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Linear synchronization and circuit implementation of chaotic system with complete amplitude control *
Abstract
Although chaotic signals are considered to have great potential applications in radar and communication engineering, their broadband spectrum makes it difficult to design an applicable amplifier or an attenuator for amplitude conditioning. Moreover, the transformation between a unipolar signal and a bipolar signal is often required. In this paper, a more intelligent hardware implementation based on field programmable analog array (FPAA) is constructed for chaotic systems with complete amplitude control. Firstly, two chaotic systems with complete amplitude control are introduced, one of which has the property of offset boosting with total amplitude control, while the other has offset boosting and a parameter for partial control. Both cases can achieve complete amplitude control including amplitude rescaling and offset boosting. Secondly, linear synchronization is established based on the special structure of chaotic system. Finally, experimental circuits are constructed on an FPAA where the predicted amplitude control is realized through only two independent configurable analog module (CAM) gain values.
... Recently, a large number of studies have been reported on third order chaotic systems with some special properties including variable-boostable [1], conditional symmetry [2,3], total or partial amplitude control [4][5][6], and systems with adjustable symmetry and nonlinearity [7][8][9] just to name a few. Above the amplitude control property, multistability is another inherent feature which can be found in several nonlinear systems. ...
... Remark that the proposed snap system in Eq. (1) is such that the variable x 1 appears only in the fourth equation implying that our model is a variable boostable chaotic snap system [1]. Also note that in Eq. (1), parameters d and n are used to realize a total control of the signal (x 1 ) amplitude [1,5,6]. This later property shows that system (1) has a distinctive characteristic of dual-parameter total amplitude control which has not been investigated previously. ...
... More interestingly, it is also possible to combine total amplitude control with offset boosting. An example was recently investigated in [6]. However, to our best knowledge, the investigations of this interesting feature in a chaotic snap system are relatively little studied. ...
In this paper, we introduce a novel snap system with a unique parameterized piecewise quadratic nonlinearity in the form ψn (x) = x − n |x| − dx |x|where n controls the symmetry of the system and serves as total amplitude control of the signals. The model is described by a continuous time 4D autonomous system (ODE) with smooth conditional nonlinearity. We study the chaos mechanism with respect to system parameters both in the symmetric and asymmetric modes of oscillations by exploiting bifurcation diagrams, basin of attractions and phase portraits as main tools. In particular, forn = 0, the system displays a perfect symmetry and develops rich dynamics including period doubling sequences, merging crisis, hysteresis, and coexisting multiple symmetric attractors. Forn = 0, the system is non-symmetric and the space magnetization induced more complex and striking effects including asymmetric double scroll strange attractors, parallel branches, and asymmetric basin boundary leads to many coexisting asymmetric stable states and so on. Apart from all these complex and rich phenomena, many others including offset-boosting with total amplitude control, and antimonotonicity are also presented. Finally, Pspice based simulations of the proposed system are included.
... Recently, some papers reported with linear and composite control of chaotic systems. [54,55] Among these control techniques, an SMC is considered as a better and effective controller because of its many inherent advantages. [56] Recently, more attention has been paid for overcoming the inherent disadvantages of an SMC like high gain and the chattering problem. ...
... It is visual from Fig. 15 and Fig. 16 that the reaching time to the sliding surfaces is less and also chattering is negligible. 7. Comparison of the performances of the controller in Ref. [54] and the controller of the present work ...
... This section presents a comparison of the performances of the controller of the present work and the controller reported in Ref. [54]. ...
This paper presents a new four-dimensional (4D) autonomous chaotic system which has first Lyapunov exponent of about 22 and is comparatively larger than many existing three-dimensional (3D) and 4D chaotic systems. The proposed system exhibits hyperbolic curve and circular paraboloid types of equilibria. The system has all zero eigenvalues for a particular case of an equilibrium point. The system has various dynamical behaviors like hyperchaotic, chaotic, periodic, and quasi-periodic. The system also exhibits coexistence of attractors. Dynamical behavior of the new system is validated using circuit implementation. Further an interesting switching synchronization phenomenon is proposed for the new chaotic system. An adaptive global integral sliding mode control is designed for the switching synchronization of the proposed system. In the switching synchronization, the synchronization is shown for the switching chaotic, stable, periodic, and hybrid synchronization behaviors. Performance of the controller designed in the paper is compared with an existing controller.
... Moreover, complete amplitude control can be realized using two independent configurable analog modules of an FPAA. Hence, despite the characteristic broadband spectrum of chaotic signals, FPAA can be used to realize an attenuator of amplitude conditioning [29], making chaotic signals potentially applicable in radar and communication engineering [30,31,32]. Additionally, starting from the 3-dimensional diffusionless Lorenz system [33], a linear feedback control input to one of the three equations can be used to extend the system to a hyperchaotic one for some appropriate range of parametric values. ...
... Now, choosing the time scale κ = 1000 and C = 1nF, then R = 1MΩ according to (29). Also, we can choose R 9 = R 10 = 100kΩ. ...
This paper presents a new chaotic system that has four attractors including two fixed point attractors and two symmetrical chaotic strange attractors. Dynamical properties of the system, viz. sensitive dependence on initial condition, Lyapunov spectrum, measure of strangeness, basin of attraction including the class and size of it, existence of strange attractor, bifurcation analysis, multistability, electronic circuit design, and hardware implementation are rigorously treated. It is found by numerical computations that the system has a far-reaching composite basin of attraction, which is important for engineering applications. Moreover, a circuit model of the system is realized using analog electronic components. A procedure is detailed for converting the system parameters into corresponding values of electronic components such as the circuital resistances while ensuring the dynamic ranges are well contained. Besides, the system is used as the source of control inputs for independent navigation of a differential drive mobile robot, which is subject to the Pfaffian velocity constraint. Due to innate properties of the system such as sensitivity on initial condition and topological mixing, the robot’s path becomes unpredictable and guaranteed to scan the workspace, respectively.
... Moreover, complete amplitude control can be realized using two independent configurable analog modules of an FPAA. Hence, despite the characteristic broadband spectrum of chaotic signals, FPAA can be used to realize an attenuator of amplitude conditioning [29], making chaotic signals potentially applicable in radar and communication engineering [30,31,32]. Additionally, starting from the 3-dimensional diffusionless Lorenz system [33], a linear feedback control input to one of the three equations can be used to extend the system to a hyperchaotic one for some appropriate range of parametric values. ...
... Now, choosing the time scale κ = 1000 and C = 1nF, then R = 1MΩ according to (29). Also, we can choose R 9 = R 10 = 100kΩ. ...
This paper presents a new chaotic system having four attractors, including two fixed point attractors and two symmetrical chaotic strange attractors. Dynamical properties of the system, viz. sensitive dependence on initial conditions, Lyapunov spectrum, strangeness measure, attraction basin, including the class and size of it, existence of strange attractor, bifurcation analysis, multistability, electronic circuit design, and hardware implementation, are rigorously treated. Numerical computations are used to compute the basin of attraction and show that the system has a far-reaching composite basin of attraction. Such a basin of attraction is vital for engineering applications. Moreover, a circuit model of the system is realized using analog electronic components. A procedure is detailed for converting the system parameters into corresponding electronic component values such as the circuital resistances while ensuring the dynamic ranges are bounded. Besides, the system is used as the source of control inputs for independent navigation of a differential drive mobile robot, which is subject to the Pfaffian velocity constraint. Due to the properties of sensitivity on initial conditions and topological mixing, the robot's path becomes unpredictable and guaranteed to scan the workspace, respectively.
