Chaotic Windows Research Articles

Research on impact vibration response of hinged fluid-conveying pipe with bilateral gap constraints

Factors such as loose supports or barriers over the fluid-conveying pipe may cause bilateral gap constraints. Flow-induced vibration will occur during pumping, which leads to collision between the pipe and the constraints. A piecewise linear spring model with large stiffness is employed to simulate the collision process between the pipe and the constraints, and numerical simulation is carried out for the collision of hinged fluid-conveying pipe with the bilateral gap constraints. The dynamic characteristics of hinged fluid-conveying pipe with pulsating internal flow excitation and bilateral gap constraints during impact vibration are studied. Influences of various parameters including frequency, average flow rate of the pulsed internal flow excitation, position of the bilateral constraints, and gap dimensions on the system are investigated. Three paths are found for the pipe system entering chaotic window from stable periodic motion: entering chaotic window after the occurrence of period-double bifurcation, transitioning into chaotic window after developing almost periodic motion, and jumping into chaotic window directly. If the distance between two constraints is closer, the pipe between the constraints will be subject to stronger constraints during the collision process, which can cause the system to enter chaos easily. Many dynamic phenomena have been observed, such as a sliding bifurcation when periodic incomplete chattering-impact developing periodic complete chattering-impact, strong centrosymmetric properties in the bifurcation diagram and the Poincaré map at the mid-point of the pipe, jumping between stable periods, period-double vibration response, inverse period-double vibration response, grazing motion, and viscous motion.

Complete synchronization of three-layer Rulkov neuron network coupled by electrical and chemical synapses.

This paper analyzes the complete synchronization of a three-layer Rulkov neuron network model connected by electrical synapses in the same layers and chemical synapses between adjacent layers. The outer coupling matrix of the network is not Laplacian as in linear coupling networks. We develop the master stability function method, in which the invariant manifold of the master stability equations (MSEs) does not correspond to the zero eigenvalues of the connection matrix. After giving the existence conditions of the synchronization manifold about the nonlinear chemical coupling, we investigate the dynamics of the synchronization manifold, which will be identical to that of a synchronous network by fixing the same parameters and initial values. The waveforms show that the transient chaotic windows and the transient approximate periodic windows with increased or decreased periods occur alternatively before asymptotic behaviors. Furthermore, the Lyapunov exponents of the MSEs indicate that the network with a periodic synchronization manifold can achieve complete synchronization, while the network with a chaotic synchronization manifold can not. Finally, we simulate the effects of small perturbations on the asymptotic regimes and the evolution routes for the synchronous periodic and the non-synchronous chaotic network.

A non-autonomous approach to study the impact of environmental toxins on nutrient-plankton system

The interaction between phytoplankton and zooplankton has a significant impact on the marine ecology. The interplay between these two species is the building blocks for most of the food webs operating in an aquatic ecosphere. The environmental toxins released by different external sources also affect the phytoplankton-zooplankton dynamics. In the present study, we propose a model to explore the kinetics of a nutrient-phytoplankton-zooplankton-environmental toxins (NPZT) system. The defence mechanism of phytoplankton against zooplankton is reflected through modified Holling type IV response, whereas the consumption of nutrients by phytoplankton is outlined by Holling type II response. The external toxins are assumed to have the capability of reducing the birth rate of phytoplankton species after coming into contact with their cells. To make our model more pragmatic, seasonal variation in the parameters is also taken into account. Firstly, we do the analysis related to the autonomous model (non-seasonal) like; its boundedness, existence of equilibrium points, their stability analysis, and occurrence of Hopf-bifurcation. Further, for the non-autonomous model (seasonal), we analyze the existence of positive periodic solution and its global stability. Through numerical simulations, we observe that for the non-seasonal model, increasing the rate of suppressing phytoplankton’s growth by environmental toxin, and rate at which environmental toxin is added to system make it unstable through Hopf-bifurcation. These oscillations can be removed by raising phytoplankton’s inhibitory effect against zooplankton, and this increment also leads to the extinction of the zooplankton population, making zooplankton free equilibrium a stable one. Both models, non-seasonal as well as seasonal manifest different types of multistability, and this is an exciting character associated with non-linear models. We also note that the inclusion of seasonality in our system promotes the coexistence of all populations. Further, through numerical simulations, we show that making some of the parameters seasonal can cause the emergence of chaos in the system. To verify chaos, we sketch the Poincaré map and evaluate the maximum Lyapunov exponent. The seasonal model also shows the switching of stability through different periodic and chaotic windows on varying the maximum intrinsic growth rate for phytoplankton, and contact rate between environmental toxin and phytoplankton. To substantiate our results, we picture several time-series graphs, basins of attraction, one and two-parametric bifurcation diagrams. Thus we expect that the present work can assist biologists and mathematicians in studying nutrient-plankton systems in a more detailed and realistic manner. This study can also help researchers in the estimation of non-seasonal as well as seasonal parameters while studying these types of complex non-linear models. Therefore, the present work seems to be enriched from a mathematical and biological point of view.

