Caos

14 February 2000

Physics Letters A 266 Ž2000. 19–23 www.elsevier.nlrlocaterphysleta

A new class of chaotic circuit
J. C. Sprott
) Department of Physics, UniÕersity of Wisconsin, Madison, WI 53706, USA Received 30 November 1999; received in revised form 4 January 2000; accepted 4 January 2000 Communicated by C.R. Doening

Abstract A new class of chaotic electrical circuit using onlyresistors, capacitors, diodes, and inverting operational amplifiers is { described. This circuit solves the equation x q Ax q x s GŽ x ., where GŽ x . is one of a number of elementary piecewise ¨ ˙ linear functions. These circuits are easy to construct and to scale over a wide range of frequencies. They exhibit a variety of dynamical behaviors and offer an excellent opportunity for detailed comparisonwith theory. q 2000 Published by Elsevier Science B.V. All rights reserved.
PACS: 05.45.Ac; 02.30.Hq; 02.60.Cb; 47.52.q j; 84.30.Ng Keywords: Chaos; Electrical circuit; Operational amplifier; Jerk; Differential equations

After three decades of study, the sufficient conditions for chaos in a system of autonomous ordinary differential equations ŽODEs. remain unknown. For continuous flows, thePoincare–Bendixson theorem ´ w1x implies the necessity of three variables and at least one nonlinearity. The Rossler attractor w2x is a ¨ standard example of such a system with a single quadratic nonlinearity. Systems with one nonlinearity can generally be written as a third-order ODE in a single scalar variable, suggesting a means to catalog and quantify the complexity of such systems w3x. In thisscheme, the Rossler system is relatively compli¨ cated w4x, and the algebraically simplest dissipative quadratic form w5x is { x s yAx q x 2 y x , Ž 1. ¨ ˙ which exhibits chaos for values of A equal to or slightly greater than 2.017. Systems of the form
)

{

x s F Ž x, x, x . have been called jerk equations Žtime ¨ ˙ derivative of acceleration. w6x. The discovery of this and other suchsimple systems w7x prompted a search w8x for similar examples in which the quadratic nonlinearity is replaced by N x N or another elementary piecewise linear function. Although Eq. Ž1. with N x N in place of x 2 does not ˙ ˙ appear to have chaotic solutions for any A and initial conditions, chaos was found w9x in the system

{

x s yAx y x q N x N y1 ¨ ˙

Ž 2.

with A equal to or slightlygreater than 0.6. Eq. Ž2. is a special case of the more general system

{

x q Ax q x s G Ž x . , ¨ ˙

Ž 3.

Fax: q 1-608-2627205. E-mail address: sprott@juno.physics.wics.edu ŽJ.C. Sprott..

where GŽ x . is a nonlinear function with the properties discussed below. Integrating each term in Eq. Ž3. reveals that it is a damped harmonic oscillator driven by a nonlinear memory term involving theintegral of GŽ x . w4x. Such an equation often arises in the feedback control of an oscillator in which the experimen-

0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 0 2 6 - 8

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J.C. Sprottr Physics Letters A 266 (2000) 19–23

yC required to solve Eq. Ž2. is produced by the circuit in Fig. 2Ža.w10x. Fig. 2 shows three other cases with chaotic solutions. In each case, the constant A is taken as 0.6 and B is selected to be well

Fig. 1. A general circuit for solving Eq. Ž3. using one of the nonlinear feedback elements in Fig. 2 for GŽ x ..

tally accessible variable is a transformed and integrated version of the fundamental dynamical variable. Eq. Ž3. with a piecewise linear GŽ x .suggests a class of chaotic electrical circuit that is simple to construct, analyze, and scale to most any desired frequency. It is new in the sense that it does not involve analog multiplication, it uses only resistors, capacitors, diodes and operational amplifiers, and the governing equation is simpler than any previously modeled electronically. The most straightforward implementation w8x involves...