... ∀ in a neighborhood of = 0. Theorem 1 [26,44] If the system (11) is a minimum phase system, the system (12) will be equivalent to a passive system and asymptotically stabilized at an equilibrium point if we let the local feedback control as follows: ...
... Chaos phenomenon is widely applied in the field of engineering. Specifically, electronic circuits [44][45][46][47][48][49][50], secure communication [51], robotic [52], random bits generator [53], and voice encryption [54]. In this section, we describe a possible circuit to implement new chaotic system with line of equilibria (1) as presented in Figure 4. ...
A new chaotic system with line equilibrium is introduced in this paper. This system consists of five terms with two transcendental nonlinearities and two quadratic nonlinearities. Various tools of dynamical system such as phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation diagram and Poincarè map are used. It is interesting that this system has a line of fixed points and can display chaotic attractors. Next, this paper discusses control using passive control method. One example is given to insure the theoretical analysis. Finally, for the new chaotic system, an electronic circuit for realizing the chaotic system has been implemented. The numerical simulation by using MATLAB 2010 and implementation of circuit simulations by using MultiSIM 10.0 have been performed in this study. 1. INTRODUCTION Discovering chaotic attractor is an important issue in chaotic systems. We can classify two kinds of chaotic attractors: self-excited attractors and hidden attractors [1-2]. The chaotic system such as Lorenz system [3], Rí µí±̈ ssler [4], Lí µí±¢̈ [5], Chen [6], Rucklidge [7] Sprott [8] etc, belongs to the self-excited attractors. The chaotic systems with hidden attractors are divided into three parts: (a) system with no equilibria [9] (b) system with stable equilibria [10] and (c) system with infinite number of equilibria [11]. Hidden attractors have been used in applied models such as a model of the phase-locked loop (PLL) [12], aircraft flight control systems [13], drilling system actuated by induction motor [14], Lorenz-like system describing convective fluid motion in rotating cavity [15] and a multilevel DC/DC converter [16]. Motivated by the major work of Jafari and Sprott, researchers focused on chaotic systems with line of equilibria. The nine simple chaotic flows with line of equilibria were proposed by Jafari and Sprott [17]. Five novel chaotic system with a line of equilibria and two parallel lines were proposed by Li and Sprott [18]. Li and Sprott have presented chaotic systems with a line of equilibria and two perpendicular lines of equilibria by using signum functions and absolute-value functions [19]. In addition, Li et al reported a hyperchaotic system with an infinite number of equilibria and circuit design [20]. Hyperchaos and horseshoe in a 4D memristive system with a line of equilibria were proposed in [21]. The simplest 4-D chaotic system
... Coherent oscillatory behaviour was also investigated in biological processes (Fróhlich, 1968). Designing of chaotic circuits is also an important area in chaos modelling Sprott, 2013, 2014;Li et al., 2017;Sambas et al., 2015Sambas et al., , 2016aSambas et al., , 2016bSambas et al., 2017;Volos et al., 2017;Pham et al., 2016c;Tacha et al., 2016). ...
... MultiSIM is a popular software for simulating electronic chaotic circuits Sambas et al., 2015Sambas et al., , 2016aSambas et al., , 2016b. In this section, four state variables of the biological snap system (4) x, y, z, w are rescaled with amplitude control methods Sprott, 2013, 2014;Li et al., 2017). In this study, a linear scaling is considered as follows: By applying Kirchhoff's circuit laws into the circuit in Figures 18-21, we get its circuital equations: ...
In this work, we build a new four-dimensional autonomous biological snap oscillator model for enzyme-substrate reactions in a brain waves model. We investigate the process modelling of the new autonomous biological snap oscillator via phase portraits, simulations, dissipativity, symmetry, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation analysis, Poincaré map, etc. In addition, it is interesting that the electronic circuit model of the new biological snap oscillator is also investigated and phase portraits are analysed in MultiSIM.
... Coherent oscillatory behaviour was also investigated in biological processes (Fróhlich, 1968). Designing of chaotic circuits is also an important area in chaos modelling Sprott, 2013, 2014;Li et al., 2017;Sambas et al., 2015Sambas et al., , 2016aSambas et al., , 2016bSambas et al., 2017;Volos et al., 2017;Pham et al., 2016c;Tacha et al., 2016). ...
... MultiSIM is a popular software for simulating electronic chaotic circuits Sambas et al., 2015Sambas et al., , 2016aSambas et al., , 2016b. In this section, four state variables of the biological snap system (4) x, y, z, w are rescaled with amplitude control methods Sprott, 2013, 2014;Li et al., 2017). In this study, a linear scaling is considered as follows: By applying Kirchhoff's circuit laws into the circuit in Figures 18-21, we get its circuital equations: ...
In this work, we build a new four-dimensional autonomous biological snap oscillator model for enzyme-substrate reactions in a brain waves model. We investigate the process modelling of the new autonomous biological snap oscillator via phase portraits, simulations, dissipativity, symmetry, Lyapunov exponents, Kaplan-Yorke dimension, bifurcation analysis, Poincaré map, etc. In addition, it is interesting that the electronic circuit model of the new biological snap oscillator is also investigated and phase portraits are analysed in MultiSIM.
... In addition to the multistability mentioned above, symmetric dynamical systems exhibit many interesting behaviors, such as spontaneous symmetry breaking, instantaneous chaos, attractor doubling, and merger crisis [38][39][40][41][42]. However, the interesting dynamic behavior of symmetry breaking is rarely studied in the literature [43][44][45][46]. ...
A new 3D offset-boostable symmetric system is proposed by an absolute value function introduced. The system seems to be more fragile and easier to the state of broken symmetry. Coexisting symmetric pairs of attractors get closer and closer, and finally get emerged together. Basins of attraction show how these coexisting attractors are arranged in phase space. All these coexisting attractors can be easily offset boosted in phase space by a single constant when the initial condition is revised accordingly. PSpice simulations prove all the phenomena.
... g a and g b are the gains of analog multipliers M a and M b respectively. The main circuit consists of two channels to integrate the two ordinary differential equations of (4a) and (4b) (Bao et al. 2017;Li et al. 2017], as drawn at the bottom of Fig. 8. The main circuit in Fig. 8 has two dynamic elements, corresponding to two state variables of V v , and V r . ...
Neurons can exhibit abundant electrical activities due to physical effects of various electrophysiology environments. The electromagnetic induction flows can be triggered by changes in neuron membrane potential, which can be equivalent to a memristor applying on membrane potential. To imitate the electromagnetic induction effects, we propose a three-variable memristor-based Wilson neuron model. Using several kinetic analysis methods, the memristor parameter- and initial condition-related electrical activities are explored intensively. It is revealed that the memristive Wilson neuron model can display rich electrical activities, including the asymmetric coexisting electrical activities and antimonotonicity phenomenon. Finally, using off-the-shelf discrete components, an analog circuit on a hardware level is implemented to verify the numerically simulated coexisting electrical activities. Studying these rich electrical activities in neurons can build the groundwork to widen the neuron-based engineering applications.
... Nonsingular terminal sliding mode control is developed to guarantee the synchronization between complex variable chaotic systems in a given finite time [26,27]. Some authors introduce amplitude control of chaotic systems although chaotic signals could have great applications in radar and communication engineering [28][29][30]. In Ref. [31] a free-controlled chaotic oscillation is presented having a great potential in electronic circuits where chaotic signal can be controlled by two independent potentiometers. ...
