quasiperiodic, and chaotic attractors. The latter are scale-free on critical curves that produce power laws and critical exponents. Here, we investigate the scale-free chaos phase transition in two space dimensions. We show that the shape of the chaotic swarm on the critical curve reflects the split between the core and the vapor of insects observed in midge swarms and that the dynamic correlation function collapses only for a finite interval of small scaled times. We explain the algorithms used to calculate the largest Lyapunov exponents, the static and dynamic critical exponents, and compare them to those of the three-dimensional model.

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.

Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties

The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system.

High-dimensional memristive neural network and its application in commercial data encryption communication

With the development of economic globalization, demands for digital business data sharing among different subsidiary companies are constantly emerging. How to achieve secure transmission of commercial data in the Internet of Things (IoT) is a challenging task. In this paper, we propose a memristive neural network-based method for encrypted communication of commercial data. First, a high-dimensional memristive Hopfield neural network (MMHNN) model with three memristive synapses and seven neurons is proposed and its complex dynamical behaviors are deeply analyzed. The results of theoretical analysis and numerical simulations show that the proposed MMHNN can generate quasi-periodic, periodic, chaotic and hyperchaotic attractors, as well as some controllable 1-direction (1D), 2D, 3D hyperchaotic multi-scroll attractors and other types of parametric regulatory dynamics and initial value regulation dynamics. In addition, an analog equivalent circuit for the MMHNN is constructed, and the circuit simulations experimental results demonstrate the correctness and feasibility of the multi-scroll attractors phenomena. Finally, a novel MMHNN-based encryption scheme for IoT commercial data images is proposed by applying hyperchaotic spatial multi-scroll attractors, and the performance analysis shows that the proposed system achieves good encryption performances, especially in terms of keyspace, information entropy, correlation and differential attacks. The hardware experiments further verify the effectiveness and superiority of the proposed commercial data encryption scheme based on the MMHNN.

Accurate Computations up to Breakdown of Quasi-Periodic Attractors in the Dissipative Spin–Orbit Problem

We consider a Celestial Mechanics model: the spin–orbit problem with a dissipative tidal torque, which is a singular perturbation of a conservative system. The goal of this paper is to show that it is possible to maintain the accuracy and reliability of the computation of quasi-periodic attractors for parameter values extremely close to the breakdown and, therefore, it is possible to obtain information on the breakdown mechanism of these quasi-periodic attractors. The method uses at the same time numerical and rigorous improvements to provide (i) a very accurate computation of the time-1 map of the spin–orbit problem (which reduces the dimensionality of the problem); (ii) a very efficient KAM method for maps which computes the attractor and its tangent spaces (by quadratically convergent, low storage requirements, and low operation count); (iii) explicit algorithms backed by a rigorous a posteriori KAM theorem, which establishes that if the algorithm is successful and produces a small residual, then there is a true solution nearby; and (iv) guaranteed algorithms to reach arbitrarily close to the border of existence as long as there are enough computer resources. As a by-product of the accuracy that we maintain till breakdown, we study several scale-invariant observables of the tori used in the renormalization group of infinite-dimensional spaces. In contrast with previously studied simple models, the behavior at breakdown of the spin–orbit problem does not satisfy standard scaling relations which implies that the spin–orbit problem is not described by a hyperbolic fixed point of the renormalization operator.

A Novel Four-Dimensional Memristive Hyperchaotic Map Based on a Three-Dimensional Parabolic Chaotic Map with a Discrete Memristor

Recently, the application of memristors in chaotic systems has been extensively studied. Unfortunately, there is limited literature on the introduction of discrete memristors into chaotic maps, especially into non-classical multidimensional maps. For this reason, this paper establishes a new three-dimensional parabolic chaotic map model; in order to improve the complexity and randomness of the map, it is coupled with a square-charge-controlled discrete memristor to design a new four-dimensional memristive hyperchaotic map. Firstly, the stability of the two maps is discussed. And their dynamical properties are compared using Lyapunov exponential spectra and bifurcation diagrams. Then, the phase diagram and iteration sequence of the 4D memristive hyperchaotic map are obtained. Meanwhile, we investigate the hyperchaotic states, the transient chaos, state transfer and attractor coexistence phenomena of the four-dimensional memristive map. In particular, the special state transfer phenomenon of switching from a periodic attractor to a quasi-periodic attractor and the special coexistence phenomenon of a quasi-periodic attractor coexisting with a quasi-periodic attractor around fixed points are found, which have not been observed in other systems. Finally, the phase-track diagrams and iterative sequence diagrams of the four-dimensional memristive map are verified on a digital experimental platform, revealing its potential for practical applications.

