A novel bounded 4D chaotic system

This paper presents a novel bounded four-dimensional (4D) chaotic system which can display hyperchaos, chaos, quasiperiodic and periodic behaviors, and may have a unique equilibrium, three equilibria and five equilibria for the different system parameters. Numerical simulation shows that the chaotic attractors of the new system exhibit very strange shapes which are distinctly different from those of the existing chaotic attractors. In addition, we investigate the ultimate bound and positively invariant set for the new system based on the Lyapunov function method, and obtain a hyperelliptic estimate of it for the system with certain parameters.     

Generalized Darboux transformation and parameter-dependent rogue wave solutions to a nonlinear Schrödinger system

Objectives of the paper are (1) to design two new real and complex no equilibrium point hyperchaotic systems, (2) to design synchronisation technique for the new systems using the contraction theory and (3) to validate the results by using circuit realisation. First a new no equilibrium point hyperchaotic system is developed using a 3-D generalised Lorenz system; then using the new system a new complex no equilibrium point hyperchaotic system is reported. Both the new systems have hidden chaotic attractors. Various dynamical behaviours are observed in the new systems like chaotic, periodic, quasi-periodic and hyperchaotic. Both the systems have inverse crisis route to chaos with the variation of parameter a and crisis route to chaos with the variation of parameters \(b,\ c\) and d. These phenomena along with hidden attractors in a complex hyperchaotic system are not seen in the literature. Synchronisation between the identical new hyperchaotic systems is achieved using the contraction theory. Further the synchronisation between the identical new complex hyperchaotic systems is achieved using adaptive contraction theory. The proposed synchronisation strategies are validated using the MATLAB simulation and circuit implementation results. Further, an application of the proposed system is shown by transmitting and receiving an audio signal.     

Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system

In this paper, we introduce a new chaotic complex nonlinear system and study its dynamical properties including invariance, dissipativity, equilibria and their stability, Lyapunov exponents, chaotic behavior, chaotic attractors, as well as necessary conditions for this system to generate chaos. Our system displays 2 and 4-scroll chaotic attractors for certain values of its parameters. Chaos synchronization of these attractors is studied via active control and explicit expressions are derived for the control functions which are used to achieve chaos synchronization. These expressions are tested numerically and excellent agreement is found. A Lyapunov function is derived to prove that the error system is asymptotically stable.     

一双峰混沌系统非线性动力学行为

通过对一双峰混沌系统的非线性动力学行为的研究,发现随着系统参数的变化,双峰混沌系统由混沌状态开始,经阵发性混沌、不动点、倍周期分岔到受初始值的影响两个混沌吸引子,而后又收敛为另一个不动点,最后再次进入混沌状态。该系统呈现出复杂的非线性动力学行为。     

Designing a multi-scroll chaotic system by operating Logistic map with fractal process

Recently, chaotic systems have been widely investigated in several engineering applications. This paper presents a new chaotic system based on Julia’s fractal process, chaotic attractors and Logistic map in a complex set. Complex dynamic characteristics were analyzed, such as equilibrium points, bifurcation, Lyapunov exponents and chaotic behavior of the proposed chaotic system. As we know, one positive Lyapunov exponent proved the chaotic state. Numerical simulation shows a plethora of complex dynamic behaviors, which coexist with an antagonist form mixed of bifurcation and attractor. Then, we introduce an algorithm for image encryption based on chaotic system. The algorithm consists of two main stages: confusion and diffusion. Experimental results have proved that the proposed maps used are more complicated and they have a key space sufficiently large. The proposed image encryption algorithm is compared to other recent image encryption schemes by using different security analysis factors including differential attacks analysis, statistical tests, key space analysis, information entropy test and running time. The results demonstrated that the proposed image encryption scheme has better results in the level of security and speed.     

Dynamical analysis of the generalized Sprott C system with only two stable equilibria

A generalized Sprott C system with only two stable equilibria is investigated by detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations, and routes to chaos. In the parameter space where the equilibria of the system are both asymptotically stable, chaotic attractors coexist with period attractors and stable equilibria. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear.     

Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

The chaotic vibrations of a bimetallic shallow shell of revolution under time-varying temperature excitation are investigated in the present study. The governing equations are established in forms similar to those of classical single-layered shell theory by re-determination of reference surface. The nonlinear differential equation in time-mode is derived by variational method following an assumed spatial-mode. The Melnikov function is established theoretically to estimate regions of the chaos, and the Poincaré map, phase portrait, Lyapunov exponent, and Lyapunov dimension are used to determine if a chaotic motion really appears. Further investigations are developed by means of detailed numerical simulation, and both the bifurcation diagrams and corresponding maximum Lyapunov exponent are illustrated. The influence of static and time-dependent temperature parameters, height parameter of the shell, and damping parameter on the dynamic characteristics is examined. Interesting phenomena such as the onset of chaos, transient chaotic motion, chaos with interior crisis and period window, period-doubling scenario and reversed period-doubling bifurcation leading to chaos, jump phenomena, and chaos suddenly converting to period orbit have been observed from these figures.     

