Fast–Slow Dynamics and Bifurcation Mechanism in a Novel Chaotic System

2019 ◽  
Vol 29 (10) ◽  
pp. 1930028
Author(s):  
Lan Huang ◽  
Guoqing Wu ◽  
Zhengdi Zhang ◽  
Qinsheng Bi
Keyword(s):  
Natural Frequency ◽  
Chaotic System ◽  
Three Dimensional ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Fast Subsystem ◽  
Bursting Oscillations ◽  
Exciting Frequency ◽  
Different Types ◽  
Multiple Coexisting Attractors

This paper proposes a novel three-dimensional chaotic system with multiple coexisting attractors, where different values of a constant control parameter may drive the chaotic behaviors to evolve from single-scroll to double-scroll attractors. When the controlling term is replaced by a periodic harmonic excitation where the exciting frequency is far less than the natural frequency, chaotic movement may disappear, while periodic bursting oscillations will take place. Based on the fact that during a period defined by the natural frequency, the exciting term keeps almost a constant, the whole exciting term can be regarded as a slow-varying parameter resulting in a generalized autonomous system, its equilibrium branches as well as the related bifurcations occurring with the variation of the slow-varying parameter are derived. With the increase of the exciting amplitude, asymmetric and symmetric bursting attractors can be observed, for which the mechanism can be analyzed by the overlap of the equilibrium branches and the transformed phase portraits. With different values of the exciting amplitude corresponding to the change region of the slow-varying parameter, different bifurcations such as fold and Hopf bifurcations may involve the bursting structures, leading to different types of bursting oscillations. Furthermore, the phase space can be divided into two regions by a line boundary because of the symmetry of the vector field. When the trajectory from one region returning to the region arrives at the boundary, two asymmetric bursting attractors located in different regions coexist, which are symmetric to each other. However, when the trajectory passes across the boundary, an enlarged symmetric bursting attractor can be observed, whose trajectory connects the two original asymmetric attractors. Furthermore, it is found that when the trajectory runs along a stable equilibrium branch to the bifurcation point, it may move almost strictly along an unstable equilibrium branch of the fast subsystem because of the delay influence of the bifurcation.

Nonlinear Dynamics and Application of the Four Dimensional Autonomous Hyper-Chaotic System

2014 ◽  
Author(s):  
Ge Kai ◽  
Wei Zhang
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Simulation ◽  
Differential Equations ◽  
Chaotic System ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Chaotic Motions ◽  
Finance System

In this paper, we establish a dynamic model of the hyper-chaotic finance system which is composed of four sub-blocks: production, money, stock and labor force. We use four first-order differential equations to describe the time variations of four state variables which are the interest rate, the investment demand, the price exponent and the average profit margin. The hyper-chaotic finance system has simplified the system of four dimensional autonomous differential equations. According to four dimensional differential equations, numerical simulations are carried out to find the nonlinear dynamics characteristic of the system. From numerical simulation, we obtain the three dimensional phase portraits that show the nonlinear response of the hyper-chaotic finance system. From the results of numerical simulation, it is found that there exist periodic motions and chaotic motions under specific conditions. In addition, it is observed that the parameter of the saving has significant influence on the nonlinear dynamical behavior of the four dimensional autonomous hyper-chaotic system.

scholarly journals Chaos-Based Application of a Novel Multistable 5D Memristive Hyperchaotic System with Coexisting Multiple Attractors

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-19 ◽  
Author(s):  
Fei Yu ◽  
Li Liu ◽  
Shuai Qian ◽  
Lixiang Li ◽  
Yuanyuan Huang ◽  
...  
Keyword(s):  
Image Encryption ◽  
Chaotic System ◽  
Random Number Generator ◽  
Phase Portraits ◽  
Hyperchaotic System ◽  
The Novel ◽  
Multiple Attractors ◽  
Period 2 ◽  
Different Types ◽  
System Designs

