A new 5D hyperchaotic system with stable equilibrium point,transient chaotic behaviour and its fractional-order form

In this paper, we construct a novel 4D autonomous chaotic system with four cross-product nonlinear terms and five equilibria. The multiple coexisting attractors and the multiscroll attractor of the system are numerically investigated. Research results show that the system has various types of multiple attractors, including three strange attractors with a limit cycle, three limit cycles, two strange attractors with a pair of limit cycles, two coexisting strange attractors. By using the passive control theory, a controller is designed for controlling the chaos of the system. Both analytical and numerical studies verify that the designed controller can suppress chaotic motion and stabilise the system at the origin. Moreover, an electronic circuit is presented for implementing the chaotic system.     

Bistability in a hyperchaotic system with a line equilibrium

A hyperchaotic system with an infinite line of equilibrium points is described. A criterion is proposed for quantifying the hyperchaos, and the position in the three-dimensional parameter space where the hyperchaos is largest is determined. In the vicinity of this point, different dynamics are observed including periodicity, quasi-periodicity, chaos, and hyperchaos. Under some conditions, the system has a unique bistable behavior, characterized by a symmetric pair of coexisting limit cycles that undergo period doubling, forming a symmetric pair of strange attractors that merge into a single symmetric chaotic attractor that then becomes hyperchaotic. The system was implemented as an electronic circuit whose behavior confirms the numerical predictions.     

Phase resetting effects for robust cycles between chaotic sets

In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated, owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena, including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. In this paper we introduce and discuss an instructive example of an ordinary differential equation where one can observe and analyze robust cycling behavior. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a R?ssler system), and/or saddle equilibria. For this model, we distinguish between cycling that includes phase resetting connections (where there is only one connecting trajectory) and more general non(phase) resetting cases, where there may be an infinite number (even a continuum) of connections. In the nonresetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability, whereas more general cases may give rise to "stuck on" cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase resetting and connection selection.     

Dynamical analysis of a new multistable chaotic system with hidden attractor: Antimonotonicity,coexisting multiple attractors,and offset boosting

Analyzing chaotic systems with coexisting and hidden attractors has been receiving much attention recently. In this article, we analyze a four dimensional chaotic system which has a plane as the equilibrium points. Also this system is of the group of systems that have coexisting attractors. First, the system is introduced and then stability analysis, bifurcation diagram and Largest Lyapunov exponent of this system are presented as methods to analyze the multistability of the system. These methods reveal that in some ranges of the parameter, this chaotic system has three different types of coexisting attractors, chaotic, stable node and limit cycle. Some interesting dynamics properties such as reversals of period doubling bifurcation and offset boosting are also presented.     

Dynamic analysis and synchronization control of an unusual chaotic system with exponential term and coexisting attractors

This paper reports a new four-dimensional chaotic system consisting of an exponential nonlinear term, two quadratic nonlinear terms and five linear terms. The system has only one equilibrium and performs stability, periodicity and chaos with the variation of the parameters. It losses its stability with the occurrence of Hopf bifurcation and goes into chaos via period-doubling bifurcation. One more interesting feature of the system is that it can generate multiple coexisting attractors for different initial conditions, such as two strange attractors with one limit cycle, one strange attractor with two limit cycles, etc. The dynamic properties of the system are presented by numerical simulation includes bifurcation diagrams, Lyapunov exponent spectrum and phase portraits. An electronic circuit is constructed to implement the chaotic attractor of the system. Based on the linear quadratic regulator (LQR) method, the synchronization control of the system is investigated.     

Embedding any desired number of coexisting attractors in memristive system

A simple variable-boostable system is selected as the structure for hosting an arbitrarily defined memristor for chaos producing. The derived three-dimensional (3-D) memristive chaotic system shows its distinct property of offset, amplitude and frequency control. Owing its merits any desired number of coexisting attractors are embedded by means of attractor doubling and self-reproducing based on function-oriented offset boosting. In this circumstance two classes of control gates are found:one determines the number of coexisting attractors resorting to the independent offset controller while the other is the initial condition selecting any one of them. Circuit simulation gives a consistent output with theoretically predicted embedded attractors.     

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.     

共存吸引子个数可调的混沌系统

运用Silnikov定理构建一个具有共存吸引子且个数可调的混沌系统.首先在经典混沌系统基础上构建一个结构简单的混沌系统,分析系统的动力学特性,验证系统马蹄意义下的混沌特性.在此基础上,将多零点分段函数引入该系统,以扩展系统平衡点的方式来增加系统的不变集,进而建立具有共存吸引子个数可调的混沌系统,由于共存吸引子的复杂性,...     

Expansion Around the Mean Field in Quantum Magnetic Systems

We introduce a new definition of ordered phase in a magnetic system based on properties of the local spin state probability. A statistical functional associated to this quantity depends both on amplitude and direction of the local magnetization. We show that it is possible to introduce an expansion at fixed magnetization amplitude in the inverse of lattice coordination number if the direction is selected by an extremum condition. In the limit of infinite coordination number we recover the mean field results. First order corrections are derived for the Ising model in the presence of a transverse field and for the XY model. Our results concerning critical temperature and order parameter compare favorably with other approaches.     

From Limit Cycles to Strange Attractors

We define a quantitative notion of shear for limit cycles of flows. We prove that strange attractors and SRB measures emerge when systems exhibiting limit cycles with sufficient shear are subjected to periodic pulsatile drives. The strange attractors possess a number of precisely-defined dynamical properties that together imply chaos that is both sustained in time and physically observable.     

非线性电路通向混沌的演化过程

给出了四阶非线性电路通向复杂性的两种演化模式,指出这两种模式与三个共存的平衡点有关.在第一种模式中,不稳定的平衡点由Hopf分岔导致了稳定的周期运动,经过倍周期分岔通向混沌,其所有的吸引子都保持对称结构;而在第二种模式中,另两个平衡点由Hopf分岔产生相互对称的极限环,并分别导致了两个混沌吸引子,其分岔过程步调一致,而且所有的吸引子都相互对称.随着参数的变化,这两个混沌吸引子相互作用形成一个扩大的混沌吸引子,导致与第一种分岔模式中定性一致的混沌运动.     

The Painlevé property,integrability and chaotic behaviour of a two-coupled Duffing oscillators

Integrability and chaotic behaviour in a two-coupled Duffing oscillators are studied. The coupling is nonlinear. Painlevé test is performed to identify integrable cases of damped- and force-free system. Exact analytical solutions are given for the integrable cases. Effect of external periodic forces for (i) single well with infinite height potential, (ii) potential with a hump at the centre and (iii) single well with finite height hump potential are numerically investigated. Occurrence of multiple attractors and period doubling cascades of coexisting attractors is presented.     

A simple feedback control for a chaotic oscillator with limited power supply

We proposed a simple feedback control method to suppress chaotic behavior in oscillators with limited power supply. The small-amplitude controlling signal is applied directly to the power supply system, so as to alter the characteristic curve of the driving motor. Numerical results are presented showing the method efficiency for a wide range of control parameters. Moreover, we have found that, for some parameters, this kind of control may introduce coexisting periodic attractors with complex basins of attraction and, therefore, serious problems with predictability of the final state the system will asymptote to.     

