1 Chua’s oscillator

1.1 State equations

The (unnormalized) state equations are:

dv1- = -1-[(v - v )G - f(v )] dt C1 2 1 1 { dv2- -1- dt = C2 [(v1- v2)G + i3] di3 1- dt = - L [v2 + R0i3]

with

f (v1) = Gbv1 + 0.5(Ga - Gb)[| v1 + Bp |- |v1- Bp|]

1.2 Dimensionless state equations

Dimensionless state equations can be derived from the original ones scaling time, voltages, and currents.

1.2.1 The classical case

In the original paper [1], where the circuit was extensively analyzed, time, voltages, and currents are scaled by RC₂, B_p, and B_pG, respectively. The new state variables become:

x = v1- , y = v2- , z = -i3-- Bp Bp BpG

and the normalized time is given by:

t t = ----- RC2

With these assumptions, we obtain the following dimensionless state equations:

dx dt- = a(- x + y - f (x)) { dy dt- = x - y + z dz --- = - by - gz dt

with

f(x) = m1x + 0.5(m0 - m1)[ |x + 1|- |x - 1 |] , m0 = GaR , m1 = GbR

where

C2- 2 a = C , b = R C2/L , g = RR0C2/L 1

1.2.2 The applet case

The applet uses a different set of dimensionless state equations, as indicated below. Assume G_a - the slope of the inner segment of the nonlinear resistor characteristic - is negative and set R_a = -1/G_a > 0 . Then scale time, voltages, and currents by C₁R_a , B_p , and B_p/R_a , respectively. If we use as state variables:

X = -v1 ; Y = -v2 ; Z = i3Ra- Bp Bp Bp

with normalized time given by:

t t = ------,(Ra = - 1/Ga > 0) RaC1

the following dimensionless state equations are obtained:

dX-- dt = a(-X + Y )- f(X) { dY-- dt = s[a(X - Y ) + Z] dZ dt- = -c(Y + rZ)

with

f(X) = - bX + 0.5(b - 1)[|X + 1 |- |X - 1 |]

and where

Ra- Gb- C1- 2 R0- a = R ,(Ra = - 1/Ga > 0) ; b = Ga ; s = C2 ; c = C1R a/L ; r = Ra

After this scaling, the nonlinear resistor has breakpoints for X = ±1 and the slopes of the piecewise linear characteristic are -1 in the inner segment and -b in the outer segments. Finally, note that the case = 0 corresponds to the (classical) Chua’s circuit, where R₀ = 0.

References

[1] T. Matsumoto, L. 0. Chua, and M. Komuro, “The double scroll”, IEEE Trans. Circuits Syst., vol. CAS-32, pp. 797-818, 1985.