Leon Chua

Many physical devices are modeled in such a way that the equations defining the system have \( \mathbb{Z}_2 \)-symmetry (invariance under the change of the sign in the state variables). In these systems, the origin is always an equilibrium point. Its simple, well arranged and transparent form expressed as a compact set of first-order ordinary differential equations (ODE) is also very convenient for modeling chaotic systems considered as determined systems exhibiting complex and unpredictable behavior.

There is no generally accepted definition of chaotic phenomena. Roughly speaking, a chaotic system is one that is deterministic but at the same time exhibits irregular, or random, behavior. A distinctive and readily observable property of chaotic systems is sensitive dependence on initial conditions: the trajectories starting from two arbitrarily close points diverge at an exponential rate and become uncorrelated in a short period of time. Chaos, or strange behavior, is currently one of the most exciting areas in the research of nonlinear systems.

The Chua circuit was invented in the fall of 1983 by the American electrical engineer and computer scientist Leon Ong Chua (born in 1936) in response to two unfulfilled quests among many researchers on chaos concerning two wanting aspects of the Lorenz equations (Lorenz, 1963). The first quest was to devise a laboratory system which can be realistically modeled by the Lorenz equations in order to demonstrate chaos is a robust physical phenomenon, and not merely an artifact of computer round-off errors. The second quest was to prove that the Lorenz attractor, which was obtained by computer simulation, is indeed chaotic in a rigorous mathematical sense. The existence of chaotic attractors from the Chua circuit had been confirmed numerically by Matsumoto (1984), observed experimentally by Zhong and Ayrom (1985), and proved rigorously in (Chua, et al, 1986). The basic approach of the proof is illustrated in a guided exercise on Chua’s circuit in the well-known textbook by Hirsch, Smale and Devaney (2003).

The Chua's equation is a model of one of the simplest electronic circuits, exhibiting a wide range of complex dynamical behaviors. Let us consider the Chua's equation with a cubic nonlinearity (see [Pivka et al., 1996]):

\[ \begin{split} \dot{x} &= \alpha \left( y - a\,x^3 -c\,x \right) , \\ \dot{y} &= x-y+z , \\ \dot{z} &= - \beta \, y - \gamma \, z . \end{split} \]
In this equation a parameter γ is included in order to take into account small resistive e ects in the inductance.
      We plot the chua circuit with matlab:

The following matlab code is used for animation of the chua circuit:


Classical Chua circuits.


disp("Non-linear Differential Equations");
disp("Chua's Circuit")
disp('Background:')
disp("Chua's circuit was invented by Leon Chua in 1983.")
disp('The circuit is an autonomous nonperiodic oscillator that must contain a locallly active resistor')
disp('which is a device that has a negative resistance which amplifies current')
disp('3 differential equations:')
disp('(dx/dt)= a[y-x-g(x)]')
disp('(dy/dt)= x-y+z')
disp('(dz/dt)= -By')

%solving the 3 differential equtions and graphing the results
% the initial x position is 1 while y and z are 0
% graphs for t from 0 to 150 seconds
% the ode45 command is used to solve differential equations
[t,y] = ode45(@chua_function,[0 150],[1 0 0]);
%the solution is graphed
plot3(y(:,1),y(:,2),y(:,3),'r')
title("Chua's circuit")
xlabel('Capasitor 1 voltage')
ylabel('Capasitor 2 voltage')
zlabel('Curent in the inductor')
grid

function out = chua_function(~,p)
% t represents time and p represents position
a = 16; % 'a' represents alpha
B = 30; % 'B' represents beta
m = -1.2; % m and n are slopes
n = -0.7;

% these values are parameters set in the main script. x is 1 and y and z are 0
x = p(1);
y = p(2);
z = p(3);

g = n*x+0.5*(m-n)*(abs(x+1)-abs(x-1));
% these equations are derivatives with respect to time of x, y, and z
xdot = a*(y-x-g);
ydot = x - y+ z;
zdot = -B*y;
% these are the values that will be graphed
out = [xdot ydot zdot]';
end