The dynamics of higher order MIMO cascade nonlinear systems, MIMOn, (n>3), using simulation of bifurcation and Lyapunov diagrams and spatial phase portraits is analyzed in this paper. One of the characteristics of these systems is the possibility of spatial hyperchaos appearance. Control of spatial hyperchaos in MIMOn systems using modified Pyragas method is analyzed, also. The results are illustrated by example of MIMO6 system.
... There is an increasing tendency that field programmable analog arrays (FPAA) is used in many areas: system modeling, signal processing, feature extraction, for example. As with FPGA, FPAAbased circuit implementation has the advantages in both hardware and software of fast response, reduced cost and simplicity of design for the programmable architecture [36] . Moreover, there is no need to make a discretization with the variables in FPAA, which is different from FPGA. ...
Chaos is a paradigm shift of all science, which provides a collection of concepts and methods to analyze a novel behavior that can arise in a wide range of disciplines. However, most of researches in simulations and applications of chaos are performed on finite-state automata, which inevitably causes chaos to collapse. Here we present a hybrid model by controlling digital system with continuous chaotic system to construct chaos on finite-state automata. A new concept and method named Generalized Symbolic Dynamics (GSD) is proposed to target the hybrid system. Based on GSD, a rigorous proof is given that the controlled digital system is chaotic in the sense of Devaney. Moreover, analog-digital hybrid circuit is built for the digital chaotic system. Finally, a simple pseudorandom number generator is designed as a proof of concept. Results show that the proposed generator has good performance for cryptography. Such digital chaotic systems, which are not subject to degradation, could pave the way for widespread applications of chaos.
... Further exploration shows that sometimes two types of coexisting attractors may exist in a normal system, but it is difficult to divide them clearly and find which the dominant factor is. When the system is constructed in circuit [17][18][19][20][21][22], the increasing process of the chaotic signal may be hidden for the saturation of the IC chip. ...
... Besides, the system has a large maximum lyapunov exponent when the is very small, which indicates the system is more complex. These characteristics will make the SFOH system have good application prospects in image encryption [36]. Besides, with the increase of the derivative order, the maximum Lyapunov exponent (MLE) is gradually decreasing, and the system is changed (2) Fix , and changes at the range . ...
In this paper, a simplest fractional-order hyperchaotic (SFOH) system is obtained when the fractional calculus is applied to the piecewise-linear hyperchaotic system, which possesses seven terms without any quadratic or higher-order polynomials. The numerical solution of the SFOH system is investigated based on the Adomian decomposition method (ADM). The methods of segmentation and replacement function are proposed to solve this system and analyze the dynamics. Dynamics of this system are demonstrated by means of phase portraits, bifurcation diagrams, Lyapunov exponent spectrum (LEs) and Poincaré section. The results show that the system has a wide chaotic range with order change, and large Lyapunov exponent when the order is very small, which indicates that the system has a good application prospect. Besides, the parameter a is a partial amplitude controller for the SFOH system. Finally, the system is successfully implemented by digital signal processor (DSP). It lays a foundation for the application of the SFOH system. Keywords: Chaos, Fractional calculus, Simplest fractional-order hyperchaotic system, Adomian decomposition method DSP implementation
... After that, Fatma Yildirim Dalkiran and J. C. Sprott realized a fourth-order hyperjerk system based on FPAA [14]. Chunbiao Li designed and implemented chaotic systems with complete amplitude control and constructed infinitely many attractors in a programmable chaotic circuit based on FPAA [15,16]. But the above researchers had not designed and implemented time-delay chaotic systems based on the FPAA. ...
Time-delay chaotic systems can have hyperchaotic attractors with large numbers of positive Lyapunov exponents, and can generate highly stochastic and unpredictable time series with simple structures, which is very suitable as a secured chaotic source in chaotic secure communications. But time-delay chaotic systems are generally designed and implemented by using analog circuit design techniques. Analog implementations require a variety of electronic components and can be difficult and time consuming. At this stage, we can now solve this question by using FPAA (Field-Programmable Analog Array). FPAA is a programmable device for implementing multiple analog functions via dynamic reconfiguration. In this paper, we will introduce two FPAA-based design examples: An autonomous Ikeda system and a non-autonomous Duffing system, to show how a FPAA device is used to design programmable analog time-delay chaotic systems and analyze Shannon entropy and Lyapunov exponents of time series output by circuit and simulation systems.
... Although pure mathematical chaotic systems have been investigated widely in the literature (e.g. simplest chaotic systems [Li et al., 2016;Sprott, 1994Sprott, , 1997aSprott, , 1997bSprott, , 2000, chaotic systems with hidden attractors [Dudkowski et al., 2016;Jafari et al., 2017;Pham et al., 2017b;Pham et al., 2018;Ren et al., 2018], fractional order [Ruan et al., 2018;He et al., 2016a;He et al., 2016b;He et al., 2017;He et al., 2018a;Wang et al., 2014a;Wang et al., 2014b], hyperchaotic systems [Yu et al., 2018;Zhang et al., 2017a;Zhang et al., 2017b;Ruan et al., 2016;He et al., 2015;Pan et al., 2011;Rajagopal et al., 2017a;Rajagopal et al., 2017c], multiwing systems [Ai et al., 2018;He et al., 2013], especial form of equilibrium Sprott et al., 2015;Pham et al., 2017a;Pham et al., 2017cPham et al., , 2017dJafari et al., 2018c], controllable chaotic systems [Li et al., 2015b;Li et al., 2017a;Li et al., 2017d], especial features Li et al., 2015a;Rajagopal et al., 2017b], and so on), real-world chaotic systems and models have been paid less attention. Gilbert Walker identified pressure difference between the eastern and western sides of the Pacific Ocean and coined the term Southern Oscillation. ...
... Another distinguishing property of system (1) is offset boosting by a single constant. From the physical view point, the possibility of amplitude control represents an important feature of a potential chaos generator [50][51][52][53][54][55][56][57]. The introduced feedback state can be viewed as a useful alternative to control the amplitude of the variables. ...
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, ifurcation
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.
The activation function is crucial in the Hopfield neural network (HNN) to restrict the input–output relation of each neuron. The physical realizability and simplicity of the hardware circuit of activation function are beneficial to promote the practical engineering application of the HNN. However, the HNN commonly used hyperbolic tangent activation function involves a complex hardware circuit implementation. This paper discusses a piecewise-linear activation function (PWL-AF) with simplified circuit implementation and a tri-neuron small-world HNN is built as a paradigm. The hardware implementation circuit of the HNN is greatly simplified, benefited from the PWL-AF with a simple analog circuit. Meanwhile, the dynamics related to the PWL-AF and initial conditions are numerically explored. The numerical results demonstrate that the PWL-AF-based HNN can produce dynamical behaviors like the HNN based on the hyperbolic tangent activation function. Nevertheless, the multistability with up to six kinds of coexisting multiple attractors emerged because of the PWL-AF breakpoint. This can give more flexible and potential aspects in multistability-based engineering applications. Especially, the PWL-AF breakpoint value simultaneously acts as the offset booster and amplitude controller in regulating the offset boosting and amplitude rescaling of neuron states. Afterwards, an analog circuit with three straightforward operational amplifiers (op-amp)-based circuit modules is designed for the PWL-AF, and a PCB-based analog circuit is thereby implemented for the tri-neuron small-world HNN. The hardware experiments agree with the numerical simulations, implying the feasibility of the PWL-AF simplification for the HNN.