Lag synchronization for fuzzy chaotic system based on fuzzy observer

A new fuzzy observer for lag synchronization is given in this paper. By investi- gating synchronization of chaotic systems, the structure of drive-response lag synchronization for fuzzy chaos system based on fuzzy observer is proposed. A new lag synchronization criterion is derived using the Lyapunov stability theorem, in which control gains are obtained under the LMI condition. The proposed approach is applied to the well-known Chen＇s systems. A simulation example is presented to illustrate its effectiveness.     

A new chaotic attractor from hybrid optical bistable system

In this work, a new three-dimensional autonomous chaotic system has been introduced by modifying a hybrid optical system. The single quadratic nonlinearity is replaced by a single cubic nonlinearity; the new system can display two 1-scroll chaotic attractors simultaneously or one 2-scroll chaotic attractor. The bifurcation diagram is obtained and Lyapunov spectrum is calculated for the proposed system. The results show that the new system exhibits rich complexity features such as stable, periodic, and chaotic dynamics.     

Resonance and Bifurcation in a Nonlinear Duffing System with Cubic Coupled Terms

The dynamic behaviors of two-degree-of-freedom Duffing system with cubic coupled terms are studied. First, the steady-state responses in principal resonance and internal resonance of the system are analyzed by the multiple scales method. Then, the bifurcation structure is investigated as a function of the strength of the driving force F. In addition to the familiar routes to chaos already encountered in unidimensional Duffing oscillators, this model exhibits symmetry-breaking, period-doubling of both types and a great deal of highly periodic motion and Hopf bifurcation, many of which occur more than once. We explore the chaotic behaviors of our model using three indicators, namely the top Lyapunov exponent, Poincaré cross-section and phase portrait, which are plotted to show the manifestation of coexisting periodic and chaotic attractors.     

Periodic and chaotic dynamics of a sliding driven oscillator with dry friction

We have performed a numerical study of the dynamics of a harmonically forced sliding oscillator with two degrees of freedom and dry friction. The study of the four-dimensional dynamical system corresponding to the two non-linear motion equations can be reduced, in this case, to the study of a three-dimensional Poincaré map. The behaviour of the system has been investigated calculating bifurcation diagrams, time series, periodic and chaotic attractors and basins of attraction. Furthermore, a systematic study of the stability of periodic solutions and their bifurcations has been carried out applying the Floquet theory. The results show rich dynamics being very sensitive to the changes in forcing amplitudes (control parameter), where periodic and chaotic states alternatively appear. It is shown how the system exhibits different types of bifurcational phenomena (saddle-node, symmetry-breaking, period-doubling cascades and intermittent transitions to chaos) into relatively narrow intervals of the control parameter. Moreover, a collection of chaotic attractors was computed to show the evolution of the chaotic regime. Finally, basins of attraction were calculated. In all the cases studied, the basins exhibit fractal structure boundaries and, when more of two attractors are coexisting, we have found Wada basin boundaries.     

Dynamical analysis and FPGA implementation of a chaotic oscillator with fractional-order memristor components

Memristor-based chaotic and hyperchaotic systems are of great interest in the recent years, and addition of meminductor and memcapacitors to the family has widened the applications. In this paper, we propose a new chaotic system with fractional-order memristor and memcapacitor components. Nonlinear chaotic properties of the proposed system are investigated with equilibrium points, eigenvalues, Lyapunov exponents, bifurcation and bicoherence plots. We show that a small model disturbance can make the system to show self-excited and hidden attractors. We use the Adomian Decomposition method for implementing the proposed system in Field Programmable Gate Arrays.     

Analysis of a new 3D smooth autonomous system with different wing chaotic attractors and transient chaos

This letter proposes a new 3D quadratic autonomous chaotic system which displays an extremely complicated dynamical behavior over a large range of parameters. The new chaotic system has five real equilibrium points. Interestingly, this system can generate one-wing, two-wing, three-wing and four-wing chaotic attractors and periodic motion with variation of only one parameter. Besides, this new system can generate two coexisting one-wing and two coexisting two-wing attractors with different initial conditions. Furthermore, the transient chaos phenomenon happens in the system. Some basic dynamical behaviors of the proposed chaotic system are studied. Furthermore, the bifurcation diagram, Lyapunov exponents and Poincaré mapping are investigated. Numerical simulations are carried out in order to demonstrate the obtained analytical results. The interesting findings clearly show that this is a special strange new chaotic system, which deserves further detailed investigation.     

Gaussian mixture model and receding horizon control for multiple UAV search in complex environment

Intriguing as the discovery of new chaotic maps is, some new maps also bring new nonlinear phenomena of iterative map behavior. In this paper, we present a simple two-dimensional chaotic map which has three totally separated regions. The twin regions, creating strange and interesting attractors, are close to each other and vertically reflected however not identical in shape, while the distant region, generating a Hénon-like attractor, starts with period-doubling until complete chaos. Given the unusual behavior of the map introduced in this paper, we initially presented linear stability and bifurcation analysis per regions, with Lyapunov exponents and largest exponent computation. Besides the standardized calculations, what we focus here is to find out how a simple map can exhibit different chaotic behaviors in different regions.     

