Chaos in Electronics - download pdf or read online

By M. A. van Wyk, W.-H Steeb (auth.)

ISBN-10: 904814857X

ISBN-13: 9789048148578

ISBN-10: 9401589216

ISBN-13: 9789401589215

Many dynamical platforms in physics, chemistry and biology convey complicated be­ haviour. The it appears random movement of a fluid is the simplest recognized instance. How­ ever additionally vibrating constructions, digital oscillators, magnetic devices,lasers, chemical oscillators, and inhabitants kinetics can behave in a classy demeanour. you will see that abnormal oscillations, that's referred to now as chaotic behaviour. The learn box of nonlinear dynamical platforms and particularly the examine of chaotic structures has been hailed as one of many very important breaktroughs in technology this century. The sim­ plest cognizance of a procedure with chaotic behaviour is an digital oscillator. the aim of this ebook is to supply a entire advent to the applying of chaos thought to digital structures. The publication offers either the theoretical and experimental foundations of this study box. every one digital circuit is defined intimately including its mathematical version. Controlling chaos of digital oscilla­ tors can also be integrated. finish of proofs and examples are indicated by means of •. inside of examples the tip of proofs are indicated with O. we want to convey our gratitude to Catharine Thompson for a severe interpreting of the manuscript. Any valuable feedback and reviews are welcome. e-mail tackle of the 1st writer: MVANWYK@TSAMAIL. TRSA. AC. ZA e-mail handle of the 1st writer: WHS@RAU3. RAU. AC. ZA domestic web page of the authors: http://zeus. rau. ac. za/steeb/steeb. html xi bankruptcy 1 creation 1.

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The y-components of the system 's fixed points are = 3/2 . y~ = 0, y; We restrict our attention to the region of the (a, (3)-plane which satisfy the following inequality [273] , Two predicted (stable) limit cycles are found by numerical evaluation of the terms No and N I . These predicted limit cycles are symmetrical with respect to the origin of the phase space and their projections onto the y-axis enclose the fixed points Y:'l 28 CHAPTER 2. ANALYSIS OF CHAOTIC SYSTEMS and v; Therefore, their interaction with the fixed point (y-component ) considered .

When r3 < a (respectively r3 > 0) the bifurcation at r = r( f) is said to be supercritical (respectively subcriiicals. 43 Use the Hopf bifurcation theorem to study the bimap defined by the system (see [499]) XI,n+1 ) ( where f :I x I -t = ( f( Xl,n, X2,n) ) , f( X2 ,n, XI ,n) X2,n+! I x I is defined by bER . 6. HOPF BIFURCATION 45 We only consider the fixed points on the diagonal here. Notice that these fixed points exist only for b ~ 3/4. For b :» 3/4 a stable period-2 orbit exists, given by the points x, and X2, XT = I T _ x2 - (2b(b + 1) - Jb(b + 1)(4b2 b(4b+3) (2b(b + 1) + Jb(b + 1)(4b2 b(4b+3) - - 3) 2b(b + 1) + Jb(b + 1)(4b2 , b(4b+3) 3) 2b(b + 1) - Jb(b + 1)(4b2 , b(4b+3) - 3)) , - 3)) .

ANALYSIS OF CHAOTIC SYSTEMS 4) The remaining N - 2 eigenvalues of the critical Jacobian Dxf(x'(ro); ro) are contained inside the unit circle {zECllzl 0) then for r < a (respectively r > 0), the fixed point x*(r(f)) is stable (respectively unstable) and for r > a (respectively r < 0) the fixed point x*(r( f)) is unstable (respectively stable) and the surrounding manifold H(r( f)) is attracting (respectively repelling).

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Chaos in Electronics by M. A. van Wyk, W.-H Steeb (auth.)


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