Periodically controlled switches are added to chuas original, piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in size. The applet shows a simulation of chuas circuit, plotting the voltage measured across c1 against the voltage measured across c2. The chuas circuit brings a verity of complex dynamical nature with chaos that provides a wide range of bifurcation phenomena and attractors. Chuas circuit is a paradigm for chaos which is considered as the simplest nonlinear electrical circuit that exhibits almost known chaotic phe. Dec 30, 2005 an improved implementation of chua s circuit using dual output current conveyor is presented. The main purpose of the paper is to explore the influence of the coupling of two scales on the dynamics of a nonsmooth dynamical system. Chuas circuit chaus circuit is a simple electronic circuit that demonstrates chaotic behavior. In section 4, we show how the changes in the dynamics of the circuit affects first return maps, symbolic planes and fragmentation patterns. The basic properties of chaotic systems, speci cally, sensitivity to initial conditions and dense topology, intuitively suggest that in the physical world a. Pdf chuas circuit is a nonlinear dynamic circuit that has assumed a paradigmatic role in mathematical, physical and experimental demonstrations of. Our study focuses on phenomena that can be observed directly in the numerical simulation of the model, and on phenomena which are revealed by a more elaborate analysis based on continuation techniques and bifurcation theory.
Irregulary shaped satellites like hyperion moon of saturn tumble chaotically. An autonomous circuit made from standard components resistors, capacitors, inductors must satisfy three criteria before it can display chaotic behaviour. Synchronization of chuas circuits with multiscroll. Chuas circuit is the simplest possible electronic circuit that can exhibit chaotic behavior. This approach to chaos with memristors is unique and radically different from previous proposals 22, 2830 that are based on modified chuas circuits 31, 32 with active elements.
The paper treats the question of robust control of chaos in chuas circuit based on the internal. Robust op amp realization of chuas circuit eecs at uc berkeley. Chua, who was a visitor of waseda university japan at that time matsumoto 1984. Applying kcl and kvl, the chua s circuit is described by three differential equations. An inductorfree realization of the chuas circuit based. Simulation results show that the proposed methods are valid. Using an oscilloscope you can witness a chua s circuit create the strange attractor known as the double scroll. The initial values of the parameters used in the applet correspond to the component values in the circuit diagram, and show a simple periodic orbit oscillation.
Thisisbecause it is capable of realizing the behavior of every member of the chuas circuit. Chua circuit equations california institute of technology. We emphasize also that our realization of chaos results from the simplest singleelement nonautonomous chaotic circuit consisting of the chaotic memristor directly. Chaos and synchronization through the study of two. Chaos synchronization and transmission of information in. The objective is achieved using a decomposition method, which allows the solution of the fractional differential equations to be written in closed form.
The ease of construction of the circuit has made it a ubiquitous realworld example of a chaotic system, leading. Chuas circuit for high school students gandhi, gauruv. In this paper, we continue our study of rank one chaos in switchcontrolled circuits. A fast and simple implementation of chuas oscillator using a cubiclike chua diode keith odonoghue. Our experiments reported in this letter are based on measurements obtained by connecting each of these two setups in place of the 2terminal nonlinear resistor in fig. Relaxation oscillations in a nonsmooth oscillator with. We also propose an effective way of adapting the coupling strengths of complex networks. Robust fault detection during the synchronization process. Even though numerical analysis can help to bring out a detailed picture of the dynamics of a nonlinear system such as the duffing oscillator equation 5.
The priciples of the chaos control procedure follow the guidelines given by 7. When it is determined how the circuit works, there will be explained to what extend the circuit is controlled and how it can be synchronized. An inductorfree realization of the chuas circuit based on electronic analogy. Because chua s circuit is endowed with virtually every bifurcation phenomena reported in the extensive literature on chaos, and because it is the only chaotic system which can be easily built by a novice, simulated in a personal computer, and tractable mathematically, it has become a paradigm for chaos, and a vehicle for illustrating this. As an expansion and extension for threeorder and fourorder chuas circuit, a new method to construct a class of highorder chuas circuit and its fpga hardware implementation has been studied in this paper.
Pdf numerical simulation of chuas circuit oriented to. Chaos in the chua circuit nyu tandon school of engineering. In this work we analyze in detail the pair of spirallike attractors for r 1. Chuas circuit is literally the simplest chaotic circuit. This corresponds to the display on an xy oscilloscope with probes connected across these capacitors. Implementation of the chua s circuit and its applications. This paper argues that chaos theory provides a useful theorectical framework for. Chua in 1983, which is characterized by its simplicity, easy implementation, robustness and a simple static nonlinearity that allows for a large and diverse set of dynamical behaviors. Among the members of chua s circuit fam ily, the autonomous canonical chuas circuit intro duced by chua and lin 1990 is of considerable importance. The circuit that chua designed has been shown to be the simplest autonomous electronic circuit capable of exhibiting chaotic behavior.