Scale-free chaos in the confined Vicsek flocking model.

The Vicsek model encompasses the paradigm of active dry matter. Motivated by collective behavior of insects in swarms, we have studied finite-size effects and criticality in the three-dimensional, harmonically confined Vicsek model. We have discovered a phase transition that exists for appropriate noise and small confinement strength. On the critical line of confinement versus noise, swarms are in a state of scale-free chaos characterized by minimal correlation time, correlation length proportional to swarm size and topological data analysis. The critical line separates dispersed single clusters from confined multicluster swarms. Scale-free chaotic swarms occupy a compact region of space and comprise a recognizable "condensed" nucleus and particles leaving and entering it. Susceptibility, correlation length, dynamic correlation function, and largest Lyapunov exponent obey power laws. The critical line and a narrow criticality region close to it move simultaneously to zero confinement strength for infinitely many particles. At the end of the first chaotic window of confinement, there is another phase transition to infinitely dense clusters of finite size that may be termed flocking black holes.

Chordwise flexible aft-tail suppresses jet-switching by reinstating wake periodicity in a flapping foil

The effect of a chordwise flexible aft-tail of a rigid heaving aerofoil on the dynamical transitions of the trailing-wake is studied here. The two-way coupled fluid–solid dynamics is simulated using an in-house fluid–structure interaction (FSI) platform, comprising a discrete forcing immersed boundary method based incompressible Navier–Stokes solver, weakly coupled with a finite difference method based structural solver. The FSI dynamics is studied in comparison to the corresponding rigid tail configuration. For the latter, mild jet-switching due to quasi-periodic movement of the wake vortices gives way to vigorous jet-switching as the dynamics transitions to a state of intermittency, where the quasi-periodic behaviour gets interspersed with chaotic windows. Introduction of a moderately flexible tail regularises this intermittent dynamics, eliminating jet-switching. The wake exhibits a deflected reverse Kármán pattern with fluctuating angles, governed by quasi-periodicity. With a highly flexible tail (very low rigidity), the wake shows almost a symmetric reverse Kármán street as periodicity is restored. Flexibility of the aft-tail is next controlled by changing its length, and flow is regularised and periodicity retained for moderate rigidity for increased length. Different dynamical states are established through robust nonlinear dynamical tools. The underlying flow-field behaviour, instrumental in suppressing the jet-switching phenomenon, is identified through a detailed investigation of the near-field vortex interactions dictated by the dynamics. A suite of measures has also been derived from the unsteady flow field to quantify the interactions of the key near-field vortices with a view to understanding the mechanism of switching and its subsequent suppression through flexibility.

Effect of Double-Frequency Excitation on a Fractional Model of Cerebral Aneurysm

This paper investigates the effect of double-frequency excitation on a fractional cerebral aneurysm model at the circle of Willis, where two kinds of external excitation are caused by the heart and arrhythmia respectively. Firstly, numerical analysis is performed on the proposed fractional model by using bifurcation diagrams and phase diagrams, and it is found that the periodic solution windows and chaotic solution windows switch more frequently than that resulted from single-frequency excitation, as a consequence of simultaneous resonances. Then, by using the method of multiple scales, the primary resonance response equation is obtained, and the effects of the system parameters on the amplitude of the fractional cerebral aneurysm model is analytically investigated. It is shown that double-frequency excitation changes the vibration patterns of the system under single-frequency excitation, and makes the primary resonance curve of the system deviate. Finally, the flexibility of using the amplitude-frequency equation to estimate the amplitude of the resonant solutions is verified numerically.