Novel memristive hyperchaotic system designs and their engineering applications have received considerable critical attention. In this paper, a novel multistable 5D memristive hyperchaotic system and its application are introduced. The interesting aspect of this chaotic system is that it has different types of coexisting attractors, chaos, hyperchaos, periods, and limit cycles. First, a novel 5D memristive hyperchaotic system is proposed by introducing a flux-controlled memristor with quadratic nonlinearity into an existing 4D four-wing chaotic system as a feedback term. Then, the phase portraits, Lyapunov exponential spectrum, bifurcation diagram, and spectral entropy are used to analyze the basic dynamics of the 5D memristive hyperchaotic system. For a specific set of parameters, we find an unusual metastability, which shows the transition from chaotic to periodic (period-2 and period-3) dynamics. Moreover, its circuit implementation is also proposed. By using the chaoticity of the novel hyperchaotic system, we have developed a random number generator (RNG) for practical image encryption applications. Furthermore, security analyses are carried out with the RNG and image encryption designs.

scholarly journals Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Xindong Ma ◽  
Shuqian Cao
Keyword(s):  
Natural Frequency ◽  
Coupling Effect ◽  
Excitation Frequency ◽  
Time Interval ◽  
Periodic Excitation ◽  
Bursting Oscillations ◽  
Fold Bifurcation ◽  
Exciting Frequency ◽  
Bursting Oscillation ◽  
Parameter Condition

The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slow-varying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric “homoclinic/homoclinic” bursting oscillation, symmetric “fold/Hopf” bursting oscillation, symmetric “fold/fold” bursting oscillation, and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop, are revealed with different values of the parameterbby means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency.

THE CUSP–HOPF BIFURCATION

2007 ◽  
Vol 17 (08) ◽  
pp. 2547-2570 ◽  
Author(s):  
J. HARLIM ◽  
W. F. LANGFORD
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium Points ◽  
Invariant Tori ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Planar System ◽  
Bursting Oscillations ◽  
Codimension Two ◽  
Truncated Normal ◽  
Comprehensive Study

The coalescence of a Hopf bifurcation with a codimension-two cusp bifurcation of equilibrium points yields a codimension-three bifurcation with rich dynamic behavior. This paper presents a comprehensive study of this cusp-Hopf bifurcation on the three-dimensional center manifold. It is based on truncated normal form equations, which have a phase-shift symmetry yielding a further reduction to a planar system. Bifurcation varieties and phase portraits are presented. The phenomena include all four cases that occur in the codimension-two fold–Hopf bifurcation, in addition to bistability involving equilibria, limit cycles or invariant tori, and a fold–heteroclinic bifurcation that leads to bursting oscillations. Uniqueness of the torus family is established locally. Numerical simulations confirm the prediction from the bifurcation analysis of bursting oscillations that are similar in appearance to those that occur in the electrical behavior of neurons and other physical systems.

Relaxation Oscillations in a Nonsmooth Oscillator with Slow-Varying External Excitation

2019 ◽  
Vol 29 (07) ◽  
pp. 1930019 ◽  
Author(s):  
Zhixiang Wang ◽  
Zhengdi Zhang ◽  
Qinsheng Bi
Keyword(s):  
Original System ◽  
Relaxation Oscillations ◽  
External Excitation ◽  
Bursting Oscillations ◽  
Nonlinear Resistor ◽  
Periodic Movement ◽  
Exciting Frequency ◽  
Different Types ◽  
Nonsmooth Dynamical System ◽  
Electric Source

The main purpose of the paper is to explore the influence of the coupling of two scales on the dynamics of a nonsmooth dynamical system. Based on a typical Chua’s circuit, by introducing a nonlinear resistor with piecewise characteristics as well as a harmonically changed electric source, a modified nonsmooth model is established, in which the coupling of two scales in frequency domain exists. Different types of bursting oscillations, appearing in the combination of large-amplitude oscillations, called spiking oscillations ([Formula: see text]), and small-amplitude oscillations or at rest, denoted by quiescent states ([Formula: see text]), can be observed with the variation of the exciting amplitude. When the exciting frequency is relatively small, by regarding the whole exciting term as a slow-varying parameter, the original system can be transformed into a generalized autonomous system. The phase space can be divided into three regions by the nonsmooth boundaries, in which the trajectory is governed by three different subsystems, respectively. Based on the analysis of the three subsystems as well as the behaviors on the nonsmooth boundaries, all the equilibrium branches and their bifurcations can be obtained, which can be employed to investigate the mechanism of the bursting oscillations. It is found that, for relatively small exciting amplitude, since no bifurcation on the equilibrium branches can be realized with the variation of the slow-varying parameter, the system behaves in periodic movement, which may evolve to bursting oscillations when a pair of fold bifurcations occurs with the increase of the exciting amplitude. Further increase of the exciting amplitude may lead to more complicated bursting oscillations, which may bifurcate into two coexisted asymmetric bursting attractors via symmetric breaking. Interaction between the two attractors may result in an enlarged symmetric bursting attractor, in which more forms of bifurcations at the transitions between the quiescent states and repetitive spiking states can be observed.