Multiscroll attractors also called nscroll attractor include the lu chen attractor, the modified chen chaotic attractor, pwl duffing attractor, rabinovich fabrikant attractor, modified chua chaotic attractor, that is, multiple scrolls in a single attractor. With the results obtained, we conclude that this circuit can entrances the chaotic circuit family, extending the knowledge of memristor behaviors and chaotic circuits. Chaos in the orbits of giant planets jupiter, saturn and uranus. Introduction to chuas circuit a schematic of chuas circuit is shown in figure 1. Based on a typical chuas circuit, by introducing a nonlinear resistor with piecewise characteristics as well as a harmonically changed electric source, a modified nonsmooth model is established, in which the coupling of two scales in frequency domain exists. Fundamental theory and applications, ieee transactions on author. Chaos in nonlinear electronic circuits springerlink. Furthermore, a novel chaos based sequential gate that behaves like.
New sufficient conditions are provided using a lyapunov approach and the well known s procedure. The circuit is very sensitive to vibration especially on a breadboard, and good solder connections are essential to a reliable circuit. The asymptotical stability is proven with lyapunov theory for both of the proposed methods, and the system can be dragged to one of its three unstable equilibrium points respectively. Synchronization problem is solved using linear matrix inequality tools. Implementation of chuas circuit with a cubic nonlinearity. Chuas circuit chuas circuit is a simple electronic circuit that exhibits classic chaos theory behavior.
In this paper, the milestones of chaos history, from the use. Building this circuit requires only basic soldering skills and a few common circuit components. A fast and simple implementation of chuas oscillator using a. A general method allowing to construct nonlinear resistors with arbitrary currentvoltage iv characteristics is proposed. The initial values of the parameters used in the applet correspond to the component values in the circuit. For the modified chuas circuit, there have been several controllers proposed to suppress the chaos. Revised february 5, 1997 bifurcations and the structure of limit sets are studied for a threedimensional chuas circuit system with a cubic nonlinearity. The chua circuit is the simplest electronic circuit exhibiting. The existence of chaotic attractors from the chua circuit had been confirmed numerically by. The applet shows a simulation of chua s circuit, plotting the voltage measured across c1 against the voltage measured across c2. Ayrom arracr this paper reports a perioddoubling route to chaos as ob served from a laboratory model of the simplest possible chaotic autono mous circuit.
By means of computer simulations, we have illustrated chaotic synchronization of generalized chuas circuits coupled in masterslave configuration. Chaos synchronization and transmission of information. This is because of its simplicity, inexpensive and because electronic devices are well understood, so chuas circuit is an example of a chaotic circuit madam, 1993. These beautiful patterns are truly chaotic and can be modeled by relatively. Based on the structure of a typical thirdorder chuas circuit, a fifthorder, sixthorder and seventhorder chuas circuit have been constructed through the series in thetype sub. Adaptive control and synchronization of a modified chuas. All the components for chuas circuit are standard, offtheshelf items except for the negative nonlinear resistor. Synchronization of chaos between the driver and the driven system is demonstrated. Dec 07, 2014 chuas circuit is the simplest electronic circuit that produces chaosthe output of this circuit never repeats the same sequence, and is a truly random signal. A feedback control method and an adaptive feedback control method are proposed for chua s circuit chaos system, which is a simple 3d autonomous system. Lyapunov exponents from chuas circuit chuas circuit. Multistability in chuas circuit with two stable nodefoci b. The circuit that chua designed has been shown to be the simplest autonomous electronic circuit. Chen and dong 4 applied the linear feedback control for guiding the chaotic trajectory of the circuitsystem to alimit cycle.
Sangiovannivincentelli, spice3 version 3e users manual, erl. Chaos synchronization was applied to encrypted transmission of audio and jpg image information. Improved realization of canonical chuas circuit with synthetic. The example of a cubic iv characteristic is presented showing a perfect agreement between the theoretical desired resistor and its. Kennedy eecs department university of california, berkeley technical report no. Chua tbe genesis of chua s circuit following a nontechnical personal reminiscence of the author s conception of chua s circuit, this paper presents the systematic sequence of technical steps which the author used to design his chaotic circuit. Circuit simulation of an analog secure communication based on. Cluster synchronization of nonlinearly coupled complex. Chaos and synchronization through the study of two coupled. The easiest way to observe chaos is in electronic circuits.