Slow–Fast Dynamics of a Coupled Oscillator with Periodic Excitation

We study the influence of the coexisting steady states in high-dimensional systems on the dynamical evolution of the vector field when a slow-varying periodic excitation is introduced. The model under consideration is a coupled system of Bonhöffer–van der Pol (BVP) equations with a slow-varying periodic excitation. We apply the modified slow–fast analysis method to perform a detailed study on all the equilibrium branches and their bifurcations of the generalized autonomous system. According to different dynamical behaviors, we explore the dynamical evolution of existing attractors, which reveals the coexistence of a quasi-periodic attractor with diverse types of bursting attractors. Further investigation shows that the coexisting steady states may cause spiking oscillations to behave in combination of a 2D torus and a limit cycle. We also identify a period-2 cycle bursting attractor as well as a quasi-periodic attractor according to the period-2 limit cycle.

On the emergence of secondary tones in airfoil noise

We present the results of direct numerical simulations of a NACA 0012 airfoil, with Mach number 0.3 and angle of attack of $3^\circ$ , examining the dynamics of the flow with increasing Reynolds numbers. Two-dimensional simulation results are obtained with chord-based Reynolds numbers in the range $3.2 \times 10^3 \leq Re \leq 2.70 \times 10^4$ , where each simulation uses the last time step of the previous one as a starting point, to capture the evolution of dynamics as a function of $Re$ . The development of the pressure fluctuations with time shows a transition from periodic to quasi-periodic attractor for $2.38 \times 10^4 \leq Re \leq 2.42 \times 10^4$ , leading to the emergence of secondary tones in the wall and acoustic field pressure spectra, different from peaks related to the fundamental frequency $f_1$ and the respective harmonics; a second, incommensurate frequency $f_2$ appears, leading to several secondary tones with frequency $af_1 + bf_2$ , with $a$ and $b$ integers. Further increase of the Reynolds number leads to the emergence of a tertiary frequency, $f_3$ , indicating a route to chaos of the Ruelle–Takens–Newhouse type. Such a mechanism is related to the ladder-type characteristic structure of the tones, indicating that dynamic systems theory is an important tool for understanding airfoil tonal noise.

Vibration Analysis of a Nonlinear Absorber Coupled to a Hand-Held Impact Machine

Abstract Exposure of human hands to harmful levels of vibration can lead to severe injuries. To attenuate these hand transmitted vibrations, a nonlinear vibration absorber inerter (NVAI) is proposed in this work. The proposed NVAI is attached to a coupled nonlinear system of a hand-held impact machine and a hand-arm system. The combined coupled nonlinear system is modeled as a lumped parameter model with a combination of cubic and linear stiffness components, linear viscous dampers, and lumped masses. The governing equations of motion are solved analytically using the method of harmonic balance and validated using numerical simulations. A numerical bifurcation diagram of the system reveals the existence of complex solutions such as quasi-periodic and chaotic attractors. The appearances of quasi-periodic and chaotic attractors are later confirmed by Lyapunov exponents. Further, we explore the ability of the proposed NVAI to decrease the area corresponding to unstable quasi-periodic and chaotic motion in the excitation amplitude–frequency space. This observation further implies the delay in the onset of quasi-periodic and chaotic motion for a range of forcing amplitude using Lyapunov exponents. Finally, parametric analyses are carried out to identify the critical design parameters of the NVAI. These analyses reveal that an increase in the damping, mass, and inertance of the absorber ameliorates the performance of the NVAI. Furthermore, the critical value of the external excitation, corresponding to a sudden change in the response of the system, can be controlled using an appropriate selection of absorber parameters.

A 3D chaotic system with multi-transient behavior and its application in image encryption

Based on the special structure of the variable-boostable chaotic system VB19, that includes a smooth secondary flux-controlled memristor, a three-dimensional memristive chaotic system with only one equilibrium point is constructed. Numerical analysis reveals that the system has novel dynamical characteristics of controlling multiple types of attractors by a single parameter. At the same time, the existence of multiple transient behaviors and the coexistence of periodic and quasi-periodic attractors significantly increase the complexity of the system. Moreover, the system has a large range of high complexity phenomena, which is relatively rare in low-dimensional chaotic systems. A circuit is designed and implemented for this system in the physical circuit. Finally, this memristive system is applied to image encryption, whose excellent performance is demonstrated by key space, key sensitivity analysis, histogram analysis, and correlation coefficient analysis, which provides a reference for the further research of low dimensional memristive systems in the field of image encryption.

Complex dynamics in an unexplored simple model of the peroxidase-oxidase reaction.