Generating Four-Wing Hyperchaotic Attractor and Two-Wing, Three-Wing, and Four-Wing Chaotic Attractors in 4D Memristive System

2017 ◽  
Vol 27 (02) ◽  
pp. 1750027 ◽  
Author(s):  
Ling Zhou ◽  
Chunhua Wang ◽  
Lili Zhou
Keyword(s):  
Chaotic System ◽  
Three Dimensional ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Hyperchaotic System ◽  
Multiple Attractors ◽  
System A ◽  
Dynamical Behaviors ◽  
Memristive System ◽  
Line Of Equilibria

By adding only one smooth flux-controlled memristor into a three-dimensional (3D) pseudo four-wing chaotic system, a new real four-wing hyperchaotic system is constructed in this paper. It is interesting to see that this new memristive chaotic system can generate a four-wing hyperchaotic attractor with a line of equilibria. Moreover, it can generate two-, three- and four-wing chaotic attractors with the variation of a single parameter which denotes the strength of the memristor. At the same time, various coexisting multiple attractors (e.g. three-wing attractors, four-wing attractors and attractors with state transition under the same system parameters) are observed in this system, which means that extreme multistability arises. The complex dynamical behaviors of the proposed system are analyzed by Lyapunov exponents (LEs), phase portraits, Poincaré maps, and time series. An electronic circuit is finally designed to implement the hyperchaotic memristive system.

Constructing a Class of Four-Dimensional Correlative and Switchable Hyperchaotic Systems

2010 ◽  
Vol 44-47 ◽  
pp. 1802-1806
Author(s):  
Fan Yang ◽  
Dong Li ◽  
Hong Qing Tu
Keyword(s):  
Lyapunov Exponent ◽  
Chaotic System ◽  
Chaotic Attractor ◽  
Three Dimensional ◽  
Phase Portraits ◽  
State Variables ◽  
Nonlinear Functions ◽  
Basic Properties ◽  
Hyperchaotic Systems

A class of four-dimensional correlative and switchable hyperchaotic systems were built by adding state variables, nonlinear functions or using the method of anti-control the three-dimensional chaotic system. We studied detailedly some of its basic properties, such as the feature of equilibrium, the phase portraits of hyper chaotic attractor, Lyapunov exponent and the evolutive course of systemic dynamical action.

scholarly journals New Fractional Order Chaotic System: Analysis, Synchronization, and it’s Application

2021 ◽  
Vol 17 (1) ◽  
pp. 1-9
Author(s):  
Zain_Aldeen Rahman ◽  
Basil Jassim ◽  
Yasir Al_Yasir
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Secure Communication ◽  
System Analysis ◽  
Three Dimensional ◽  
Nonlinear Dynamic System ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Communication Scheme ◽  
New System

In this paper, a new nonlinear dynamic system, new three-dimensional fractional order complex chaotic system, is presented. This new system can display hidden chaotic attractors or self-excited chaotic attractors. The Dynamic behaviors of this system have been considered analytically and numerically. Different means including the equilibria, chaotic attractor phase portraits, the Lyapunov exponent, and the bifurcation diagrams are investigated to show the chaos behavior in this new system. Also, a synchronization technique between two identical new systems has been developed in master- slave configuration. The two identical systems are synchronized quickly. Furthermore, the master-slave synchronization is applied in secure communication scheme based on chaotic masking technique. In the application, it is noted that the message is encrypted and transmitted with high security in the transmitter side, in the other hand the original message has been discovered with high accuracy in the receiver side. The corresponding numerical simulation results proved the efficacy and practicability of the developed synchronization technique and its application

Bursting Oscillations and the Mechanism with Sliding Bifurcations in a Filippov Dynamical System