Chuas circuit consists of a linear inductor l, two linear capacitors. Robust fault detection during the synchronization process of nonlinear lure dynamical networks. This circuit, known for its rich repertoire of nonlinear dynamical phenomena and has become a universal paradigm for chaos. An inductorfree realization of the chuas circuit based on electronic analogy 391 of wienbridge 15 and inductance emulator circuits based on transistors 16, voas 17, cfoas 12, otas 18, 19 and ftfn 20. A useful list of links related to chuas circuit and chaos.
The coupling between the vertices of the network is nonlinear, and sufficient conditions are derived analytically for the attainment of cluster synchronization. Robust op amp realization of chuas circuit eecs at uc. Practical implementation of chua s circuit with a cubic nonlinearity since the desired l i characteristic of the nonlinear resistor lv in chuas circuit is an oddsymmetric function with respect to origin, here we use the cubic polynomial 7 with the coefficients. In addition, w e presen t exp erimen tal results and spice sim ulations for a w orking circuit using o theshelf comp onen ts. Hartley and mossayebi showed the control of modified chuas circuit systems and demonstrated how to design a controller for tracking the capacitor voltage of the system by applying the input.
The estimation of lyapunov exponents from experimental data is demonstrated by means of a chaotic time series from chuas circuit. Periodicity and chaos in chuas circuit guoqun zhong and f. The conservatism of previous existing works is relaxed. The analogous circuit is implemented and used to perform an experimental mapping of a large variety of attractors, where the perfect match between experiment and.
We present the bifurcation analysis of chuas circuit equations with a smooth nonlinearity, described by a cubic polynomial. This is the standard chuas circuit used in research and a number of experiments, but there are many different ways to realize the full circuit. Numerical simulation of chuas circuit oriented to circuit synthesis. Xu1 1school of information science and engineering, changzhou university, changzhou 2164. Chaos in the chua circuit project for the course on dynamical systems department of electrical engineering eindhoven university of technology siep weiland abstract. Chua s circuits are some of the simplest kinds of chaotic circuits. A paradigm for chaos, world scientific, singapore, 1993. Because chuas circuit is endowed with virtually every bifurcation phenomena reported in the extensive literature on chaos, and because it is the only chaotic system which can be easily built by a novice, simulated in a personal computer, and tractable mathematically, it has become a paradigm for chaos, and a vehicle for illustrating this ubiquitous phenomenon.
However, when building this circuit at home be aware that, as a chaotic circuit. To appear in the international journal of bifurcation and chaos in dec. Chuas circuit, canonical chuas circuit, cfoa, chaos. Chuas circuit also known as a chua circuit is a simple electronic circuit that exhibits classic chaotic behavior. Insection4, weshowhowthechangesinthedynamics of the circuit affects. This paper investigates bifurcation and chaos in the fractionalorder chua and chen systems from the timedomain point of view. Finally, some concluding remarks are given in section 4. One of the most important chaotic circuits was created by leon o.
Chen and dong, applied the linear feedback control for guiding the chaotic trajectory of the circuit to limit cycle. In particular, this paper illustrates the parallel implementation of two combinatorial logic gates using a single chua s circuit. The circuit consists of three energy storing elements. The circuit 4schematic is simple but the nature of. Online identification and control of chaos in a real chua s circuit 429 system dynamics near chosen periodic orbit.
For studying a more complex chaotic behavior called hyperchaos, whichhas beenreported in hydrodynamicsand semiconductor devices, one has to combine the behavior of several chua s circuits and couple them in various ways. Be sure to check out the wikipedia article on chuas circuit for a brief overview of the circuits functionality and a rabbit trail of information on chaos theory. There are many variations on how to build chua s circuit, but figure a shows the basic design. Finally, in section 5, we summarize our conclusions. Among the members of chuas circuit family, the autonomous canonical chuas circuit introduced by chua and lin 1990 is of considerable importance. The ease of construction of the circuit has made it a ubiquitous realworld example of a chaotic system, leading some to declare it a paradigm for chaos. However, chuas circuit provides only 3dimensional chaos, as figs. An improved chuas circuit and its use in hyperchaotic.