A previously overlooked version of the so-called Olsen model of the peroxidase-oxidase reaction has been studied numerically using 2D isospike stability and maximum Lyapunov exponent diagrams and reveals a rich variety of dynamic behaviors not observed before. The model has a complex bifurcation structure involving mixed-mode and bursting oscillations as well as quasiperiodic and chaotic dynamics. In addition, multiple periodic and non-periodic attractors coexist for the same parameters. For some parameter values, the model also reveals formation of mosaic patterns of complex dynamic states. The complex dynamic behaviors exhibited by this model are compared to those of another version of the same model, which has been studied in more detail. The two models show similarities, but also notable differences between them, e.g., the organization of mixed-mode oscillations in parameter space and the relative abundance of quasiperiodic and chaotic oscillations. In both models, domains with chaotic dynamics contain apparently disorganized subdomains of periodic attractors with dinoflagellate-like structures, while the domains with mainly quasiperiodic behavior contain subdomains with periodic attractors organized as regular filamentous structures. These periodic attractors seem to be organized according to Stern-Brocot arithmetics. Finally, it appears that toroidal (quasiperiodic) attractors develop into first wrinkled and then fractal tori before they break down to chaotic attractors.

Existence of Self-Excited and Hidden Attractors in the Modified Autonomous Van Der Pol-Duffing Systems

This study investigates the multistability phenomenon and coexisting attractors in the modified Autonomous Van der Pol-Duffing (MAVPD) system and its fractional-order form. The analytical conditions for existence of periodic solutions in the integer-order system via Hopf bifurcation are discussed. In addition, conditions for approximating the solutions of the fractional version to periodic solutions are obtained via the Hopf bifurcation theory in fractional-order systems. Moreover, the technique for hidden attractors localization in the integer-order MAVPD is provided. Therefore, motivated by the previous discussion, the appearances of self-excited and hidden attractors are explained in the integer- and fractional-order MAVPD systems. Phase transition of quasi-periodic hidden attractors between the integer- and fractional-order MAVPD systems is observed. Throughout this study, the existence of complex dynamics is also justified using some effective numerical measures such as Lyapunov exponents, bifurcation diagrams and basin sets of attraction.

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.

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.

Dynamics of a charged Thomas oscillator in an external magnetic field

In this letter, we provide a detailed numerical examination of the dynamics of a charged Thomas oscillator in an external magnetic field. We do so by adopting and then modifying the cyclically symmetric Thomas oscillator to study the dynamics of a charged particle in an external magnetic field. These dynamical behaviours for weak and strong field strength parameters fall under two categories; conservative and dissipative. The system shows a complex quasi-periodic attractor whose topology depends on initial conditions for high field strengths in the conservative regime. There is a transition from adiabatic motion to chaos on decreasing the field strength parameter. In the dissipative regime, the system is chaotic for weak field strength and weak damping but shows a limit cycle for high field strengths. Such behaviour is due to an additional negative feedback loop that comes into action at high field strengths and forces the system dynamics to be stable in periodic oscillations. For weak damping and weak field strength, the system dynamics mimic Brownian motion via chaotic walks. We claim that the modified Thomas oscillator is a prototypical model to understand the dynamics of an active particle.

Hyperchaotic Oscillation and Multistability in a Fourth Order Smooth Chua System with Implementation Using No Analog Multipliers

Abundant dynamical behaviors and succinct realization forms are of great significance for nonlinear circuits and systems. Hyperchaos represents more complex oscillation mode compared with chaos, while multistability means more complex essential features compared with robustness. And the coexistence of the two is rare but interesting. Based on the classic Chua model, a fourth order smooth Chua system is constructed through utilizing linear feedback control method in this article. The introduction of hyperbolic sine nonlinearity remarkably simplifies the physical realization form for the new system. The two interesting nonlinear phenomena of hyperchaotic oscillation and multistability are discovered by adjusting system parameters. Dynamic analyses including bifurcation diagrams and Lyapunov exponent spectra comprehensively reveal the evolution of oscillatory behaviors brought by the variation of system parameters, whereupon more attractors such as periodic attractors, chaotic attractors in different oscillation scales, coexisting quasi-periodic attractors, coexisting multiple attractors and coexisting periodic attractors are found. And the attractor morphologies of this system are rare and significantly different from those in the reported Chua circuits and systems. Hardware circuit experiments without using analog multipliers are performed from a four-layer printed circuit board, which efficiently verify the simulated dynamical behaviors within the acceptable deviation.

