A simple Lorenz circuit and its radio frequency implementation

Abstract

A remarkably simple electronic circuit design based on the chaotic Lorenz system is described. The circuit consists of just two active nonlinear elements (high-speed analog multipliers) and a few passive linear elements. Experimental implementations of the circuit exhibit the classic butterfly attractor and the hysteretic transition from steady state to chaos observed in the Lorenz equations. The simplicity of the circuit makes it suitable for radio frequency applications. The power spectrum of the observed oscillations displays a peak frequency as high as 930 kHz and significant power beyond 1 MHz.

A simple electronic circuit realization of the tent map

“…In several implementations of this kind of circuits [9,10] analog multipliers have been employed with a normalization of the signal by a factor of about ten. This normalization was necessary because of the physical restrictions in the analog multiplier.…”

Emergence of chaos in transistor circuits evolved towards maximization of approximate signal entropy

“…Five circuits were constructed around 2 BJTs in Darlington or pseudo-Darlington (one BJT reversed) configuration (no. 2,7,8,11,12), 1 circuit included 2 BJTs connected in series (no. 1) and 2 circuits included 2 BJTs connected in parallel through inductors and capacitors (no.…”

“…In the last decade, a wide range of other circuits have been proposed, based on single-transistors through to op-amp oscillators and phase-locked loops; research in this area has principally aimed at optimizing specific implementation features such as spectral content and power consumption [11][12][13]. Despite these developments, a formal design methodology allowing the general synthesis of arbitrary chaotic oscillators remains lacking; these are in practice either heuristically obtained by adding energy-storing components to well-known circuits such as Colpitts's oscillator, or designed by implementing differential equations known a-priori to be chaotic, such as Lorentz's system [11][12][13][14][15]. Occasionally, novel topologies have been discovered through experimentation [16,17].…”

Designing Chaotic Mathematical Circuits for Solving Practical Problems

“…The variables, x, y, z of (4) are the voltages across capacitors C1, C2, and C3 after a suitable rescaling. Mi configured as a current output device [8] An hardware electronic circuit based on slightly modified Rössler equations [9] is shown in Fig. 5.…”