Autonomous Silva Young Type Chaotic Oscillator with Flat Power Spectrum

E. Tamaseviciute Department of Mechanical and Process Engineering, Swiss Federal Institute of Technology in Zürich, CH-8092, Switzerland, phone: +41 7990 37076, e-mail: elenata@student.ethz.ch G. Mykolaitis Department of Physics, Vilnius Gediminas Technical University, Saulėtekio av. 11, LT-10223 Vilnius, Lithuania, phone: +370 5 2619589, e-mail: gytis@pfi.lt A. Tamasevicius Division of Electronics, Center for Physical Sciences and Technology, A. Goštauto str. 11, LT-01108 Vilnius, Lithuania, phone: +370 5 2619589, e-mail: tamasev@pfi.lt


Introduction
Chaotic oscillations have been observed and intensively investigated in a variety of nonlinear electrical circuits [1,2].In contrast to common noise originated from random fluctuations, chaos is generated by purely deterministic systems, similar to the periodic waveform oscillators.Therefore chaotic oscillators are notable for their high power and high efficient performance.Unlikely to periodic oscillations characterized by discrete spectra, chaotic oscillations exhibit broadband continuous power spectra.The fields of practical engineering application of chaotic oscillators include, for example, spread spectrum communication systems, information scrambling for secure transmission, also noise radar systems.
Chaotic oscillations can be generated in the third order or higherorder autonomous nonlinear electrical circuits, for example oscillators described in [313].In the case of non-autonomous circuits, i.e. nonlinear damped oscillators driven by external periodic voltage, the minimal order of the circuit can be reduced to two.However, most of chaotic circuits designed so far have noticeable unevenness of the spectral density.This undesirable feature is observed not only in the non-autonomous secondorder nonlinear circuits, but also in the autonomous thirdorder [37], in the fourthorder [811] chaotic, in the sixthorder [12] and in higherorder [13] hyperchaotic oscillators.
Using the well-known in nonlinear dynamics the secondorder nonlinear DuffingHolmes equations [14]   Silva and Young [15,16] invented a non-autonomous electronic circuit generating chaotic oscillations with a broadband noiselike spectrum.Later Kandangath et al. described lowfrequency version of the oscillator [17].A simplified SilvaYoung oscillator [18] was used in various physical experiments demonstrating the performance of electronic analogue controllers designed for stabilizing dynamical systems [19,20].The main shortcoming of the nonautonomous chaotic oscillators in many practical applications is that they have sharp 2030 dB height peaks at the external drive frequency f 1 = /2 and its higher harmonics [15].To get around the problem Lindberg et al. recently suggested the thirdorder autonomous Duffing Holmes (or SilvaYoung) type chaotic oscillator [21] described by the following set of equations: where  f is the cut-off frequency of the low-pass RC filter inserted in the negative feedback loop of the oscillator.Though sharp peaks are avoided in the power spectrum, the spectral density still exhibits large undesirable unevenness of about 1015 dB [21,22].
In this paper, we describe both, experimentally and numerically the fourthorder nonlinear circuit exhibiting unusually (for deterministic chaotic oscillators) flat power spectrum.In contrast to the Lindberg's [21] chaotic oscillator the novel oscillator includes an additional LC resonant tank in the positive feedback loop.

Electronic circuitry
The novel fourth-order autonomous circuit is presented in Fig. 1.

Experimental results
The typical output signal and the corresponding power spectrum are shown in Fig. 2

Equations
Using the Kirchhoff's laws the following set of differential equations for the circuit in Fig. 1 is derived: where the V C1 and V C2 are the voltages across the capacitor C 1 and C 2 , the I L1 and I L2 are the currents through the inductor L 1 and L 2 , respectively.The k is the gain of OA1 based stage with respect to the non-inverting input.The V D (V C1 ) is the voltage across the diodes D 1 and D 2 , depending on the voltage V C1 , applied via resistor R 4 , and on the state of the diodes (either OFF, or ON).The r 1 and r 2 are the internal resistances of the inductive coil L 1 and L 2 , respectively (specifically, r 1 = r 2 = 2 ).Other notations are evident from the circuit diagram in Fig. 1.Equations for the correction circuit L 3 R 8 C 3 are rather trivial and are not presented here for simplicity.Moreover, the OA3 based stage is an external unit used for correction only, i.e. it is not involved in the generation of chaotic oscillations.
Let us introduce a set of the dimensionless state varia- bles x, y, z, v, dimensionless time and parameters a, b, c,  d,  k, (V* is the forward voltage drop across the diodes).At small forward currents of 10 to 20 A the V* is typically about 0.5 V for silicon p-n junction diodes; thus 2V* 1 V.The characteristic resistance of the resonance loop is   650 .Then the set of four differential equations, convenient for numerical integration, is obtained: (5)

Numerical results
Equations ( 3) have been integrated numerically using the Wolfram MATHEMATICA and FREEPASCAL computational software programs.The typical results for chaotic waveform and power spectrum are presented in Fig. 4 and Fig. 5, respectively.

Conclusions
We have designed and built a novel SilvaYoung type the fourthorder chaotic oscillator characterized by the spectral unevenness of less than 10 dB (Fig. 3a).Moreover, very simple secondorder external resonant circuit L 3 R 8 C 3 (Fig. 1) inserted in the output follower allows decreasing this value to less than 3 dB (Fig. 3b).
On one hand, the developed fourthorder oscillator can be considered as an extended version of the previously proposed thirdorder chaotic oscillator [22] in the sense that its order is simply increased from three to four.On the other hand, the novel oscillator is similar from the structural point of view to the secondorder nonautonom-ous SilvaYoung circuit [15,16].The only difference of the suggested fourthorder circuit is that the external periodic driving oscillator in the original SilvaYoung circuit is replaced with a passive internal secondorder resonance circuit L 2 R 5 C 2 (Fig. 1), inserted in the second positive feedback loop.Nevertheless, this replacement yields fairly flat power spectrum of the output signal with no peaks at the resonant frequencies.

Fig. 1 .
Fig. 1. Circuit diagram of the fourth-order autonomous Sil-vaYoung type oscillator The oscillator comprises an OA1 based nonlinear resonant stage with the diodes D1, D2, the L 1 C 1 tank, and resistor R 4 in the first positive feedback loop, also an OA2 based linear resonant stage with the L 2 C 2 tank and the load resistor R 5 in the second positive feedback loop.The OA3 based stage is an external output follower with the spectral correction circuit L 3 R 8 C 3 .

Fig. 2 .Fig. 3 .
Fig. 2. Experimental snapshot of chaotic output waveform.Vertical scale 1 V/div.Horizontal scale 1 ms/div.R 6 = 490  function F(x) involving the currentvoltage characteristic of the diodes is approximated by a three-segment piecewise linear function: