Chaos control and chaos synchronization for multi-scroll chaotic attractors generated using hyperbolic functions

In this paper, we design a series of chaotic systems that can generate one-directional, two-directional and three-directional multi-scroll chaotic attractors. Then, based upon the properties of these chaotic systems, we construct appropriate Lyapunov functions and design simple linear feedback controls to globally exponentially stabilize and synchronize these chaotic systems. Numerical simulation results are also presented to show the applicability of the proposed control laws.     

Generation and circuit implementation of fractional-order multi-scroll attractors

This paper considers the generating of multi-scroll chaotic attractors for a new fractional-order linear system by using the piecewise-linear function. Multi-scroll chaotic attractors are generated by extending the number of saddle equilibrium points with index 2. Poincaré map and maximum Lyapunov exponents are applied to verifying the chaotic behaviors of the generated multi-scroll chaotic attractors. A circuit for the multi-scroll attractor is designed and simulated. Moreover, physical experiment of 3-scroll attractors and 5-scroll attractors are implemented. The numerical simulation, the circuit simulation and hardware experimental results are in accordance with each other, which verifies the effectiveness and physical realization of the approach.     

Multi-stable chaotic attractors in generalized synchronization

In this work, for given driving and response systems, the phenomenon of multi-stable chaotic attractors existing in generalized synchronization is studied. Consider the driving system descried by a Rössler system, and the response system being a multi-scroll chaotic system, some numerical simulations are proposed. The results show that by choosing suitable coupled parameters, there are multi-stable chaotic attractors in the response system, and each of them synchronizes with the driving system. Moreover, the basins of attraction on the parameter plane and initial condition plane are analyzed.     

Maximal Unstable Dissipative Interval to Preserve Multi-scroll Attractors via Multi-saturated Functions

In this paper, we present families of piecewise linear systems which are controlled by a continuous piecewise monoparametric control function for the generation of monoparametric families of multi-scroll attractors. Thus, the maximum range of values that the parameter set can take in order to preserve the useful dynamics for generating of multi-scroll attractors is found and it will be called maximal robust dynamics interval. This class of dynamical systems is the result of combining two or more unstable “one-spiral” trajectories. We give necessary and sufficient conditions in order to preserve multi-scroll attractors in terms of a parameter, i.e., a family of multi-scroll attractors is generated by means of a family of switching systems with multiple monoparametric companion matrices. Lastly, we provide an example to show how the developed theory works.     

A special full-state hybrid projective synchronization in symmetrical chaotic systems

A special full-state hybrid projective synchronization type is proposed in this paper. The anti-synchronization and complete synchronization can be achieved simultaneously in this new synchronization phenomenon. We point out how to realize this synchronization in chaotic systems: anti-synchronization in symmetrical coordinate subspace and complete synchronization in its normal coordinate subspace. Two illustrative examples, multi-scroll chaotic system by the partial Lyapunov stability theory, and a four-dimensional chaotic system by the invariance principle of differential equation are presented to exhibit this new synchronization.     

Multi-scroll and hypercube attractors from a general jerk circuit using Josephson junctions

In this paper Josephson junctions are used in order to generate n-scroll and n-scroll hypercube attractors. The design and realization of multi-scroll attractors depends on synthesizing the nonlinearity with an electrical circuit. Therefore we propose to use of the Josephson junction in a general jerk circuit in such a way that there is no need for synthesizing the nonlinearity towards chaotic n-scroll and hyperchaotic n-scroll hypercube attractors. The results are illustrated with computer simulations.     

复杂多卷波混沌系统的理论设计与电路实现

双卷波Chua电路的发明第一次在混沌理论与非线性电路之间建立了直接联系.复杂多卷波混沌系统在混沌理论与非线性电路之间架起了桥梁.复杂多卷波混沌系统具有明确的工程应用背景,它的理论设计与电路实现在过去三十年里得到迅猛发展.本文简要的回顾国内外过去三十年在复杂多卷波混沌系统的理论设计与电路实现上的主要研究进展,包括基本理论,设计方法与典型的工程应用,试图推进国内复杂多卷波混沌系统的研究.     

