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Analog Lorenz Attractor Computer

An analog computer for solving sets of Lorenz attractor differential
equations

CheeenNPPCheeenNPP
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Analog Computer Chaotic circuits

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  * Op Amp Challenge

This project was created on 05/17/2023 and last updated 7 days ago.

Description

After finding a good high linearity multiplier (NJM4200, log
architecture), I built this Lorentz chaotic circuit, the essence of
which is to use multipliers and integrators to solve the Lorentz
equations. The Lorentz equation describes the principle of
atmospheric dynamics, which is commonly known as the "butterfly
effect", and the oscilloscope shows the Lissajous figure, just like
the elusive drifting wings.

Details

Analog Lorenz Attractor Computer

1. The Origin of Analog Computer

One of the main purposes of analog circuits is to solve mathematical
problems, such as building a circuit corresponding to a nonlinear
differential equation and analyzing the phase plane characteristics
of it by observing its output voltage with an oscilloscope or analog
plotter. I will take a famous nonlinear differential equation, the
Lorenz Attractor, as an example, and show the whole process of
solving it with an analog circuit.

2.Lorenz Attractor

The Lorentz attractor is a set of equations describing the dynamical
behavior of the atmosphere, which reveals the chaotic phenomena
contained in meteorological changes and is known as the "butterfly
effect". The Lorentz attractor consists of three nonlinear
differential equations:

Among them, sigma, b and r are the three constants that determine the
characteristics of the system, commonly taken as sigma = 10, r = 28
and b = 8/3. Before building the analog circuit, we can simulate it
with software to understand some basic characteristics (sounds a bit
strange~~). The software used is Octave, and the Lorentz attractor
function is declared in lorenz.m, where x is a three-dimensional
vector set, and x(1), x(2), and x(3) represent x, y, and z in the
original set of equations, respectively, while a scale factor k is
added for further usage.

% Lorenz.m

function dx = lorenz(x, s, r, b, k)
    dx = zeros(3, 1);
    dx(1) = k*(s*(x(2) - x(1)));
    dx(2) = k*(-x(1)*x(3) + r*x(1) - x(2));
    dx(3) = k*(x(1)*x(2) - b*x(3));
end

 The lorenz function is called in test.m and solved numerically. Here
the fourth-order Runge-Kutta method is used and the initial values of
the equations are set to (1, 1, 1). Finally, the trace of  x is
presented in three and one dimensions, respectively:

% test.m

clc; clear all;

dt = 1e-3;
t = 0 : dt : 100 - dt;

s = 10;
r = 28;
b = 8/3;
k = 1;

x = zeros(3, length(t));
x(:, 1) = [1; 1; 1];

for tn = 1 : 1 : length(t) - 1
    k1 = lorenz(x(:, tn),           s, r, b, k);
    k2 = lorenz(x(:, tn) + dt*k1/2, s, r, b, k);
    k3 = lorenz(x(:, tn) + dt*k2/2, s, r, b, k);
    k4 = lorenz(x(:, tn) + dt*k3,   s, r, b, k);
    x(:, tn + 1) = x(:, tn) + dt/6*(k1 + 2*k2 + 2*k3 + k4);
end

figure;
plot3(x(1, :), x(2, :), x(3, :), '-'); box on; grid on; drawnow;

figure;
subplot(3, 1, 1); plot(t, x(1, :)); title("x");
subplot(3, 1, 2); plot(t, x(2, :)); title("y");
subplot(3, 1, 3); plot(t, x(3, :)); title("z");

The simulation results are shown in the following figure:

3.How to Design an Analog Computer

How can this set of equations be "translated" into an analog circuit?
This is illustrated by a diagram:

In analog circuits, the most convenient circuits used to perform
calculations are those designed based on inverting amplifiers, which
are simpler in form than non-inverting amplifiers and also avoid
distortion and noise from common-mode operation, which was
particularly important in the early years of op-amp development.

In the first step, determine the implementation of the various
operations as:

  * Integral: use the inverting integrator circuit, but temporarily
    discard the resistor, which is equivalent to integrating the
    input current and multiplying by -1.
  * Multiplying by -1: use the inverting amplifier and set the two
    resistors with the same value.
  * Addition: use current summation instead of voltage, so the form
    is simpler.
  * Multiplying by a constant k: use a resistor with a resistance of
    1/k, the voltage across it is converted into a current and output
    as a result.
  * Multiplication: implemented using a dedicated voltage multiplier.

In the second step, the integrator is cascaded with the inverting
amplifier, and the output of the integrator is set to -x, the output
of the inverter is x, and the input of the integrator is dx/dt. Then
calculate the equivalence of dx/dt and input it at dx/dt. The x and
-x signals needed in the operation are obtained from the above
output. y and z signals are obtained in the same way. The signal flow
diagram can be drawn in this...

Read more >>

View all details

Files

lorenz_attractor_1.asc
                                      
LTSpice simulation file              Download

asc - 5.59 kB - 05/17/2023 at 13:18

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[3357631424]
Anthony wrote 2 days ago * point

Hey, though such devices have been built before I thought this
project was really cool and exceptionally documented. I feel I can
mostly -- though perhaps not 'entirely'-- follow you
through transition between the formulae, and then deriving the
eventual circuit. Do you have a textbook you could suggest where I
would be able to learn more about this?

  Are you sure? yes | no

[7123831681]
CheeenNPP wrote 2 days ago * point

Thank you for your reply! Although I didn't refer to a book
dedicated to analog computer design, the book LINEAR CIRCUIT DESIGN
HANDBOOK helped me a lot:

 https://www.analog.com/en/education/education-library/
linear-circuit- design-handbook.html

In addition, there are many examples of circuit design ideas and
techniques in the linear technology application notes, which you can
find at analog.com.

  Are you sure? yes | no

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