A novel four-element RCLM hyperchaotic circuit based on current-controlled extended memristor

Due to complex frequency-driven and amplitude-driven nonlinear characteristics, various memristor emulators have been used for design of hyperchaotic oscillators and their security communication applications. It is essential to design the simplest possible oscillator with rich chaotic and hyperchaotic attractors for convenience of practical use. In this paper, only using one capacitor, one inductor, two resistors, and two diodes, a novel passive current-controlled second-order generalized memristor is presented. The model state equation and frequency/amplitude-driven voltage–current characteristics loci of the memristor are studied, and a simple four-element RCLM hyperchaotic circuit is constructed. The symmetry, dissipativity, stability of equilibrium points and dynamical behaviors are analyzed using nonlinear system theory and numerical tools. Various periodic, quasiperiodic, chaotic, and hyperchaotic attractors are coined. Finally, circuit simulations are executed and the results validate the correctness of the theoretical analysis.

Adaptively Adjusted Beliefs About Prices in an Evolutive Muthian Cobweb Model

In the present work, starting from the 1D evolutionary Muthian cobweb model in [Hommes & Wagener, 2010], where the economy is populated by biased firms and unbiased fundamentalist firms, we investigate the effects generated by adaptively adjusted beliefs on the long-period price of the goods produced. As in [Hommes & Wagener, 2010], we focus on the case in which the Muthian model is globally stable in the sense of Guesnerie [2002], stable under naive expectations. Coupling into the details, we first assume that just biased producers, being aware of the systematic errors they make in their forecasts, partially rely on an adaptive adjustment of beliefs, obtaining a 3D model, which has a unique steady state. Although in [Hommes & Wagener, 2010] the fundamental steady state is always stable and at most it can coexist with a period-two cycle, we show that the complexity degree increases in our case, where on increasing the reactivity of the evolutionary mechanism, there may be up to two stability thresholds for the steady state, both corresponding to flip bifurcations, and the equilibrium can coexist with a quasiperiodic attractor. On checking the robustness of such outcomes, we contrast them with those we find for the 4D setting, obtained by assuming that unbiased fundamentalists update their belief about the fundamental value partially relying on the same adaptive adjustment mechanism used by biased producers. The outcomes are similar to those of the 3D framework. For instance, in both cases, there exists just the fundamental equilibrium, which can coexist with a quasiperiodic attractor.

Multistable dynamics in a Hopfield neural network under electromagnetic radiation and dual bias currents

This paper investigates a Hopfield neural network under the simulation of external electromagnetic radiation and dual bias currents, in which the fluctuation of magnetic flux across the neuron membrane is used to emulate the influence of electromagnetic radiation. Utilizing conventional analytical methods, the basic properties of the proposed Hopfield neural network are discussed. Due to the addition of electromagnetic radiation and dual bias currents, the Hopfield neural network shows high sensitivity to system parameters and initial conditions. The proposed Hopfield neural network possesses multistability with periodic attractor, quasi-periodic attractor, chaotic attractor and transient chaotic attractor, and all of the attractors are hidden attractors because there is no equilibrium point in the system. In particular, when the neuron membrane magnetic flux is different, the system can present transient chaos with different chaotic times. More interestingly, with the change of system parameters, the proposed Hopfield neural network can exhibit parallel bifurcation behaviors. Finally, the Multisim simulation and hardware experiment results based on discrete electronic components are conducted to support the numerical ones. These results could give useful information to the study of nonlinear dynamic characteristics of the Hopfield neural network.

Nonlinear oscillations of a collapsible tube subjected to unsteady external pressure

The non-linear dynamics of an extremely thin-walled collapsible tube with internal flow subjected to a time-varying external pressure are studied experimentally and theoretically. For the constant chamber pressure case, we observe the existence of a fixed-point attractor, period-1 attractor, and quasiperiodic attractor. The period-1 limit cycle oscillations are essentially relaxation oscillations with up-down asymmetry in the time domain, and as the Reynolds number increases, the asymmetry becomes greater. With the forcing (varying chamber pressure), the system has no fixed points; its response can be period-n, quasiperiodic, or chaotic, depending upon the Reynolds number, driving amplitude, and frequency. For the forced system, at a low Reynolds number, the external forcing dominates the self-excited oscillations and symmetric oscillations are observed; at a higher Reynolds number, the reverse is true. In experiments and theory, aperiodic oscillations for the forced system are always observed in regimes beyond the Hopf bifurcation point of the unforced system. Distended and collapsed cases, under forcing, exhibit only 1:1 synchronous oscillation. These suggest that a natural oscillation timescale of the system must be present for the external forcing to induce aperiodicity. In the experiments, the forced system exhibits signs of quasiperiodic route to chaos at lower driving amplitude, while period-doubling route to chaos at higher driving amplitude. When the system is forced near its natural frequency, an aperiodic response is totally suppressed.