Mathematical analysis of memristor through fractal-fractional differential operators: A numerical study

The newly generalized energy storage component, namely, memristor, which is a fundamental circuit element so called universal charge-controlled mem-element, is proposed for controlling the analysis and coexisting attractors. The governing differential equations of memristor are highly nonlinear for mathematical relationships. The mathematical model of memristor is established in terms of newly defined fractal-fractional differential operators so called Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractal-fractional differential operator. A novel numerical approach is developed for the governing differential equations of memristor on the basis of Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractal-fractional differential operator. We discussed chaotic behavior of memristor under three criteria such as (i) varying fractal order, we fixed fractional order; (ii) varying fractional order, we fixed fractal order; and (ii) varying fractal and fractional orders simultaneously. Our investigated graphical illustrations and simulated results via MATLAB for the chaotic behaviors of memristor suggest that newly presented Atangana-Baleanu, Caputo-Fabrizio, and Caputo fractal-fractional differential operators generate significant results as compared with classical approach.     

A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system

This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic attractors and a pair of diagonal double-wing chaotic attractors. More importantly, the system can generate a four-wing chaotic attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic attractors, major and minor diagonal double-wing chaotic attractors, and a four-wing chaotic attractor. Poincaré-map analysis shows that the system has extremely rich dynamics. The physical existence of the four-wing chaotic attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

Modeling of memristor-based chaotic systems using nonlinear Wiener adaptive filters based on backslash operator

Memristor-based chaotic systems have complex dynamical behaviors, which are characterized as nonlinear and hysteresis characteristics. Modeling and identification of their nonlinear model is an important premise for analyzing the dynamical behavior of the memristor-based chaotic systems. This paper presents a novel nonlinear Wiener adaptive filtering identification approach to the memristor-based chaotic systems. The linear part of Wiener model consists of the linear transversal adaptive filters, the nonlinear part consists of nonlinear adaptive filters based on the backslash operator for the hysteresis characteristics of the memristor. The weight update algorithms for the linear and nonlinear adaptive filters are derived. Final computer simulation results show the effectiveness as well as fast convergence characteristics. Comparing with the adaptive nonlinear polynomial filters, the proposed nonlinear adaptive filters have less identification error.     

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.     

Occurrence and underlying mechanism of multi-stripe chaotic attractors

This paper is concerned with the generation of multi-stripe chaotic attractors. Simple periodic nonlinear functions are employed to transform the original chaotic attractors to a pattern with multiple “parallel” or “rectangular” stripes. The relationship between the system parameters related to some periodic functions and the shape of the generated attractor is analyzed. Theoretic analysis about the underlying mechanism of generating the parallel stripes in the attractors is given. A general creation mechanism of multi-stripe attractors of the Lorenz system and other well-known chaotic systems is derived from the proposed unified approach.     

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.     

A new butterfly-shaped attractor of Lorenz-like system

In this letter a new butterfly-shaped chaotic attractor is reported. Some basic dynamical properties, such as Poincare mapping, Lyapunov exponents, fractal dimension, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied. Furthermore, we clarify that the chaotic attractors of the system is a compound structure obtained by merging together two simple attractors through a mirror operation.     

A novel chaotic attractor

In this letter, a novel chaotic attractor is reported. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincare mapping, the continuous spectrum and chaotic behavior of this new transverse butterfly attractor are studied. Meanwhile, the forming mechanism of its compound structure obtained by merging together two simple attractors after performing one mirror operation has been investigated by detailed numerical as well as theoretical analysis. Furthermore, the complex chaotic dynamical behavior of the system has been also proofed by experimental simulation of a designed electronic oscillator based on EWB.     

Generations and mechanisms of multi-stripe chaotic attractors of fractional order dynamic system

In this paper, the generations of multi-stripe chaotic attractors of fractional order system are considered. The original fractional order chaotic attractors can be turned into a pattern with multiple “parallel” or “ rectangular” stripes by employing certain simple periodic nonlinear functions. The relationships between the parameters relate to the periodic functions and the shape of the generated attractors are analyzed. Theoretical investigations about the underlying mechanisms of the parallel striped attractors of fractional order system are presented, with the fractional order Lorenz, Rössler and Chua’s systems as examples. Moreover, the periodic doubling striped route to chaos of fractional order Rössler system and maximum Lyaponov exponent calculations are also given.     

新三维非线性系统的动力学分析

通过代数方法,构造出来一个具有复杂混沌吸引子的非线性混沌自治三维系统．从理论和数值两方面对吸引子进行了分析和仿真,得到了系统在平衡点处不稳定的参数范围．通过分岔图和Lyapunov指数谱进一步揭示了系统丰富的动力学行为.     

Chaos generation via a switching fractional multi-model system

This paper introduces a system with switching multi-model structure which can generate chaos. Sub-models in this structure are fractional-order linear systems with any desired commensurate order less than 1. It shows that this system is capable of demonstrating chaotic behavior if its parameters and switching rule are suitably chosen. The structure of the proposed system is defined in a general form; consequently various chaotic attractors can be created by this system with different choices of order, parameters and switching rule. Numerical simulations illustrate behavior of the introduced system in some different situations.     