In order to make the peak and offset of the signal meet the requirements of artificial equipment, dynamical analysis and geometric control of laser system has become indispensable. In this paper, a locally active memristor with non-volatile memory is introduced into a complex-valued Lorenz laser system. By using numerical measures, complex dynamical behaviors of the memristive laser system are uncovered. It appears the alternating appearance of quasi-periodic and chaotic oscillations. The mechanism of transformation from a quasi-periodic pattern to a chaotic pattern is revealed from the perspective of Hamilton energy. Interestingly, initial-values-oriented extreme multi-stability patterns are found, where the coexisting attractors have the same Lyapunov exponents. In addition, the introduction of a memristor greatly improves the complexity of the laser system. Moreover, to control the amplitude and offset of the chaotic signal, two kinds of geometric control methods including amplitude control and rotation control are designed. The results show that these two geometric control methods have revised the size and position of the chaotic signal without changing the chaotic dynamics. Finally, a digital hardware device is developed and the experiment outputs agree fairly well with those of the numerical simulations.
Kaotik sinyaller kendi içerisinde bir düzeni olan ancak düzensiz davranış sergileyen tahmin edilmesi zor sinyallerdir. Kaotik sinyallerin en önemli kullanım alanlarından biri güvenli haberleşme sistemleridir. Haberleşme sisteminin güvenilirliği kaotik işaretin karmaşıklığına bağlıdır. Kaotik sinyallerin karmaşıklığını artıran birçok yöntem ile haberleşme sistemleri gerçekleştirilmektedir. Bu yöntemlerin BER/SNR performansına artırmaktan çok sistemin güvenilirliğini artırmıştır. Kesir dereceli kaotik sinyallerin kaos tabanlı haberleşme sistemlerinde hem haberleşme güvenilirliğinin hem BER/SNR performansının artmasına olumlu etkisi olabileceği düşünülmektedir. Bu nedenle çalışmada kesir dereceli kaotik tabanlı haberleşme sistemlerinin BER/SNR performansının bilgisayar benzetimi ile dinamik sistemin nümerik analizi gerçekleştirilmiştir. Ayrıca benzetim analog tabanlı FPAA yapılar kullanılarak deneysel olarak tekrarlanmıştır. Elde edilen benzetim ve deneysel sonuçlar tam dereceli benzer çalışmalarla karşılaştırılmıştır.
In this letter, a compact memristor structure unit is applied for constructing the discrete chaotic system and, consequently, a memristor-type chaotic mapping is designed. Two independent system parameters are proven to be partial and total amplitude controllers. Meanwhile, the internal memristor parameter returns the map a typical bifurcation. Finally, a hardware experiment based on STM32 is carried out by verifying the theoretical finding. To the best of our knowledge, the memristor-type chaotic mapping has not been previously reported.
Carbon nanotubes (CNTs) are used in various nano-electromechanical systems (NEMS), and the parameters (including the system parameters and the excitation parameters) may result in chaos in these systems. Thus, understanding the mechanism of the chaos arising from NEMS is vital for CNT’s applications. Motivated by this need, the chaotic properties of a single-walled carbon nanotube system resulting from parametric excitation and external excitation are investigated in this paper. The criteria for the existence of the chaotic behavior in the system with periodic and quasi-periodic perturbations are obtained by the homoclinic Melnikov and the second-order average methods. Furthermore, in order to show the connection between periodic motion and complex behavior, the subharmonic periodic solutions, inside and outside the homoclinic loop, are analyzed. The global structure and the saddle-node bifurcation of the unperturbed averaged system are also considered. Finally, the Poincaré section and the transversal intersection of the unstable and stable manifolds are presented to verify the occurrence of chaos or subharmonic solution. The simulation results confirm the correctness of the theoretical analysis.
As the competition for marine resources is increasingly fierce, the security of underwater acoustic communication has attracted a great deal of attention. The information and location of the communicating platform can be leaked during the traditional underwater acoustic communication technology. According to the unique advantages of chaos communication, we put forward a novel communication scheme using complex parameter modulation and the complex Lorenz system. Firstly, we design a feedback controller and parameter update laws in a complex-variable form with rigorous mathematical proofs (while many previous references on the real-variable form were only special cases in which the imaginary part was zero), which can be realized in practical engineering; then we design a new communication scheme employing parameter modulation. The main parameter spaces of the complex Lorenz system are discussed, then they are adopted in our communication scheme. We also find that there exist parametric attractors in the complex Lorenz system. We make numerical simulations in two channels for digital signals and the simulations verify our conclusions.
The cooperative behaviors resulted from the interaction of coupled identical oscillators have been investigated intensively. However, the coupled oscillators in practice are nonidentical, and there exist mismatched parameters. It has been proved that under certain conditions, complete synchronization can take place in coupled nonidentical oscillators with the same equilibrium points, yet other cooperative behaviors are not addressed. In this paper, we further consider two coupled nonidentical oscillators with the same equilibrium points, where one oscillator is convergent while the other is chaotic, and explore their cooperative behaviors. We find that the coupling mode and the coupling strength can bring the coupled oscillators to different cooperation behaviors in unidirectional or undirected couplings. In the case of directed coupling, death islands appear in two-parameter spaces. The mechanism inducing these transitions is presented.
This paper proposes a universal method for a cascade chaotic system (CCS). CCSs may own better performances, including larger maximum Lyapunov exponents, extended full mapping range of chaos, and more coefficient variations. The chaos-cascade theorems had been proposed in our previous papers, which are more suitable for discrete chaotic systems with the same domain. In this paper, we further improve a universal method to normalize arbitrary discrete chaotic systems for constructing a series of CCSs. Besides, the inheritance and enhancement of robustness from the subsystem are first explored for CCSs. Finally, the designed CCS is implemented on field programmable gate array board. The generated chaotic sequences are obtained by an oscilloscope and tested by NIST software.
In this paper, we propose a new hyperchaotic complex system and systematically study its basic dynamic characteristics such as dissipation, Lyapunov exponent, attractor, parameter changes, and ${Poincar}\acute{e}$ cross-section. The system is based on the complex Lorenz chaotic system, and the chaotic system has higher dimensionality and more parameters than complex Lorenz system. We provide a general method for constructing complex chaotic systems. We use the state linear feedback method to realize the synchronization control of the chaotic system as the driving system and the response system and simulate all of the systems by Simulink. The chaotic system and its synchronization simulation are simulated by Simulink to verify their hardware feasibility.
In this paper, the effects of a bias term modeling a constant excitation force on the dynamics of an infinite-equilibrium chaotic system without linear terms are investigated. As a result, it is found that the bias term reduces the number of equilibrium points (transition from infinite equilibria to only two equilibria) and breaks the symmetry of the model. The nonlinear behavior of the system is highlighted in terms of bifurcation diagrams, maximal Lyapunov exponent plots, phase portraits, and basins of attraction. Some interesting phenomena are found including, for instance, hysteretic dynamics, multistability, and coexisting bifurcation branches when monitoring the system parameters and the bias term. Also, we demonstrate that it is possible to control the offset and amplitude of the chaotic signals generated. Compared to some few cases previously reported on systems without linear terms, the plethora of behaviors found in this work represents a unique contribution in comparison with such type of systems. A suitable analog circuit is designed and used to support the theoretical analysis via a series of Pspice simulations.
In this review, we categorize chaotic systems with special properties in their equilibrium into eight groups: Systems with no equilibrium (NE), systems with stable equilibrium (SE), systems with line equilibrium (LE), systems with curve equilibrium (CE), systems with plane equilibrium (PE), systems with surface equilibrium (ES), systems with unstable equilibrium (US), and systems which belong to more than one of the above categories (Chameleon systems). We introduce the pioneer works from each group. Also, we review some recent important papers related to those groups.