Design, Analysis, and Application of Flipped Product Chaotic System

In this paper, a novel method is proposed to build an improved 1-D discrete chaotic map called flipped product chaotic system (FPCS) by multiplying the output of one map with the output of a vertically flipped second map. Two variants, each with nine combinations, are shown with trade-off between computational cost and performance. The chaotic properties are explored using the bifurcation diagram, Lyapunov exponent, Kolmogorov entropy, and correlation coefficient. The proposed schemes offer a wider chaotic range and improved chaotic performance compared to the constituent maps and several prior works of similar nature. Wide chaotic window and improved chaotic complexity are two desired characteristics for several security applications as these two characteristics ensure enhanced design space with elevated entropic properties. We present a general Field-Programmable Gate Array (FPGA) design framework for the hardware implementation of the proposed flipped-product schemes and the results show good qualitative agreement with the numerical results from MATLAB simulation. Finally, we present a new Pseudo Random Number Generator (PRNG) using the two variants of the proposed chaotic map and validate their excellent randomness property using four standard statistical tests, namely NIST, FIPS, TestU01, and Diehard.

Bifurcation Structures of Nested Mixed-Mode Oscillations

In recently published work [Inaba & Kousaka, 2020a; Inaba & Tsubone, 2020b], we discovered significant mixed-mode oscillation (MMO) bifurcation structures in which MMOs are nested. Simple mixed-mode oscillation-incrementing bifurcations (MMOIBs) are known to generate [Formula: see text] oscillations for successive [Formula: see text] between regions of [Formula: see text]- and [Formula: see text]-oscillations, where [Formula: see text] and [Formula: see text] are adjacent simple MMOs, e.g. [Formula: see text] and [Formula: see text], where [Formula: see text] is an integer. MMOIBs are universal phenomena of evidently strong order and have been studied extensively in chemistry, physics, and engineering. Nested MMOIBs are phenomena that are more complex, but have an even stronger order, generating chaotic MMO windows that include sequences [Formula: see text] for successive [Formula: see text], where [Formula: see text] and [Formula: see text] are adjacent MMOIB-generated MMOs, i.e. [Formula: see text] and [Formula: see text] for integer [Formula: see text]. Herein, we investigate the bifurcation structures of nested MMOIB-generated MMOs exhibited by a classical forced Bonhoeffer–van der Pol oscillator. We use numerical methods to prepare two- and one-parameter bifurcation diagrams of the system with [Formula: see text], and 3 for successive [Formula: see text] for the case [Formula: see text]. Our analysis suggests that nested MMOs could be widely observed and are clearly ordered phenomena. We then define the first return maps for nested MMOs, which elucidate the appearance of successively nested MMOIBs.

Dynamic Characteristics Analysis of Cracked Magnetic Rotor-Bearing System

A dynamic model of the rotor system supported by active magnetic bearings at both sides is established, the fourth-order Runge-Kutta method is used to simulate. The periodic motion transition of the system and its evolution to chaotic motion are discussed from bifurcation diagrams and phase diagrams. It emphatically analyse the change of crack influencing factor and eccentricity on the system response and stability. The results show that the system presents period motion with appropriate parameter conditions. Large crack fatigue damage and value of eccentricity increases the amplitude and chaotic motion window in the low speed region, decrease the system stability. When the rotor crosses the critical speed range and reaches the high rotational speed region, the stable period-1 motion is dominant.

Essential chaotic dynamics of chatter in turning processes.