2018 ◽  
Vol 28 (12) ◽  
pp. 1850146 ◽  
Author(s):  
Rui Qu ◽  
Yu Wang ◽  
Guoqing Wu ◽  
Zhengdi Zhang ◽  
Qinsheng Bi
Keyword(s):  
Dynamical System ◽  
Natural Frequency ◽  
Stable Equilibrium ◽  
Multiple Scales ◽  
Hopf Bifurcations ◽  
Periodic Excitation ◽  
Nonsmooth Boundary ◽  
Bursting Oscillations ◽  
First Case ◽  
Exciting Frequency

The main purpose of the paper is to investigate the effect of multiple scales in frequency domain on the complicated oscillations of Filippov system with discontinuous right-hand side. A relatively simple model based on the Chua’s circuit with periodic excitation is introduced as an example. When the exciting frequency is far less than the natural frequency, implying that an order gap between the exciting frequency and the natural frequency exists, the whole exciting term can be considered as a slow-varying parameter, based on which the bifurcations of the two subsystems in different regions divided by the nonsmooth boundary are presented. Two typical cases are considered, which correspond to different distributions of equilibrium branches as well as the related bifurcations. In the first case, periodic symmetric Hopf/Hopf-fold-sliding bursting oscillations can be obtained, in which Hopf bifurcations may cause the alternations between the quiescent states and the spiking states, while fold bifurcations connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. While the second case is the periodic symmetric fold/fold-fold-sliding bursting, where the fold bifurcations not only lead to the alternations between the quiescent states and the spiking states, but also connect the two quiescent states moving along the stable equilibrium branches and sliding along the nonsmooth boundary, respectively. It is pointed out that, different from the bursting oscillations in smooth dynamical systems in which the bifurcations may cause the alternations between quiescent states and spiking states, in the nonsmooth system, bifurcations may not only lead to the alternations, but also connect different forms of quiescent states. Furthermore, in the Filippov system, sliding movement along the nonsmooth boundary can be observed, the mechanism of which is presented based on the analysis of the two subsystems in different regions.

scholarly journals On occurrence of sudden increase of spiking amplitude via fold limit cycle bifurcation in a modified Van der Pol-Duffing system with slow-varying periodic excitation

2021 ◽  
Author(s):  
Xiaofang Zhang ◽  
Bin Zhang ◽  
Xiujing Han ◽  
Qinsheng Bi
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Multiple Equilibrium ◽  
Sudden Increase ◽  
Van Der Pol ◽  
Full System ◽  
Bursting Oscillations ◽  
Duffing System ◽  
Exciting Frequency ◽  
Different Types

Abstract The main purpose of the paper is to reveal the mechanism of certain special phenomena in bursting oscillations such as the sudden increase of the spiking amplitude. When multiple equilibrium points coexist in a dynamical system, several types of stable attractors via different bifurcations from these points may be observed with the variation of parameters, which may interact with each other to form other types of bifurcations. Here we take the modified van der Pol-Duffing system as an example, in which periodic parametric excitation is introduced. When the exciting frequency is far less than the natural frequency, bursting oscillations may appear. By regarding the exciting term as a slow-varying parameter, the number of the equilibrium branches in the fast generalized autonomous subsystem varies from one to five with the variation of the slow-varying parameter, on which different types of bifurcations, such as Hopf and pitch fork bifurcations, can be observed. The limit cycles, including the cycles via Hopf bifurcations and the cycles near the homo-clinic orbit may interact with each other to form the fold limit cycle bifurcations. With the increase of the exciting amplitude, different stable attractors and bifurcations of the generalized autonomous fast subsystem involve the full system, leading to different types of bursting oscillations. Fold limit cycle bifurcations may cause the sudden change of the spiking amplitude, since at the bifurcation points, the trajectory may oscillate according to different stable limit cycles with obviously different amplitudes. At the pitch fork bifurcation point, two possible jumping ways may result in two coexisted asymmetric bursting attractors, which may expand in the phase space to interact with each other to form an enlarged symmetric bursting attractor with doubled period. The inertia of the movement along the stable equilibrium may cause the trajectory to pass across the related bifurcations, leading to the delay effect of the bifurcations. Not only the large exciting amplitude, but also the large value of the exciting frequency may increase inertia of the movement, since in both the two cases, the change rate of the slow-varying parameter may increase. Therefore, a relative small exciting frequency may be taken in order to show the possible influence of all the equilibrium branches and their bifurcations on the dynamics of the full system.

Sign in / Sign up

Export Citation Format

Share Document