In this paper, a new two-dimensional nonlinear oscillator with an unusual sequence of rational and irrational parameters is introduced. This oscillator has endless coexisting limit cycles, which make it a megastable dynamical system. By periodically forcing this system, a new system is designed which is capable of exhibiting an infinite number of coexisting asymmetric torus and strange attractors. This system is implemented by an analog circuit, and its Hamiltonian energy is calculated.
Designing new chaotic system with specific features is an interesting field in nonlinear dynamics. In this paper, some new chaotic systems with cyclic symmetry are proposed. In order to understand the overall behavior of such systems, the dynamical analyses such as stability analysis, bifurcation and Lyapunov exponent analysis are done. The accurate examination of bifurcation plot represents that these systems are multistable which makes them more interesting. Also, the basin of attraction of these systems is investigated to detect the type of attractors of these systems which are self-excited. Finally, the circuit implementation is carried out to show their feasibility.
This paper reports the finding a new chaotic system with a conch-shaped equilibrium curve. The proposed system is a new addition to existing chaotic systems with closed curves of equilibrium points in the literature. Lyapunov exponents of the new chaotic system are studied for verifying chaos properties and phase portraits of the new system via MATLAB are unveiled. An electronic circuit simulation of the new chaotic system with conch-shaped equilibrium curve is shown using MultiSIM to check the model feasibility.
In the chaos literature, there is currently significant interest in the discovery of new chaotic systems with hidden chaotic attractors. A new 4-D chaotic system with only two quadratic nonlinearities is investigated in this work. First, we derive a no-equilibrium chaotic system and show that the new chaotic system exhibits hidden attractor. Properties of the new chaotic system are analyzed by means of phase portraits, Lyapunov chaos exponents, and Kaplan-Yorke dimension. Then an electronic circuit realization is shown to validate the chaotic behavior of the new 4-D chaotic system. Finally, the physical circuit experimental results of the 4-D chaotic system show agreement with numerical simulations.
Simple dynamical systems are of interest always. In this paper we propose a simple jerk system with one exponential nonlinearity. Dynamical properties of the proposed system are investigated. To show the practical realisability of the proposed system we implement the exponential jerk system using off the shelf components. Fractional order and time delays are considered as complex analysis patterns
of nonlinear systems. We investigate the fractional order time delayed exponential jerk system.For numerical analysis we use the modified Adomian Decomposition Method. To show the engineering
importance of the proposed system, we derive a pseudo random number generator based on it.Various test results are presented to show the randomness of the system.
To analyze the complexity of chaotic pseudo-random sequences accurately, spectral entropy (SE) algorithm is used to analyze chaotic pseudo-random sequences generated by Logistic map, Gaussian map or TD-ERCS system. The SE algorithm has few parameters, and has high robustness with the sequence length N (the only parameter) and the pseudo-random binary number K. Using sliding window method, the evolution features are analyzed, and complexity of discrete chaotic systems with different initial conditions and system parameters are calculated. The results show that SE algorithm is effective for analyzing the complexity of the chaotic pseudo-random sequences, and TD-ERCS is a high complexity system with wide parameter range, and has the best complex performance among the three chaotic systems. The complexity of the same chaotic system with different initial values fluctuates within a small range. It provides a theoretical and experimental basis for the applications of chaotic sequences in the field of information security.
The noise-like chaotic signal can be generated with very simple nonlinear circuits, and has broad bandwidth and aperiodic properties. These characteristics have drawn considerable attention in the radar community. In the past, the chaotic signal or its modulated version serves as a transmitting signal, and the traditional correlation-type receiver is used for processing. In this sense, the chaotic signal acts as a radar waveform in the noise signal radar, and hence the performance advantages are not distinct. Here, we present a scheme for processing chaotic radar signals. We find a simple relation between the target parameters (range and velocity) and the system parameters of chaos-generating system. With this relation, the measurement of the target parameters is transformed into estimation of the system parameters from the radar return signals. Equipped with high resolution parameter estimation techniques, the proposed principles provide a way to develop high resolution noise signal radars.
A general approach based on the introduction of a control function for constructing amplitude-controllable chaotic systems with quadratic nonlinearities is discussed in this paper. We consider three control regimes where the control functions are applied to different coefficients of the quadratic terms in a dynamical system. The approach is illustrated using the Lorenz system as a typical example. It is proved that wherever control functions are introduced, the amplitude of the chaotic signals can be controlled without altering the Lyapunov exponent spectrum.
Molecular rotation has attracted much attention with respect to the development of artificial molecular motors, in an attempt to mimic the intelligent and useful functions of biological molecular motors. Random motion of molecular rotators—for example the 180° flip-flop motion of a rotatory unit—causes a rotation of the local structure. Here, we show that such motion is controllable using an external electric field and demonstrate how such molecular rotators can be used as polarization rotation units in ferroelectric molecules. In particular, m-fluoroanilinium forms a hydrogen-bonding assembly with dibenzo[18]crown-6, which was introduced as the counter cation of [Ni(dmit)2]- anions (dmit2-=2-thioxo-1,3-dithiole-4,5-dithiolate). The supramolecular rotator of m-fluoroanilinium exhibited dipole rotation by the application of an electric field, and the crystal showed a ferroelectric transition at 348 K. These findings will open up new strategies for ferroelectric molecules where a chemically designed dipole unit enables control of the nature of the ferroelectric transition temperature.
In addition to exhibiting a rich variety of bifurcation and chaos via tuning parameters, a chaotic system introduced by Sprott can be modeled and realized with a fixed main system block and many different changeable nonlinear function blocks such as piecewise-linear function, cubic function and other trigonometric functions. This system is very suitable for implementing a programmable chaos generator according to its changeable nonlinearity. This paper presents a FPAA (Field Programmable Analog Array)-based programmable imple-mentation of this system. Nonlinear function blocks used in this chaotic system are modeled with FPAA programming and a model is rapidly changed for realizing other nonlinear functions.
We investigated a switchable ferroelectric diode effect and its physical
mechanism in Pt/BiFeO3/SrRuO3 thin-film capacitors. Our results of electrical
measurements support that, near the Pt/BiFeO3 interface of as-grown samples, a
defective layer (possibly, an oxygen-vacancy-rich layer) becomes formed and
disturbs carrier injection. We therefore used an electrical training process to
obtain ferroelectric control of the diode polarity where, by changing the
polarization direction using an external bias, we could switch the transport
characteristics between forward and reverse diodes. Our system is characterized
with a rectangular polarization hysteresis loop, with which we confirmed that
the diode polarity switching occurred at the ferroelectric coercive voltage.
Moreover, we observed a simultaneous switching of the diode polarity and the
associated photovoltaic response dependent on the ferroelectric domain
configurations. Our detailed study suggests that the polarization charge can
affect the Schottky barrier at the ferroelectric/metal interfaces, resulting in
a modulation of the interfacial carrier injection. The amount of
polarization-modulated carrier injection can affect the transition voltage
value at which a space-charge-limited bulk current-voltage (J-V) behavior is
changed from Ohmic (i.e., J ~ V) to nonlinear (i.e., J ~ V^n with n \geq 2).
This combination of bulk conduction and polarization-modulated carrier
injection explains the detailed physical mechanism underlying the switchable
diode effect in ferroelectric capacitors.