An inductorfree realization of the chuas circuit based on. Toward the use of chuas circuit in education, art and. A paradigm for chaos, world scienti c series on nonlinear science, series b world scienti c, singapore. Controlling and synchronization world scienti c, singapore. International journal of bifurcation and chaos, vol. For uninitiated researchers, engineers, and scientists interested in a quick entry into the subject of chaos, this book offers a timely collection of 55 carefully selected papers covering almost every aspect of this subject. Chuas circuit can be modelled by a set of nonlinear differential equations where x, y and z are plotted against time, commonly represented as follows. Chaos synchronization for master slave piecewise linear. The changes in the lyapunov spectrum upon time reversal, the local thickness of the attractor, and the approximation errors of the locally linear fits are used as diagnostic tools to identify spurious lyapunov exponents and to monitor the reliability of. For illustrative purpose, we selected a generalized chuas circuit generator of 5scroll attractors. Chua in 1983, and its ease of construction has made it a very popular, realworld example of chaotic systems. The electric circuit is composed by resistors, capacitors, inductance, diodes and an operational ampli. Highlights chaos synchronization problem for piecewise linear systems is solved. Chaos in chua s oscillator with chua s diode and memristor.
Chaos in the orbits of giant planets jupiter, saturn and uranus the location of these planets cannot be predicted on a time scale longer than. This is the standard chua s circuit used in research and a number of experiments, but there are many different ways to realize the full circuit. The original chuas circuit contains a piecewise linear resistor. Chuas circuit diagrams, equations, simlulations and how to build. A physical simulation of the chuas circuit is carried out, along with the corresponding experiments. By taking advantage of the capabilities given by timedomain analysis, the paper illustrates three remarkable.
The genesis of chuas circuit university of california. Chuas circuit diagrams, equations, simlulations and how. Michael peter kennedy peadar forbes abstract the nonlinearity in chuas oscillator is commonly implemented as a threesegment piecewiselinear resistor. Chuas circuit has been studied extensively as a prototypical electronic system. Hartley suggested that the piecewise linear function can be replaced by a cubic polynomial with no major change in the systems dynamic characteristics. Leon chua is credited with developing one of the simplest to construct chaotic circuits matsumoto, 1984, which is now termed as the chuas circuit. This means roughly that it is a nonperiodic oscillator.
The application of present circuit in designing low hardware inductorless hyperchaotic circuit is also. Since the circuit is an extremely simple system, and yet it exhibits a rich variety of bifurcations and chaos, it has been studied extensively and accepted as a paradigm for studying with chaos and chaotic signals. They are considered to be a classic example of true chaos due to their design and output. Chaos in the fractional chua and chen systems with lowestorder. And managerial implications david levy department of management, ljniversity of massachusetfs boston boston, massachusetts, u. It is solely the nonlinear resistor that makes chuas circuit nonlinear.
Chuas circuit is well known to be a paradigm for chaos 12, being a simple nonlinear electrical circuit that reveals a rich variety of phenomena. Background the chuas circuit a thirdorder autonomous, dissipative electrical circuit has been investigated thoroughly at the experimental, numerical and analytical levels. Word chaos normally means that a predictable dynamic system can give unpredictable results. Applying kcl and kvl, the chuas circuit is described by three differential equations. The chua circuit can be accurately modeled by means of a set of three nonlinear ordinary differential equations in the. We discuss the hyperbolic regime and show that the evolution of this circuit is very similar to onedimensional systems with homoclinic chaos. Chuas circuit has been studied extensively as a prototypical electronic system 711. The generalized chuas circuit consists of nonlinearity with multiple breakpoints, leading to a family of nscroll attractors 14, 15. Chuas circuits are some of the simplest kinds of chaotic circuits. The piecewiselinear nature of the element means that the. The chua circuit is among the simplest nonlinear circuits that shows most complex dynamical behavior, including chaos.
Multistability in chuas circuit with two stable nodefoci. We consider a method for driving general complex networks into prescribed cluster synchronization patterns by using pinning control. Variable selection using information entropy in time. Using an oscilloscope you can witness a chuas circuit create the strange attractor known as the double scroll. Implementation of chuas chaotic circuit using current feedback opamps article pdf available in electronics letters 349.
Because chuas circuit is endowed with virtually every bifurcation phenomena reported in the extensive literature on chaos, and because it is the only chaotic system which can be easily built by a novice, simulated in a personal computer, and tractable mathematically, it has become a paradigm for chaos, and a vehicle for illustrating this. Designing chaotic mathematical circuits for solving. This paper considers the case when the modified canonical chua s oscillator contains both chua s diode and memristor. The present proposal offers the advantages of low component count, current mode processing, availability of current across inductor and availability of voltage across inductor in form of current.