The double scroll

Matsumoto, Takashi; Chua, Leon O.; Komuro, M.

doi:10.1109/tcs.1985.1085791

Cited by 572 publications

(256 citation statements)

References 13 publications

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“…Each equilibrium has one real and two complex eigenvalues. A typical trajectory in the attractor rotates around one of the two outer equilibria [20], suppose the upper one, in a counterclockwise direction with respect to the left handed coordinate system. After each rotation the trajectory gets further from the equilibrium until a certain time after which there are two possibilities: 1) the trajectory goes back to a position closer to the equilibrium and repeats a similar process, 2) the trajectory does not go back to a point close to the equilibrium but descends downward in a spiral path and "lands" on the lower part of the attractor.…”

Section: Modeling Of the Resistive Circuitsmentioning

confidence: 99%

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Electronic circuits modeling using artificial neural networks

Andrejevic

Litovski

2003

J Autom Control

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-In this paper artificial neural networks (ANN) are applied to modeling of electronic circuits. ANNs are used for application of the black-box modeling concept in the time domain. Modeling process is described, so the topology of the ANN, the testing signal used for excitation, together with the complexity of ANN are considered. The procedure is first exemplified in modeling of resistive circuits. MOS transistor, as a four-terminal device, is modeled. Then nonlinear negative resistive characteristic is modeled in order to be used as a piece-wise linear resistor in Chua's circuit. Examples of modeling nonlinear dynamic circuits are given encompassing a variety of modeling problems. A nonlinear circuit containing quartz oscillator is considered for modeling. Verification of the concept is performed by verifying the ability of the model to generalize i.e. to create acceptable responses to excitations not used during training. Implementation of these models within a behavioural simulator is exemplified. Every model is implemented in realistic surrounding in order to show its interaction, and of course, its usage and purpose.

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Section: Modeling Of the Resistive Circuitsmentioning

confidence: 99%

“…Theoretic interpretation of the chaotic behaviour is given in [20]. The parallel connection of C 2 and L constitutes a lossless oscillatory mechanism in the (V 2 -I L )-plane, whereas the conductance G provides interactions between the (C 2 , L)-oscillatory component and the active resistor together with C 1 .…”

Section: Modeling Of the Resistive Circuitsmentioning

confidence: 99%

Electronic circuits modeling using artificial neural networks

Andrejevic

Litovski

2003

J Autom Control

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“…HGO controlled the double scroll system, also known as Chua's system, 22 to follow a prescribed symbolic sequence. The differential equations describing the double scroll system are given by…”

Section: Introductionmentioning

confidence: 99%

Controlled transitions between cupolets of chaotic systems

Morena

Short

Cooke

2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We present an efficient control scheme that stabilizes the unstable periodic orbits of a chaotic system. The resulting orbits are known as cupolets and collectively provide an important skeleton for the dynamical system. Cupolets exhibit the interesting property that a given sequence of controls will uniquely identify a cupolet, regardless of the system's initial state. This makes it possible to transition between cupolets, and thus unstable periodic orbits, simply by switching control sequences. We demonstrate that although these transitions require minimal controls, they may also involve significant chaotic transients unless carefully controlled. As a result, we present an effective technique that relies on Dijkstra's shortest path algorithm from algebraic graph theory to minimize the transients and also to induce certainty into the control of nonlinear systems, effectively providing an efficient algorithm for the steering and targeting of chaotic systems. Efficiently directing a dynamical system to a desired state is a goal of many engineering applications. This process is known as "targeting" and is often achievable using small nonlinear controls. For a chaotic system, the goal of many control methods is either to stabilize the system onto one of its many unstable periodic orbits (UPOs) or to induce the system to transition between its UPOs until a targeted orbit is attained. The set of UPOs plays a significant role in determining the dynamics of a chaotic system and is said to form the skeleton of an associated attractor. In this paper, we combine a particularly effective chaos control method with algebraic graph theory and present a new technique allowing for the efficient transitioning between periodic orbits of chaotic systems. This allows one to navigate efficiently around a chaotic attractor and in physical applications conserve energy and power.

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“…Chaos corresponds to a kind of irregular behaviour, which can be obtained in nonlinear dynamical systems in many areas, especially in physical systems, for instance nonlinear circuits. Chua's circuit [18] or Du ng oscillator [11] are classical examples, more recently, MicroElectroMechanical Systems (MEMS) [3] have been studied in detail. The rst researches about chaos have concerned its analysis, and more particularly the understanding of route to chaos by studying bifurcations giving rise to chaos.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations and Chaos Generation from 1D or 2D Circuits Including Switches

2014

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Abstract. Generation of chaos is of the highest interest for many kind of applications as secure transmissions, image processing or telecommunications. In order to obtain chaotic signals that can be used in applications, simple circuits including switches have been considered those last years. Their interest is of two types: rst, they belong to the class of hybrid systems that have recently attracted a great interest, secondly, they permit to obtain chaos in a very easy way. Such circuits are very easy to implement and robust chaos can be obtained, depending upon parameter values. For this aim, it is necessary to study and understand the bifurcation structures of the circuit model. In this paper, we propose two kinds of chaos generators obtained from simple RC circuits including switches managed using a clock (impulse waveform) and the charging/discharging of the capacitors. Our models are given via one-dimensional (1D) or two-dimensional (2D) nonlinear maps. We present the bifurcation studies permitting to understand the evolution of the behaviour of our systems and the appearance of chaos.Keywords. chaotic signal; bifurcation; switching circuit; piecewise-smooth map. AMS classi cation. 94; 37Gxx; 65Pxx.Résumé. L'utilisation du chaos présente un grand intérêt dans beaucoup d'applications en lien avec les télécommunications, le traitement d'images ou les transmissions sécurisées. Des circuits très simples comprenant des commutations peuvent être utilisés pour produire des signaux chaotiques. Leur intérêt est double, tout d'abord ils appartiennent à la classe des systèmes hybrides qui ont récemment attiré l'attention, ensuite, ils permettent de générer des signaux chaotiques robustes de manière très simple et ils sont faciles à implémenter. Néanmoins il est nécessaire d'étudier et de comprendre les structures de bifurcations sous-jacentes. Dans cet article, nous proposons deux sortes de circuits RC très simples comportant des commutations qui utilisent une horloge a n de charger ou décharger les capacités. Les modèles proposés sont basés sur des récurrences de dimension un ou deux. Les structures de bifurcation des deux circuits sont étudiées et permettent de comprendre la génération de signaux chaotiques.

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The double scroll

Cited by 572 publications

References 13 publications

Electronic circuits modeling using artificial neural networks

Electronic circuits modeling using artificial neural networks

Controlled transitions between cupolets of chaotic systems

Bifurcations and Chaos Generation from 1D or 2D Circuits Including Switches

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