A circuit implementation of the chaotic Lorenz system is
described. The chaotic behavior of the circuit closely matches the
results predicted by numerical experiments. Using the concept of
synchronized chaotic systems (SCS's), two possible approaches to secure
communications are demonstrated with the Lorenz circuit implemented in
both the transmitter and receiver. In the first approach, a chaotic
masking signal is added at the transmitter to the message, and at the
receiver, the masking is regenerated and subtracted from the received
signal. The second approach utilizes modulation of the coefficients of
the chaotic system in the transmitter and corresponding detection of
synchronization error in the receiver to transmit binary-valued bit
streams. The use of SCS's for communications relies on the robustness of
the synchronization to perturbations in the drive signal. As a step
toward further understanding the inherent robustness, we establish an
analogy between synchronization in chaotic systems, nonlinear observers
for deterministic systems, and state estimation in probabilistic
systems. This analogy exists because SCS's can be viewed as performing
the role of a nonlinear state space observer. To calibrate the
robustness of the Lorenz SCS as a nonlinear state estimator, we compare
the performance of the Lorenz SCS to an extended Kalman filter for
providing state estimates when the measurement consists of a single
noisy transmitter component
In this paper, a new approach for communication using chaotic
signals is presented. In this approach, the transmitter contains a
chaotic oscillator with a parameter that is modulated by an information
signal. The receiver consists of a synchronous chaotic subsystem
augmented with a nonlinear filter for recovering the information signal.
The general architecture is demonstrated for Lorenz and Rossler systems
using numerical simulations. An electronic circuit implementation using
Chua's circuit is also reported, which demonstrates the practicality of
the approach
Bandwidth enhancement of chaotic signal generated from chaotic laser by using continuous-wave optical injection is experimentally demonstrated. A distributed feedback semiconductor laser with optical feedback is employed as the chaotic laser. The bandwidth of the chaotic signal is enhanced roughly three times by optical injection into the chaotic laser compared with the bandwidth when there is no optical injection.
The authors describe the conditions necessary for synchronizing a
subsystem of one chaotic system with a separate chaotic system by
sending a signal from the chaotic system to the subsystem. The general
scheme for creating synchronizing systems is to take a nonlinear system,
duplicate some subsystem of this system, and drive the duplicate and the
original subsystem with signals from the unduplicated part. This is a
generalization of driving or forcing a system. The process can be
visualized with ordinary differential equations. The authors have build
a simple circuit based on chaotic circuits described by R. W. Newcomb et
al. (1983, 1986), and they use this circuit to demonstrate this chaotic
synchronization
Chaos synchronization in fractional-order unified chaotic system is disscussed in this paper. Based on the stability theory of fractional-order system, the control law is presented to achieve chaos synchronization. The advantage of the proposed controllers is that they are linear and have lower dimensions than that of the states. With this technique it is very easy to find the suitable feedback constant. Simulation results for fractional-order Lorenz, Lü and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.
This paper investigates the exponential synchronization of linearly coupled ordinary differential systems. The intrinsic nonlinear dynamics may not satisfy the QUAD condition or weak-QUAD condition. First, it gives a new method to analyze the exponential synchronization of the systems. Second, two theorems and their corollaries are proposed for the local or global exponential synchronization of the coupled systems. Finally, an application to the linearly coupled Hopfield neural networks and several simulations are provided for verifying the effectiveness of the theoretical results.
In this paper, synchronization for a class of uncertain fractional-order neural networks with external disturbances is discussed by means of adaptive fuzzy control. Fuzzy logic systems, whose inputs are chosen as synchronization errors, are employed to approximate the unknown nonlinear functions. Based on the fractional Lyapunov stability criterion, an adaptive fuzzy synchronization controller is designed, and the stability of the closed-loop system, the convergence of the synchronization error, as well as the boundedness of all signals involved can be guaranteed. To update the fuzzy parameters, fractional-order adaptations laws are proposed. Just like the stability analysis in integer-order systems, a quadratic Lyapunov function is used in this paper. Finally, simulation examples are given to show the effectiveness of the proposed method.
A meminductor is a new type of memory device. It is of importance to study meminductor model and its application in nonlinear circuit prospectively. For this purpose, we present a novel mathematical model of meminductor, which considers the effects of internal state variable and therefore will be more consistent with future actual meminductor device. By using several operational amplifiers, multipliers, capacitors and resistors, the equivalent circuit of the model is designed for exploring its characteristics. This equivalent circuit can be employed to design meminductor-based application circuits as a meminductor emulator. By employing simulation experiment, we investigate the characteristics of this meminductor driven by sinusoidal excitation. The characteristic curves of current-flux (i-φ), voltage-flux (v-φ), v-ρ (internal variable of meminductor) and φ-ρ for the meminductor model are given by theoretical analyses and simulations. The curve of current-flux (i-φ) is a pinched hysteretic loop passing through the origin. The area bounding each sub-loop deforms as the frequency varies, and with the increase of frequency, the shape of the pinched hysteretic loop tends to be a straight line, indicating a dependence on frequency for the meminductor. Based on the meminductor model, a meminductive Wien-bridge chaotic oscillator is designed and analyzed. Some dynamical properties, including equilibrium points and the stability, bifurcation and Lyapunov exponent of the oscillator, are investigated in detail by theoretical analyses and simulations. By utilizing Lyapunov spectrum, bifurcation diagram and dynamical map, it is found that the system has periodic, quasi-periodic and chaotic states. Furthermore, there exist some complicated nonlinear phenomena for the system, such as constant Lyapunov exponent spectrum and nonlinear amplitude modulation of chaotic signals. Moreover, we also find the nonlinear phenomena of coexisting bifurcation and coexisting attractors, including coexistence of two different chaotic attractors and coexistence of two different periodic attractors. The phenomenon shows that the state of this oscilator is highly sensitive to its initial valuse, not only for chaotic state but also for periodic state, which is called coexistent oscillation in this paper. The basic mechanism and potential applications of the existing attractors are illustrated, which can be used to generate robust pseudo random sequence, or multiplexed pseudo random sequence. Finally, by using the equivalent circuit of the proposed meminducive model, we realize an analog electronic circuit of the meminductive Wien-bridge chaotic system. The results of circuit experiment are displayed by the oscilloscope, which can verify the chaotic characteristics of the oscillator. The oscillator, as a pseudo random signal source, can be used to generate chaotic signals for the applications in chaotic cryptography and secret communications.
A new regime of chaotic flows is explored in which one of the variables has the freedom of offset boosting. By a single introduced constant, the DC offset of the variable can be boosted to any level, and therefore the variable can switch between a bipolar signal and a unipolar signal according to the constant. This regime of chaotic flows is convenient for chaos applications since it can reduce the number of components required for signal conditioning. Offset boosting can be combined with amplitude control to achieve the full range of linear transformations of the signal. The symmetry of the variable-boostable system may be destroyed by the new introduced boosting controller; however, a different symmetry is obtained that preserves any existing multistability.
A new synchronization scheme for chaotic (hyperchaotic) maps with different dimensions is presented. Specifically, given a drive system map with dimension n and a response system with dimension m, the proposed approach enables each drive system state to be synchronized with a linear response combination of the response system states. The method, based on the Lyapunov stability theory and the pole placement technique, presents some useful features: (i) it enables synchronization to be achieved for both cases of n < m and n > m; (ii) it is rigorous, being based on theorems; (iii) it can be readily applied to any chaotic (hyperchaotic) maps defined to date. Finally, the capability of the approach is illustrated by synchronization examples between the two-dimensional Hénon map (as the drive system) and the three-dimensional hyperchaotic Wang map (as the response system), and the three-dimensional Hénon-like map (as the drive system) and the two-dimensional Lorenz discrete-time system (as the response system).