Large-amplitude oscillations of turning operations are investigated, which can be modelled by a one degree-of-freedom damped oscillator subjected to the regenerative effect that introduces a relevant time delay in the system. In the case of large oscillations, when the cutting tool loses contact with the surface of the workpiece, the time delay is switched off, leading to a non-smooth delay differential equation. To explore the geometric structure of the global dynamics of the system, the mathematical model is approximated by means of the essential part of the spectrum in the region where the stationary cutting process may lose its stability. The trajectories embedded in the infinite-dimensional phase space are interpreted in a three-dimensional subspace and then analyzed by means of a discrete Lorenz-map. The bifurcation diagrams of the non-smooth system include chaotic windows, which are presented by numerical and semi-analytical tools and compared to the existing results in the literature.

Experimental evidence of Phase Control method in chaotic Semiconductor Laser

we study how to control the dynamics of excitable systems by using the phase control technique.We study how to control nonlinear semiconductor laser dynamics with optoelectronic feedback using the phase control method. The phase control method uses the phase difference between a small.added frequenc y and the main driving frequency to suppress chaos, which leads to various periodic orbits. The experimental studying for the evaluation of chaos modulation behavior are considered in two conditions, the first condition, when one frequency of the external perturbation is varied, secondly, when two of these perturbations are changed. The chaotic system becomes regular under one frequency or two frequencies, But in two frequencies, phase control showed an excellent ability to maintain regular behavior in chaotic window and reexcite chaotic behavior when destroyed. This dynamics of the laser output are analyzed by time series and bifurcation diagram.

Inertial effects in the dc+ac driven underdamped Frenkel-Kontorova model: Subharmonic steps, chaos, and hysteresis.

The effects of inertial terms on the dynamics of the dc+ac driven Frenkel-Kontorova model were examined. As the mass of particles was varied, the response of the system to the driving forces and appearance of the Shapiro steps were analyzed in detail. Unlike in the overdamped case, the increase of mass led to the appearance of the whole series of subharmonic steps in the staircase of the average velocity as a function of average driving force in any commensurate structure. At certain values of parameters, the subharmonic steps became separated by chaotic windows while the whole structure retained scaling similar to the original staircase. The mass of the particles also determined their sensitivity to the forces governing their dynamics. Depending on their mass, they were found to exhibit three types of dynamics, from dynamical mode-locking with chaotic windows, through to a typical dc response, to essentially a free-particle response. Examination of this dynamics in both the upforce and downforce directions showed that the system may not only exhibit hysteresis, but also that large Shapiro steps may appear in the downforce direction, even in cases for which no dynamical mode-locking occurred in the upforce direction.

Thermodynamic processes expressed in terms of nonlinear maps

In this manuscript, we propose two new ideas in describing processes of inanimate–animate systems. In general, we integrate between nonlinear discrete maps and thermodynamics concepts and by doing so we prove that temperature can characterize the type of nonlinear map: regular or chaotic. Utilizing this result and by the use of chaotic windows phenomenon, we provide a possible explanation for the narrow range of temperatures in hot blooded animals. Utilizing mathematical tools borrowed from quantum mechanics, we not only describe the inanimate–animate systems, but we also present a new “philosophy” in describing a system with interacting systems (see Sect. 2).

Bursting Oscillations and Mixed-Mode Oscillations in Driven Liénard System

We report the existence of bursting oscillations and mixed-mode oscillations in a Liénard system when it is driven externally by a sinusoidal force. The bursting oscillations transit from a periodic phase to spiking trains through chaotic windows, as the control parameter is varied. The mixed-mode oscillations appear via an alternate sequence of periodic and chaotic states, as well as Farey sequences. The primary and their associated secondary mixed-mode oscillations are detected for the suitable choices of system parameters. Additionally, the system is also found to possess multistability nature. Our investigations involve numerical simulations as well as real time hardware experiments using a simple analog electronic circuit. The experimental observations are in conformation with numerical results.

A novel parallel image encryption with chaotic windows based on logistic map

Over the course of the last two decades, secure communication has become a very important issue due to the rapid growth of information technology and the development of public communication networks in which digital images are widely transmitted. In this paper, logistic map was employed for the encryption of gray-scale images. The proposed algorithm, demonstrating a proper performance according to the experimental results, divides the image into blocks and encrypts them with XOR operation and chaotic windows. Moreover, it has a large key space and the resulted encrypted images have homogeneous histograms. The large-enough NPCR and UACI of this algorithm indicate its resistance to differential attacks, not to mention the fact that it is suitable for noisy communication networks and could be made use of in parallel processing.