Special dynamic behaviors of a temporal chaotic system

When dynamic behaviors of temporal chaotic system are analyzed,we find that a temporal chaotic system has not only genetic dynamic behaviors of chaotic reflection,but also has phenomena influencing two chaotic attractors by original values.Along with the system parameters changing to certain value,the system will appear a break in chaotic region,and jump to another orbit of attractors.When it is opposite that the system parameters change direction,the temporal chaotic system appears complicated chaotic behaviors.     

Multi-scroll and hypercube attractors from a general jerk circuit using Josephson junctions

In this paper Josephson junctions are used in order to generate n-scroll and n-scroll hypercube attractors. The design and realization of multi-scroll attractors depends on synthesizing the nonlinearity with an electrical circuit. Therefore we propose to use of the Josephson junction in a general jerk circuit in such a way that there is no need for synthesizing the nonlinearity towards chaotic n-scroll and hyperchaotic n-scroll hypercube attractors. The results are illustrated with computer simulations.     

Integrated circuit generating 3- and 5-scroll attractors

This paper introduces the experimental realization of the first integrated circuit of a multi-scroll continuous chaotic oscillator showing 3- and 5-scroll attractors. It is based on a variant of the Chua’s system. The most relevant issue is the implementation of a saw-tooth-like nonlinear function, which is designed by using floating gate MOS (FGMOS) transistors. Therefore, the realization of a voltage-to-current nonlinear cell by a piecewise-linear approach allows us to have only two external control inputs instead of numerous external voltage references, as usually done in current circuit realizations. Experimental results of the proposed integrated multi-scroll oscillator along with its corner analysis are provided.     

Simulation and circuit implementation of 12-scroll chaotic system

Based on the typical Chua’s circuit and the latest research results of multi-scroll system, a model of the new system is constructed to produce multi-scroll chaotic attractors, replacing the typical Chua’s diode with the combination of sign function. The major method is to make equilibrium point located in the center of two adjacent breakpoints, and keep scrolls and bond orbits alternated with each other. The chaos generation mechanism is studied by analyzing the symmetry and invariance, the existence of the dissipation and attractor, the system equilibrium and stability. The fractal dimension, the K–S entropy, the time domain waveform and the initial value sensitivity are applied to verifying the chaotic behaviors. The numerical simulations show that the system generates n-double (n = 1, 2, 3, 4, 5, 6) scroll chaotic attractors. Finally, the design of the hardware circuit produces at a maximum of 12-scroll hardware experimental results. Theoretical analysis, numerical simulation and hardware experimental results are full matched, which further proves the existence of the system and the physical realization.     

一类非线性混沌系统混沌吸引子的冲击控制

在国内外研究工作的基础上,给出了一类非线性混沌系统混沌吸引子的冲击控制方案,运用普适方程的冲击控制理论导出了这类混沌系统混沌吸引子的冲击控制渐进稳定的条件,利用这一条件给出了混沌吸引子渐进稳定冲击控制的区间上界,最后给出了许多数据结果,这些结果对于混沌吸引子的控制将有重要的参考价值．     

Automatic synthesis of chaotic attractors

An automatic synthesis methodology of multi-scroll chaotic attractors by using staircase nonlinear functions (SNFs) is introduced. Synthesis process is carried out by considering third-order nonlinear system parameters, such as the gain of the system and number of scrolls along with real physical active device parameters, such as the dynamic range. Therefore, it is not necessary done a scaling of the dynamic range associated to the SNFs and chaotic attractor parameters like the swings, widths, equilibrium points and breakpoints can be estimated. As a consequence, chaotic attractors in 1-direction (1-D) and 2-D n × m-grid scrolls can easily be generated. Moreover, from numerical simulations, the nonlinear system can quickly be synthesized with electronic circuits. HSPICE simulations of 9-scrolls and 4 × 3-grid scrolls by using Opamps are shown in agreement with the numerical simulations.     

一类平面D_3等变映射的混沌吸引子对称增加分歧的数值确定

本文讨论了一类平面D3等变映射的分歧和混沌性质.通过计算显示出映射随着参数的变化,从周期解走向混沌以及混饨吸引子由Z2-对称走向D3-对称的全过程.给出计算混沌吸引子的对称增加分歧扩张系统的算法,数值结果表明,两者相符.     