In this paper, successive lag synchronization (SLS) on a dynamical network with communication delay is investigated. In order to achieve SLS on the dynamical network with communication delay, we design linear feedback control and adaptive control, respectively. By using the Lyapunov function method, we obtain some sufficient conditions for global stability of SLS. To verify these results, some numerical examples are further presented. This work may find potential applications in consensus of multi-agent systems.
A novel chaotic attractor with a fractal wing structure is proposed and analyzed in terms of its basic dynamical properties. The most interesting feature of this system is that it has complex dynamical behavior, especially coexisting attractors for particular ranges of the parameters, including two coexisting periodic or strange attractors that can coexist with a third strange attractor. Amplitude and phase control methods are described since they are convenient for circuit design and chaotic signal applications. An appropriately chosen parameter in a particular quadratic coefficient can realize partial amplitude control. An added linear term can change the symmetry and provide an accessible knob to control the phase polarity. Finally, an amplitude-phase controllable circuit is designed using PSpice, and it shows good agreement with the theoretical analysis.
A general method is introduced for controlling the amplitude of the variables in chaotic systems by modifying the degree of one or more of the terms in the governing equations. The method is applied to the Sprott B system as an example to show its flexibility and generality. The method may introduce infinite lines of equilibrium points, which influence the dynamics in the neighborhood of the equilibria and reorganize the basins of attraction, altering the multistability. However, the isolated equilibrium points of the original system and their stability are retained with their basic properties. Electrical circuit implementation shows the convenience of amplitude control, and the resulting oscillations agree well with results from simulation.
Hopf bifurcation, dynamics at infinity and robust modified function projective synchronization (RMFPS) problem for Sprott E system with quadratic perturbation were studied in this paper. By using the method of projection for center manifold computation, the subcritical and the supercritical Hopf bifurcation were analyzed and obtained. Then, in accordance with the Poincare compactification of polynomial vector field in R3, the dynamical behaviors at infinity were described completely. Moreover, a RMFPS scheme of this special system was proposed and proved based on Lyapunov direct method. The simulation results demonstrate the correctness of the dynamics analysis and the effectiveness of the proposed synchronization strategy.
When the polarity information in diffusionless Lorenz equations is preserved or removed, a new piecewise linear hyperchaotic system results with only signum and absolute-value nonlinearities. Dynamical equations have seven terms without any quadratic or higher order polynomials and, to our knowledge, are the simplest hyperchaotic system. Therefore, a relatively simple hyperchaotic circuit using diodes is constructed. The circuit requires no multipliers or inductors, as are present in other hyperchaotic circuits, and it has not been previously reported.
We fabricated and characterized new ambipolar silicon nanowire (SiNW) FET transistors featuring two independent gate-all-around electrodes and vertically stacked SiNW channels. One gate electrode enables dynamic configuration of the device polarity (n or p-type), while the other switches on/off the device. Measurement results on silicon show Ion/Ioff > 106 and S ≈ 64mV/dec (70mV/dec) for p(n)-type operation in the same device. We show that XOR operation is embedded in the device characteristic, and we demonstrate for the first time a fully functional 2-transistor XOR gate.
Abstract—The ambiguity functions of a kind of direct chaotic radar system are investigated. In this radar system, a microwave chaotic Colpitts oscillator is employed to generate the source signal that is directly transmitted through a wideband antenna without modulation. The auto-ambiguity function of this radar system shows many sidelobes which makes the unambiguous detection difficult. It is because the spectrum of the chaotic signal is not very flat and smooth, with pulsation peaks in it. The cross-ambiguity functions of the direct radar system have also been investigated to evaluate the electronic counter countermeasure (ECCM) performance and the “multi-user” characteristic. It is shown that rather excellent ECCM capability can be achieved in this radar system with transmitting chaotic signals generated by circuits with same parameters but at different time or with slightly different circuit parameters. In addition, several possible methods to reshape the spectrum of the chaotic signal from microwave Colpitts oscillators to improve the unambiguous detection performance are suggested at the end of this paper. 2,Shi et al.
This work introduces hardware implementation of artificial neural networks (ANNs) with learning ability on field programmable gate array (FPGA) for dynamic system identification. The learning phase is accomplished by using the improved particle swarm optimization (PSO). The improved PSO is obtained by modifying the velocity update function. Adding an extra term to the velocity update function reduced the possibility of stucking in a local minimum. The results indicates that ANN, trained using improved PSO algorithm, converges faster and produces more accurate results with a little extra hardware utilization cost. © 2012 Elsevier B.V. All rights reserved.
In this Letter, based on robust control, we provide a general theoretical result on stochastic linear generalized synchronization (GS) of chaotic systems. Given a driving system with noise perturbations and a linear synchronization function, a response system is developed easily according to the scheme derived here. By introducing the Lyapunov stability theory and linear matrix inequalities (LMIs), the condition for synchronization is proved to be effective. Finally, the Lorenz system is taken for illustration and verification.
This paper initiates a systematic methodology for generating various grid multiwing hyperchaotic attractors by switching control and constructing super-heteroclinic loops from the piecewise linear hyperchaotic Lorenz system family. By linearizing the three-dimensional generalized Lorenz system family at their two symmetric equilibria and then introducing the state feedback, two fundamental four-dimensional linear systems are obtained. Moreover, a super-heteroclinic loop is constructed to connect all equilibria of the above two fundamental four-dimensional linear systems via switching control. Under some suitable conditions, various grid multiwing hyperchaotic attractors from the real world applications can be generated. Furthermore, a module-based circuit design approach is developed for realizing the designed piecewise linear grid multiwing hyperchaotic Lorenz and Chen attractors. The experimental observations validate the proposed systematic methodology for grid multiwing hyperchaotic attractors generation. Our theoretical analysis, numerical simulations and circuit implementation together show the effectiveness and universality of the proposed systematic methodology.
We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years to a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.
A numerical examination of third-order, one-dimensional, autonomous, ordinary differential equations with quadratic and cubic nonlinearities has uncovered a number of algebraically simple equations involving time-dependent accelerations jerks that have chaotic solutions. Properties of some of these systems are described, and suggestions are given for further study. © 1997 American Association of Physics Teachers.
The drive towards shorter design cycles for analog integrated circuits has given impetus to several developments in the area of Field-Programmable Analog Arrays (FPAAs). Various approaches have been taken in implementing structural and parametric programmability of analog circuits. Recent extensions of this work have married FPAAs to their digital counterparts (FPGAs) along with data conversion interfaces, to form Field-Programmable Mixed-Signal Arrays (FPMAs). This survey paper reviews work to date in the area of programmable analog and mixed-signal circuits. The body of work reviewed includes university and industrial research, commercial products and patents. A time-line of important achievements in the area is drawn, the status of various activities is summarized, and some directions for future research are suggested.
Over the last two decades, multiwing chaos generation has seen promising advances and becomes an active research field today. It is well known that there is a gap between theoretical design and engineering applications in multiwing chaos generation. That is, most theoretical designs of multiwing chaotic attractors with mathematical proofs or numerical verification have rather complex expressions; however, most engineering applications of multiwing chaotic attractors without theoretical supports have simple expressions. To bridge the gap between theoretical design and engineering applications in multiwing chaos generation, this paper introduces a novel practical approach for generating grid multiwing butterfly chaotic attractors from the multipiecewise Lü system by constructing heteroclinic loops. It should be particularly pointed out that the designed multiwing chaotic attractors exhibit typical heteroclinic chaos from the heteroclinic Shil'nikov theorem and also have clear potential engineering applications. The proposed method can be easily extended to the generalized Lorenz system family.
