Spectrum Software
spacer
Industrial Strength Simulation
select:

divider

 

News:

Spectrum Software has released Micro-Cap 11, the eleventh generation of our SPICE circuit simulator.

For users of previous Micro-Cap versions, check out the new features available in the latest version. For those of you who are new to Micro-Cap, take our features tour to see what Micro-Cap has to offer.

 

divider

 

Simulating the Lorenz Attractor

 

The Lorenz attractor describes how the state of a nonlinear, three dimensional, dynamic system changes over time in a chaotic fashion. The attractor was originally discovered by Ed Lorenz who derived it from a simplified model of convection rolls in the earth's atmosphere. However, it also arises within lasers and dynamos.

The three differential equations that define the Lorenz attractor are:

dx/dt = Sigma*(y-x)
dy/dt = (Rho-z)*x - y
dz/dt = x*y - Beta*z

where Sigma is often referred to as the Prandtl number, Rho is often referred to as the Rayleigh number, and Beta is a geometric factor.

In Micro-Cap, differential equations such as these can be simulated by using the behavioral models available in the Macro section of the Analog Primitives. The schematic for the Lorenz attractor appears below.

Lorenz attractor schematic

The nodes x, y, and z calculate voltages equivalent to their corresponding variables in the above equations. The x variable is calculated using an Int macro, a Sub macro, and an Amp macro. The Sub macro (X2) subtracts the voltage at node y by the voltage at node x. The difference is then fed into an Amp macro (X3) whose Gain parameter is set to Sigma. The output of the Amp macro is then input into an Int macro (X1) which calculates the integral to produce x.

The y variable is calculated using an Int macro, two Sub macros, a Mul macro, and an NFV function source. The NFV function source (E1) produces a voltage equivalent to Rho. The Sub macro (X6) subtracts the Rho value by the voltage at node z. The difference is then fed into a Mul macro (X4) where it is multiplied by the voltage at node x. This product is then fed into another Sub macro (X7) where it is subtracted by the voltage at node y. This difference is then input into an Int macro (X5) which calculates the integral to produce y.

The z variable is calculated using an Int macro, a Mul macro, an Amp macro, and a Sub macro. The voltages at nodes x and y are multiplied together using the Mul macro (X10). The voltage at node z is scaled by the factor Beta using the Amp macro (X9). These two products are input into a Sub macro (X11). The difference is then input into the Int macro (X8) which calculates the integral to produce z.

The values of Sigma, Rho, and Beta are set through define statements. The following three define statements are also present in the schematic:

.define InitX 0
.define InitY 1
.define InitZ 20

These three define statements set the initial values for the x, y, and z variables. The InitX, InitY, and InitZ are used to define the VINIT parameter for the X1, X5, and X8 Int macros respectively. The VINIT parameter of the Int macro sets an initial voltage at the output of the integrator. As with other chaotic systems, the Lorenz attractor is very sensitive to these initial conditions. Even a small change in these will produce an entirely different plot.

The common settings to show chaotic behavior with the Lorenz attractor is to set Sigma to 10, Rho to 28, and Beta to 8/3. A 200 second transient analysis is run using these values which produces the classic butterfly plot when V(Z) is plotted versus V(X) as shown below.

Lorenz attractor butterfly plot

 
Return to the main Newsletter page