Mixed-mode oscillations and chaos in return maps of an oscillatory chemical reaction

The return maps, as an element of mathematical phenomenology appropriate for general examinations of complex dynamic states of the oscillatory systems were used to detect and explain the evolution of mixed-mode oscillations and chaos in a six-dimensional nonlinear reaction system of the Bray–Liebhafsky (BL) reaction, a well-studied nonlinear chemical reaction system that exhibits complex dynamic behavior. For this purpose principally different Poincare sections were applied and different transition scenarios between periodic and aperiodic states were examined by numerical simulations. It is shown that emergence of new periodic patterns can be detected by return maps already within chaotic windows. Besides, we also show that the higher dimensionality of manifold gives the impression of having several layers of manifolds.

The Dynamics of a Parametrically Driven Damped Pendulum

Ordered and chaotic states of a parametrically driven planar pendulum with viscous damping are numerically investigated. The damping makes the number of chaotic windows fewer but with larger width. Stroboscopic maps of the chaotic motion of the pendulum, driven either subharmonically or harmonically, show strange attractors with inversion symmetry in the phase plane.

BIFURCATIONS AND SYNCHRONIZATION OF THE FRACTIONAL-ORDER SIMPLIFIED LORENZ HYPERCHAOTIC SYSTEM

In this paper, dynamics of the fractional-order simplied Lorenz hyperchaotic system is investigated. Modied Adams-Bashforth-Moulton method is applied for numerical simulation. Chaotic regions and periodic windows are identied. Dierent types of motions are shown along the routes to chaos by means of phase portraits, bifurcation diagrams, and the largest Lyapunov exponent. The lowest fractional order to generate chaos is 3.8584. Synchronization between two fractional-order simplied Lorenz hyperchaotic systems is achieved by using active control method. The synchronization performances are studied by changing the fractional order, eigenvalues and eigenvalue standard deviation of the error system.

Structured chaos in a devil's staircase of the Josephson junction

The phase dynamics of Josephson junctions under external electromagnetic radiation is studied through numerical simulations. Current-voltage characteristics, Lyapunov exponents and Poincare sections are analyzed in detail. It is found that the subharmonic Shapiro steps at certain parameters are separated by structured chaotic windows. By performing a linear regression on the linear part of the data, a fractal dimension of D = 0.868 is obtained, with an uncertainty of +-0.012. The chaotic regions exhibit scaling similarity and it is shown that the devil's staircase of the system can form a backbone that unifies and explains the highly correlated and structured chaotic behavior. These features suggest a system possessing multiple complete devil's staircases. The onset of chaos for subharmonic steps occurs through the Feigenbaum period doubling scenario. Universality in the sequence of periodic windows is also demonstrated. Finally the influence of the radiation and Josephson junction parameters on the structured chaos is investigated and it is concluded that the structured chaos is a stable formation over a wide range of parameter values.

Bifurcation and chaos of a flag in an inviscid flow

A two-dimensional model is developed to study the flutter instability of a flag immersed in an inviscid flow. Two dimensionless parameters governing the system are the structure-to-fluid mass ratio M⁎ and the dimensionless incoming flow velocity U⁎. A transition from a static steady state to a chaotic state is investigated at a fixed M⁎=1 with increasing U⁎. Five single-frequency periodic flapping states are identified along the route, including four symmetrical oscillation states and one asymmetrical oscillation state. For the symmetrical states, the oscillation frequency increases with the increase of U⁎, and the drag force on the flag changes linearly with the Strouhal number. Chaotic states are observed when U⁎ is relatively large. Three chaotic windows are observed along the route. In addition, the system transitions from one periodic state to another through either period-doubling bifurcations or quasi-periodic bifurcations, and it transitions from a periodic state to a chaotic state through quasi-periodic bifurcations.