Bifurcation analysis of some forced Lu systems

Bifurcation behaviour of a forced Lu system is analyzed as the system parameter c and a forcing parameter F are varied. The Lu system belongs to a family of generalized Lorenz system. Members of this family are known to exhibit different types of chaotic attractors. Some of these attractors have been named Lorenz type L, Lu or Transition type T, Chen type T and Transverse 8 Type S. These different types of chaotic attractors are visually distinct when the parameters are widely separated. However, there is a need for identifying the precise point where transition from one type of chaotic attractor to another takes place. We identified signatures in the return map, which could be used for determining the point of transition and classifying the different types of chaotic attractors. These signatures helped to identify the point in coordinate space associated with such transitions. We find that such transitions take place when a chaotic attractor comes very close to a one-dimensional manifold on which the time derivatives of two of the variables is zero. We also find that just before coming to this point in coordinate space associated with the transition, the trajectory had approached, very closely, the equilibrium point at the origin.     

A periodically forced,piecewise linear system,Part II: The fragmentation mechanism of strange attractors and grazing

In the first part of this work, the local singularity of non-smooth dynamical systems was discussed and the criteria for the grazing bifurcation were presented mathematically. In this part, the fragmentation mechanism of strange attractors in non-smooth dynamical systems is investigated. The periodic motion transition is completed through grazing. The concepts for the initial and final grazing, switching manifolds are introduced for six basic mappings. The fragmentation of strange attractors in non-smooth dynamical systems is described mathematically. The fragmentation mechanism of the strange attractor for such a non-smooth dynamical system is qualitatively discussed. Such a fragmentation of the strange attractor is illustrated numerically. The criteria and topological structures for the fragmentation of the strange attractor need to be further developed as in hyperbolic strange attractors. The fragmentation of the strange attractors extensively exists in non-smooth dynamical systems, which will help us better understand chaotic motions in non-smooth dynamical systems.     

When Darwin meets Lorenz: Evolving new chaotic attractors through genetic programming

In this paper, we propose a novel methodology for automatically finding new chaotic attractors through a computational intelligence technique known as multi-gene genetic programming (MGGP). We apply this technique to the case of the Lorenz attractor and evolve several new chaotic attractors based on the basic Lorenz template. The MGGP algorithm automatically finds new nonlinear expressions for the different state variables starting from the original Lorenz system. The Lyapunov exponents of each of the attractors are calculated numerically based on the time series of the state variables using time delay embedding techniques. The MGGP algorithm tries to search the functional space of the attractors by aiming to maximise the largest Lyapunov exponent (LLE) of the evolved attractors. To demonstrate the potential of the proposed methodology, we report over one hundred new chaotic attractor structures along with their parameters, which are evolved from just the Lorenz system alone.     

永磁同步电动机中的混沌现象

讨论永磁同步电动机（PMSM）的动态特性,给出常输入电压、常外部扭转条件下的系统稳态特性表达式,基于Hopf分支条件提出一种调节系统参数的方法,以使其呈现极限环或混沌行为。计算机仿真结果表明在永磁同步电动机中存在混沌现象。     

Observer-based adaptive control of chaos in nonlinear discrete-time systems using time-delayed state feedback

A new approach to adaptive control of chaos in a class of nonlinear discrete-time-varying systems, using a delayed state feedback scheme, is presented. It is discussed that such systems can show chaotic behavior as their parameters change. A strategy is employed for on-line calculation of the Lyapunov exponents that will be used within an adaptive scheme that decides on the control effort to suppress the chaotic behavior once detected. The scheme is further augmented with a nonlinear observer for estimation of the states that are required by the controller but are hard to measure. Simulation results for chaotic control problem of Jin map are provided to show the effectiveness of the proposed scheme.     

Chaotic behavior in a new fractional-order love triangle system with competition

In the present work, we first modify the Sprott's nonlinear love triangle model by introducing the competition term and find that the new system also exhibits chaotic behavior. Then, to make the model more realistic, we go further to construct its corresponding fractional-order system and get the necessary condition for the existence of chaotic attractors. Finally, based on an improved version of Adams Bashforth Moulton numerical algorithm, we validate the chaotic attractors of this new fractional-order love triangle system by computer simulations.     

Generation of multi-wing chaotic attractor in fractional order system

In this paper, a novel approach is proposed for generating multi-wing chaotic attractors from the fractional linear differential system via nonlinear state feedback controller equipped with a duality-symmetric multi-segment quadratic function. The main idea is to design a proper nonlinear state feedback controller by using four construction criterions from a fundamental fractional differential nominal linear system, so that the controlled fractional differential system can generate multi-wing chaotic attractors. It is the first time in the literature to report the multi-wing chaotic attractors from an uncoupled fractional differential system. Furthermore, some basic dynamical analysis and numerical simulations are also given, confirming the effectiveness of the proposed method.     

Dynamical behavior,chaos control and synchronization of a memristor-based ADVP circuit

This paper is devoted to study the dynamical behavior of a modified Autonomous Van der Pol-Duffing (ADVP) circuit when its nonlinear element is replaced by a flux controlled memristor. The bifurcation diagrams, Lyapunov exponents, and phase portraits of the state variables are presented. Then, the chaos which appears at certain values of the system’s parameters is controlled using linear feedback control. Finally, the synchronization between two chaotic modified ADVP circuits is achieved in the case of fully unknown parameters of the system using adaptive synchronization.