Replica-symmetry breaking in dynamical glasses

Systems of globally coupled logistic maps (GCLM) can display complex collective behaviour characterized by the formation of synchronous clusters. In the dynamical clustering regime, such systems possess a large number of coexisting attractors and might be viewed as dynamical glasses. Glass properties of GCLM in the thermodynamical limit of large system sizes N are investigated. Replicas, representing orbits that start from various initial conditions, are introduced and distributions of their overlaps are numerically determined. We show that for fixed-field ensembles of initial conditions all attractors of the system become identical in the thermodynamical limit up to variations of order 1/, and thus replica symmetry is recovered for N→∞. In contrast to this, when fluctuating-field ensembles of initial conditions are chosen, replica symmetry remains broken in the thermodynamical limit. Received 9 July 2001     

Design and implementation of a new memristive chaotic system with application in touchless fingerprint encryption

In this paper, a new memristive chaotic system was constructed from three-dimensional Lorenz-type system. The system has infinitely many equilibria and exhibits coexisting attractors. The dynamic evolution corresponds to the parameters and the coupling strength indicate that the system is easy to generate chaos. Also the bifurcation diagrams from different initial conditions determine the coexistence of multiple attractors. An electronic circuit is done for verifying the physically existence of the system. Based on this system, chaos-based random number generator and the corresponding randomness tests are studied. An algorithm for touchless fingerprint encryption is established. Some comparative tests illustrate the effectiveness of the algorithm.     

On some properties of nearly conservative dynamics of Ikeda map and its relation with the conservative case

The behavior of the well-known Ikeda map with very weak dissipation (so-called nearly conservative case) is investigated. The changes in the bifurcation structure of the parameter plane while decreasing the dissipation are revealed. It is shown that when the dissipation is very weak the system demonstrates an “intermediate” type of dynamics combining the peculiarities of conservative and dissipative dynamics. The correspondence between the trajectories in the phase space in the conservative case and the transformations of the set of initial conditions in the nearly conservative case has been obtained. The dramatic increase of the number of coexisting low-period attractors and the extraordinary growth of the transient time while the dissipation decreases have been revealed. The method of plotting a bifurcation tree for the set of initial conditions has been used to classify the existing attractors by their structure. Also it was shown that most of the coexisting attractors are destroyed by rather small external noise, and the transient time in noisy driven systems increases still more. The new method of two-parameter analysis for conservative systems was proposed.     

Multistability and coexisting transient chaos in a simple memcapacitive system

The self-excited attractors and hidden attractors in a memcapacitive system which has three elements are studied in this paper. The critical parameter of stable and unstable states is calculated by identifying the eigenvalues of Jacobian matrix. Besides, complex dynamical behaviors are investigated in the system, such as coexisting attractors, hidden attractors,coexisting bifurcation modes, intermittent chaos, and multistability. From the theoretical analyses and numerical simulations, it is found that there are four different kinds of transient transition behaviors in the memcapacitive system. Finally,field programmable gate array(FPGA) is used to implement the proposed chaotic system.     

忆阻器混沌电路产生的共存吸引子与Hopf分岔

利用荷控忆阻器和一个电感串联设计一种新型浮地忆阻混沌电路.用常规动力学分析方法研究该系统的基本动力学特性,发现系统可以产生一对关于原点对称的"心"型吸引子.将观察混沌吸引子时关注的电压、电流推广到功率和能量信号,观察到蝴蝶结型奇怪吸引子的产生.理论分析Hopf分岔行为并通过数值仿真进行验证,结果表明系统随电路参数变化能产生Hopf分岔、反倍周期分岔两种分岔行为.相对于其它忆阻混沌电路该电路采用的是一个浮地型忆阻器,并且在初始状态改变时,能产生共存吸引子和混沌吸引子与周期极限环共存现象.     

Generation and circuit implementation of multi-block multidirectional grid multi-scroll chaotic attractors

Due to the dynamic characteristics of the Chua's system, multi-scroll chaotic attractors are still confined in a single block and fail to break the limit. This paper proposes an approach for generating novel multi-block multidirectional grid multi-scroll chaotic attractors that can break the limit via novel nonlinear modulating functions. According to this method, the recursion rules used to generate multi-block multidirectional grid multi-scroll attractors are mathematically obtained. The new system is autonomous; the effectiveness of this method has been verified by theoretical analysis, numerical simulation, and circuit implementation.