A4-8 - Sedma Nacionalna Konferencija so Me|unarodno U~estvo...

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APPROXIMATIONS IN SPROTT CIRCUITS Liljana Stefanovska 1 , Sanja Glamo č anin 2 1 Faculty of Tecnology and Metallurgy, Ss Cyril and Methodius Univ., Skopje, R. of Macedonia liljana@ian.tmf.ukim.edu.mk 2 Faculty of Electrical Engineering., Ss Cyril and Methodius Univ., Skopje, R. of Macedonia sanjag@etf.ukim.edu.mk Abstract Some approximation methods are suggested to study quasiperiodic regimes in a class of Sprott “jerk” electric chaotic circuits. The idea is to replace piecewise linear characteristics of nonlinear components with piecewise polynomial ones. The methods proposed are chosen to fit two proposals: a) to be the simplest possible; b) to have one free parameter that regulates approximation cleseness. These are happening to be fulfilled by cubic Hermite splines. Several examples of chao- tic circuits with approximated characteristics are given. Key Words – Sprot “jerk” circuit, Hermite appro- ximations, chaos 1. INTRODUCTION Simple electric circuits are ideal tools for studying features of chaos. According to Poincaré-Bendixon theorem, autonomous first-order ODE with continu- ous coefficients can not have bounded chaotic solution unless it is at least three dimensional. The classic example was the famous Lorenz model of atmospheric heath conduction from 1963 () , (), , xA y x yB z x y zx y C z =− =−− ± ± ± where A, B and C are real parameters. Later, many other 3D systems were found and in 1976 Rössler came with even simpler 3D system than Lorenz’s. In 1997 Linz showed that Lorenz and Rössler system can be transformed in the following form (, , ) x Jxxx = ±±± ±± ± , (1) where J is called jerk function upon the mechanical term “jerk” for time derivative of acceleration. In this way, one can consider DE (1) instead of 3D system in order to stydy different aspects of chaos. In 1996, Gottlieb rised an interesting question: What is the simplest jerk function that gives chaos? In [7] Julian Sprott from Univ. of Wisconsin made experiments with a simple jerk function 2 A x x x = −+− ±± ± , showing that system exhibits a period-doubling rute to chaos when A runs between 2.017 and 2.082. In further series of papers (see [6]-[9] ), Sprott replaced linear term x in J with several piecewise linear fun- ctions, like | |, max( ,0), sgn( ), ( ) sgn[max( ,0)] xx x H x x = (2) ( H is Heaviside step function). Figure 1, reproduced from [8], shows the way of modeling such functions by electronic components.
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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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A4-8 - Sedma Nacionalna Konferencija so Me|unarodno U~estvo...

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