The intrinsic dynamics of the Lorenz system are confined in the positive half-space with respect to the vertical axis due to a limiting threshold effect. To break such a threshold effect, a novel piecewise Lorenz system is introduced, equipped with a staircase function and an even symmetric piecewise-linear function. The new system is autonomous, and yet, it can generate various grid multiwing butterfly chaotic attractors without requiring any external forcing. A module-based circuit is designed for implementation, with experiments reported for verification and demonstration.
We propose a new design of a chaotic signal generation and cancellation system using an all fiber optic scheme. A system consists of a standard diode laser, a fiber optic micro ring resonator, and an optical add/drop multiplexer. When light from the diode laser is input into the fiber ring resonator, the chaotic signal can be generated by using the selected fiber ring resonator parameters and the diode laser input power. The required signal is obtained in the transmission link via the add/drop device by a specific user at the drop port. Simulation results obtained have shown the potential of application, especially, when the practical ring radius is 10 μm with the optical input power is in the range of the communication standard diode laser, for instance, when the coupling coefficients of the add/drop device are κ1 = 0.01 and κ2 = 0.01–0.9. When the add port of the add/drop device is employed, such a system can also be utilized for the multi user applications.
The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper, we numerically study the bifurcations and chaotic behaviors in the fractional-order simplified Lorenz system using the time-domain scheme. Chaos does exist in this system for a wide range of fractional orders, both less than and greater than three. Complex dynamics with interesting characteristics are presented by means of phase portraits, bifurcation diagrams and the largest Lyapunov exponent. Both the system parameter and the fractional order can be taken as bifurcation parameters, and the range of existing chaos is different for different parameters. The lowest order we found for this system to yield chaos is 2.62.
Chua's circuit is very suitable as a programmable chaos generator because of its robust nonlinearity. In addition to exhibiting a rich variety of bifurcation and chaos phenomenon, this circuit can be modeled and realized with a fixed main system block and many different nonlinear function blocks such as piecewise-linear function, cubic-like function, piecewise-quadratic function and other trigonometric functions. This paper presents a FPAA (Field Programmable Analog Array) based programmable implementation of Chua's circuit. Nonlinear function blocks used in Chua's circuit are modeled with an FPAA and hence a model can be rapidly changed for realization of other nonlinear functions. In this study, four FPAA-based reconfigurable implementations of Chua's circuit have been realized. Experimental results agree with numerical simulation and results obtained from discrete electronic implementations of Chua's circuit.
This paper introduces a new application of the Field Programmable Analog Array. Ease of design and reprogrammability are the advantages offered by this class of analog circuits, making them an ideal environment for the implementation of chaotic circuits and their experimental characterization.
Chua's circuit, a well-known paradigm of nonlinear circuits, has been used as an example of application. Experimental results show the suitability of the approach, highlighting the features of the new implementation: a fully on the fly programmable Chua's circuit has been obtained.
We generalize the idea of driving a stable system to the situation when the drive signal is chaotic. This leads to the concept of conditional Lyapunov exponents and also generalizes the usual criteria of the linear stability theorem. We show that driving with chaotic signals can be done in a robust fashion, rather insensitive to changes in system parameters. The calculation of the stability criteria leads naturally to an estimate for the convergence of the driven system to its stable state. We focus on a homogeneous driving situation that leads to the construction of synchronized chaotic subsystems. We apply these ideas to the Lorenz and Rössler systems, as well as to an electronic circuit and its numerical model.
A systematic examination of general three-dimensional autonomous ordinary differential equations with quadratic nonlinearities has uncovered 19 distinct simple examples of chaotic flows with either five terms and two nonlinearities or six terms and one nonlinearity. The properties of these systems are described, including their critical points, Lyapunov exponents, and fractal dimensions.
The competition between magnetism and superconductivity was investigated for the recently discovered RNi2B2C superconductors with the magnetic rare-earth elements R=Tm,Er,Ho,Dy. The systematic decrease of Tc, approximately scaled by the de Gennes factor, implies a very weak coupling between the rare-earth magnetic moments and the conduction electrons due to a small conduction-electron density at the rare-earth site. Associated with the antiferromagnetic order of the rare-earth moments, a pronounced dip structure in the upper critical field Hc2(T) was observed; a similar structure has been seen in the previously known magnetic superconductors RRh4B4 and RMo6S8. For HoNi2B2C, the pair-breaking associated with the magnetic transition is strong enough to bring about a resistive reentrant behavior even under zero field, an effect which, to our knowledge, has not previously been observed in antiferromagnetic superconductors.
We show how to extract messages masked by a chaotic signal of a time-delay system with very high dimensions and many positive Lyapunov exponents. Using a special embedding coordinate, the infinite-dimensional phase space of the time-delay system is projected onto a special three-dimensional space, which enables us to identify the time delay of the system from the transmitted signal and reconstruct the chaotic dynamics to unmask the hidden message successfully. The message extraction procedure is illustrated by simulations with the Mackey-Glass time-delay system for two types of masking schemes and different kinds of messages.
It is demonstrated that the random nature of sea clutter may be
explained as a chaotic phenomenon. For different sets of real sea
clutter data, a correlation dimension analysis is used to show that sea
clutter can be embedded in a finite-dimensional space. The result of
correlation dimension analysis is used to construct a neural network
predictor for reconstructing the dynamics of sea clutter. The
deterministic model so obtained is shown to be capable of predicting the
evolution of sea clutter. The predictive analysis is also used to
analyze the dimension of sea clutter. Using the neural network as an
approximation of the underlying dynamics of sea clutter, a dynamic-based
detection technique is introduced and applied to the problem of
detecting growlers (small fragments of icebergs) in sea clutter. The
performance of this method is shown to be superior to that of a
conventional detector for the real data sets used here
A novel chaotic radar (CRADAR) system utilizing laser chaos is investigated both numerically and experimentally. Compared with conventional radars, the proposed CRADAR has the advantages of very-high-range resolution, unambiguous correlation profile, possibility of secure detection, low probability of intercept, and high electromagnetic compatibility. Generated by an optically injected semiconductor laser, chaotic waveforms with bandwidths larger than 10 GHz can be readily obtained. In this paper, the time series, the phase portraits, and the power spectra of the chaotic states are presented. The correlation traces between the signal and the reference waveforms are plotted. The peak sidelobe level with different correlation lengths is investigated. The capability of anti-jamming and the performance under additive white Gaussian noise are studied. To show the feasibility of CRADAR, proof-of-concept experiments using a pair of planar antennas with a 1.5-GHz bandwidth covering the range from 1.5 to 3 GHz are demonstrated. A range resolution of 9 cm is achieved, which is currently limited not by the bandwidth of the chaotic states but by the detection bandwidths of the real-time oscilloscope and the antennas used.
By considering generalized synchronizable chaotic systems, the drive-auxiliary system variables are combined suitably using encryption key functions to obtain a compound chaotic signal. An appropriate feedback loop is constructed in the response-auxiliary system to achieve synchronization among the variables of the drive-auxiliary and response-auxiliary systems. We apply this approach to transmit analog and digital information signals in which the quality of the recovered signal is higher and the encoding is more secure. Comment: 7 pages (7 figures) RevTeX, Please e-mail Lakshmanan for figures, submitted to Phys. Lett. A (E-mail: lakshman@kaveri.bdu.ernet.in)
- T Yang
Yang T 1995 Int. J. Circ. Theor. Appl. 23 611
- R X Zhang
- S P Yang
- Y L Liu
Zhang R X, Yang S P and Liu Y L 2010 Acta Phys. Sin. 59 1549 (in
Chinese)