A4-7 - Sedma Nacionalna Konferencija so Me|unarodno U~estvo...

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A CLASS OF CHAOTIC CIRCUITS Ljubiša Koci ć 1 , Sonja Gegovska-Zajkova 2 1 Faculty of Electronic Engineering, University of Niš, Niš, Serbia and Montenegro [email protected] 2 Faculty of Electrical Engineering, SS Cyril and Methodius Univ., Skopje, R. of Macedonia [email protected] Abstract A class of Sprott “jerk” chaotic circuit is explored. It is shown how the Hopf fibrations, Poincaré sections and torus-supported trajectories can be obtained from the corresponding equation. Associated phenomena such as mode-locking, Farey sequences and appearance of a deter- ministic chaos are discussed. Except the famous “golden route to chaos” defined by ratios of consecutive Fibonacci numbers, other possible quasiperiodic routes are considered as well. Examples of chaotic electric circuits with the corresponding phase diagrams and FFT analysis are given. Key Words – Sprot class circuit, routes to chaos 1. INTRODUCTION Recently, Sprott ([8]) published a new class of chaotic circuit, with the underlying “jerk” dynamics ), ( x G x x A x = + + ± ± ± ± ± ± (1) where A is a parameter and G ( x ) is a piecewise linear function (Fig. 1). This class of circuits is called “jerk” upon the term in (1) of third derivative of position by time coordinate. Fig. 1. Sprott circuit Trying to simplify earlier circuit with quadratic nonlinearity 2 0 xA xx x +−+ = ±±± ±± ± , Sprott sugested the fo- llowing piecewise linear functions for G(x): B | x | C , C B max( x , 0), Bx C sgn( x ), and C sgn( x ) Bx ( B and C are constants). In these cases, Sprott examined dynamic properties of the circuit in the series of articles ([7], [1], [2]). In [3], we present the more general nonlinear charac- teristics G* ( x ) = [( k 2 k 1 ) p – ( k 2 + k 1 ) x ( k 0 + k 2 ) | x p | + ( k 0 + k 1 )| x + p |]/2 where k 0 , k 1 , k 2 and p are real con- stants. It is easy to see that G is piecewise linear continuous function for all real x , with two break- points at x = ±p. For special values of k 0 , k 1 and k 2 , G* ( x ) reduces on the Sprott’s nonlinearites G ( x ). Fig. 2. Geometry of quasi-periodic route to chaos On the other hand, authors of [3] studied phenomena of cyncronicity in two-frequencies driven oscillatory systems from theoretical points of view. Some elements are given in Fig. 2. Our idea is thou to combine theoretical and practical experiments to stydy chaotic behaviour in generalized Sprott circuits. Such experiments that we carried up to the level of mathematical models and computer simulation uses numerical procedures implemented in programming language MATHEMATICA. 2. METHOD The modified Sprott characteristics of nonlinear element that we used in [3] with the graph at Fig. 3 is
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a three-piece continous linear function with the following equation: 21 12 02 01 1 *( ) = [( ) ( 2 () | | | p | ] . Gx k kp k kx kk xp kk x −− + −+ + + + (2) So, we have four dynamical parameters (instead of two used by Sprott) to study dynamic behaviour in the circuit: the central slope k 0 , side slopes k 1 and k 2 and breaking-point apscissa p (also – p ). All the parameters are positive real numbers.
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This note was uploaded on 02/18/2010 for the course ITK ETF113L07 taught by Professor Popovskiborislav during the Spring '10 term at Pacific.

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A4-7 - Sedma Nacionalna Konferencija so Me|unarodno U~estvo...

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