Dynamics of Stochastic Ginzburg-Landau Equations Driven by Colored Noise on Thin Domains

This work is concerned with the asymptotic behaviors of solutions to a class of non-autonomous stochastic Ginzburg-Landau equations driven by colored noise and deterministic non-autonomous terms defined on thin domains. The existence and uniqueness of tempered pullback random attractors are proved for the stochastic Ginzburg-Landau systems defined on $(n+1)$-dimensional narrow domain. Furthermore, the upper semicontinuity of these attractors is established, when a family of $(n+1)$-dimensional thin domains collapses onto an $n$-dimensional domain.     

Parameter identification and synchronization of fractional-order chaotic systems

The knowledge about parameters and order is very important for synchronization of fractional-order chaotic systems. In this article, identification of parameters and order of fractional-order chaotic systems is converted to an optimization problem. Particle swarm optimization algorithm is used to solve this optimization problem. Based on the above parameter identification, synchronization of the fractional-order Lorenz, Chen and a novel system (commensurate or incommensurate order) is derived using active control method. The new fractional-order chaotic system has four-scroll chaotic attractors. The existence and uniqueness of solutions for the new fractional-order system are also investigated theoretically. Simulation results signify the performance of the work.     

A simple multi-scroll hyperchaotic system

We propose a simple autonomous hyperchaotic system that can generate multi-scroll attractors. The proposed system has a canonical structure, one control parameter, and a switching-type nonlinearity. If multiple breakpoints are added to the system nonlinearity, multi-scroll behavior can be obtained. We numerically demonstrate hyperchaotic behavior of the proposed system, under different nonlinearities, as its control parameter is changed. Furthermore, we study hyperchaos in the proposed system when it assumes a fractional order, and demonstrate that hyperchaotic behavior can be obtained in systems less than fourth order. Throughout the study, hyperchaos is verified by examining the Lyapunov spectrum, where the presence of multiple positive Lyapunov exponents in the spectrum is indicative of hyperchaos.     

Partially nearly riddled basins in systems with chaotic saddle

We show that chaotic attractors can have partially nearly riddled basins of attraction, i.e., basins which consist both of large open sets and a set in which small open sets which belong to the basins of different attractors are intermingled. We argue that such basins are robust for systems with the chaotic saddle located between at least two attractors and in the presence of noise cause the uncertainties similar to those implied by riddled basins.     

Coexistence of multiple attractors and crisis route to chaos in a novel memristive diode bidge-based Jerk circuit

In the present paper, a new memristor based oscillator is obtained from the autonomous Jerk circuit [Kengne et al., Nonlinear Dynamics (2016) 83: 751̶765] by substituting the nonlinear element of the original circuit with a first order memristive diode bridge. The model is described by a continuous time four-dimensional autonomous system with smooth nonlinearities. Various nonlinear analysis tools such as phase portraits, time series, bifurcation diagrams, Poincaré section and the spectrum of Lyapunov exponents are exploited to characterize different scenarios to chaos in the novel circuit. It is found that the system experiences period doubling and crisis routes to chaos. One of the major results of this work is the finding of a window in the parameters’ space in which the circuit develops hysteretic behaviors characterized by the coexistence of four different (periodic and chaotic) attractors for the same values of the system parameters. Basins of attractions of various coexisting attractors are plotted showing complex basin boundaries. As far as the authors’ knowledge goes, the novel memristive jerk circuit represents one of the simplest electrical circuits (no analog multiplier chip is involved) capable of four disconnected coexisting attractors reported to date. Both PSpice simulations of the nonlinear dynamics of the oscillator and laboratory experimental measurements are carried out to validate the theoretical analysis.     

Transition to chaos in nonlinear dynamical systems described by ordinary differential equations

We investigate scenarios that create chaotic attractors in systems of ordinary differential equations (Vallis, Rikitaki, Rossler, etc.). We show that the creation of chaotic attractors is governed by the same mechanisms. The Feigenbaum bifurcation cascade is shown to be universal, while subharmonic and homoclinic cascades may be complete, incomplete, or not exist at all depending on system parameters. The existence of a saddle-focus equilibrium plays an important and possibly decisive role in the creation of chaotic attractors in dissipative nonlinear systems described by ordinary differential equations. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 73–98, 2003.