Absolute term introduced to rebuild the chaotic attractor with constant Lyapunov exponent spectrum

When positive or negative feedback of absolute terms are introduced in dynamic equations of improved chaotic system with constant Lyapunov exponent spectrum, diverse structures of chaotic attractors can be rebuilt, numbers of novel attractors found and subsequently the dynamical behavior property analyzed. Drawing on the concept of global phase reversal and its implementation methods, three main features are discussed and a systematic conclusion is made, that is, the unique class of chaotic system which utilizes merely absolute terms to realize nonlinear function possesses the following three properties: adjustable amplitude, adjustable phase reversal and constant Lyapunov exponent spectrum.     

New class of chaotic systems with equilibrium points like a three-leaved clover

This paper presents a new class of chaotic systems with infinite number of equilibrium points like a three-leaved clover. They signify an exciting class of dynamical systems which represent many major characteristics of regular and chaotic motions. These chaotic systems belong to the general class of chaotic systems with hidden attractors. By using a systematic computer search, three chaotic systems with three-leaved-clover-shaped equilibria were found which are classified into dissipative systems. Dynamics of the chaotic system with the three-leaved-clover-equilibria has been investigated by using phase portraits, bifurcation diagram, Lyapunov exponents, Kaplan–Yorke dimension and Poincaré map. Moreover, an electronic circuit implementation of the theoretical system is designed to check its effectiveness. Random number generator design has been realized with newly developed chaotic systems. The obtained random bit sequences are used for image encryption. Security analysis of image encryption processes has been performed.     

Dynamics of delay induced composite multi-scroll attractor and its application in encryption

Time delay feedback has been shown to produce chaos from non-chaotic systems. In this paper, besides the single and double scroll chaotic attractors, a new composite multi-scroll attractor is found in stable systems with time delay feedback. From the viewpoint of the local stability analysis, conservation analysis, Lyapunov exponent spectrum and power spectrum, the composite multi-scroll attractor is shown to be a hyper-chaotic attractor. The phase trajectory in the new composite hyper-chaotic multi-scroll attractor diverges in multiple eigen-directions, which improves the security of secure communication and chaotic encryption. A paradigm using the multi-scroll attractor for encryption is proposed, demonstrating its potential applicability.     

A novel non-equilibrium fractional-order chaotic system and its complete synchronization by circuit implementation

In this paper, we construct a novel four dimensional fractional-order chaotic system. Compared with all the proposed chaotic systems until now, the biggest difference and most attractive place is that there exists no equilibrium point in this system. Those rigorous approaches, i.e., Melnikov??s and Shilnikov??s methods, fail to mathematically prove the existence of chaos in this kind of system under some parameters. To reconcile this awkward situation, we resort to circuit simulation experiment to accomplish this task. Before this, we use improved version of the Adams?CBashforth?CMoulton numerical algorithm to calculate this fractional-order chaotic system and show that the proposed fractional-order system with the order as low as 3.28 exhibits a chaotic attractor. Then an electronic circuit is designed for order q=0.9, from which we can observe that chaotic attractor does exist in this fractional-order system. Furthermore, based on the final value theorem of the Laplace transformation, synchronization of two novel fractional-order chaotic systems with the help of one-way coupling method is realized for order q=0.9. An electronic circuit is designed for hardware implementation to synchronize two novel fractional-order chaotic systems for the same order. The results for numerical simulations and circuit experiments are in very good agreement with each other, thus proving that chaos exists indeed in the proposed fractional-order system and the one-way coupling synchronization method is very effective to this system.     

Hybrid phase synchronization between identical and nonidentical three-dimensional chaotic systems using the active control method

In this article, the active control method is used to investigate the hybrid phase synchronization between two identical Rikitake and Windmi systems, and also between two nonidentical systems taking Rikitake as the driving system and Windmi system as the response system. Based on the Lyapunov stability theory, the sufficient conditions for achieving the hybrid phase synchronization of two chaotic systems are derived. The active control method is found to be very effective and convenient to achieve hybrid phase chaos synchronization of the identical and nonidentical chaotic systems. Numerical simulation results which are carried out using the Runge–Kutta method show its feasibility and effectiveness for the synchronization of dynamical chaotic systems.     

单调激活函数惯性项神经耦合系统的混沌共存

混沌及其稳态共存是神经网络系统中一个重要研究热点问题．本文基于惯性项神经元模型,利用非线性单调激活函数构造了一个惯性项神经耦合系统,采用理论分析和数值模拟相结合的方法,研究了系统平衡点以及静态分岔的类型,分析了系统两种不同模式的混沌及其稳态共存．具体来说,我们通过选取不同的初始值,利用相应的相位图和时间历程图,展现了系统混沌对初值的敏感依赖性．进一步,采用耦合强度作为动力学的分岔参数,研究了混沌产生的倍周期分岔机制,得到了单调激活函数耦合下的惯性项神经元系统混沌共存现象．