Plot of the Rossler Attractor

I have made an m-file to plot the Rossler Attractor using the equations and a fourth order Runge Kutta methid but the plot lines are very pointed and do not appear smooth, has anyone any suggestions about how i can create a better Attractor? step size, h, used was 0.175. The plot of the Attractor can be found below, thank you.

Answer 1

[Zinea]

Hi Conor Pierce,

I have reproduced a similar plot of the Rossler attractor with the following script “rough_rossler_script.m”:

function rossler_attractor_RK4 
% Parameters for the Rössler attractor 
a = 0.2; 
b = 0.2; 
c = 5.7; 

% Initial conditions 
y0 = [0; 2; 0]; 

% Time span 
tStart = 0; 
tEnd = 200; 
h = 0.175; % Fixed step size 

% Time vector 
tVec = tStart:h:tEnd; 
nSteps = length(tVec); 

% Preallocate the solution array 
Y = zeros(nSteps, 3); 
Y(1,:) = y0'; 

% RK4 integration 
for i = 1:nSteps-1 
    t = tVec(i); 
    y = Y(i,:)'; 
    
    k1 = h * rossler(t, y, a, b, c); 
    k2 = h * rossler(t + 0.5*h, y + 0.5*k1, a, b, c); 
    k3 = h * rossler(t + 0.5*h, y + 0.5*k2, a, b, c); 
    k4 = h * rossler(t + h, y + k3, a, b, c); 
    
    Y(i+1,:) = y + (k1 + 2*k2 + 2*k3 + k4)/6; 
end 

% Plot 
plot3(Y(:,1), Y(:,2), Y(:,3), 'LineWidth', 1.5); 
xlabel('x'); 
ylabel('y'); 
zlabel('z'); 
title('Rössler Attractor with RK4'); 
grid on; 
end 

function dydt = rossler(t, y, a, b, c) 
% Rössler attractor differential equations 
dydt = zeros(3,1); 
dydt(1) = -y(2) - y(3); 
dydt(2) = y(1) + a*y(2); 
dydt(3) = b + y(3)*(y(1) - c); 
end 

Output plot from the “rough_rossler_script.m”:

To make the plot smoother, the step size (h) must be decreased because it will lead to the calculation of more points along the trajectory. The step size (h) is reduced to 0.01 from the earlier code where it was 0.175.

The modified script “smooth_rossler_script.m” is given below:

function rossler_attractor_RK4_smooth 
% Parameters for the Rössler attractor 
a = 0.2; 
b = 0.2; 
c = 5.7; 

% Initial conditions 
y0 = [0; 2; 0]; 

% Time span 
tStart = 0; 
tEnd = 200; 
h = 0.01; % Decreased step size for smoother plot 

% Time vector 
tVec = tStart:h:tEnd; 
nSteps = length(tVec); 

% Preallocate the solution array 
Y = zeros(nSteps, 3); 
Y(1,:) = y0'; 

% RK4 integration loop 
for i = 1:nSteps-1 
    t = tVec(i); 
    y = Y(i,:)'; 
    
    k1 = h * rossler(t, y, a, b, c); 
    k2 = h * rossler(t + 0.5*h, y + 0.5*k1, a, b, c); 
    k3 = h * rossler(t + 0.5*h, y + 0.5*k2, a, b, c); 
    k4 = h * rossler(t + h, y + k3, a, b, c); 
    
    Y(i+1,:) = y + (k1 + 2*k2 + 2*k3 + k4)/6; 
end 

% Plot 
plot3(Y(:,1), Y(:,2), Y(:,3), 'LineWidth', 1.5); 
xlabel('x'); 
ylabel('y'); 
zlabel('z'); 
title('Smooth Rössler Attractor with RK4'); 
grid on; 
end 

function dydt = rossler(t, y, a, b, c) 
% Rössler attractor differential equations 
dydt = zeros(3,1); 
dydt(1) = -y(2) - y(3); 
dydt(2) = y(1) + a*y(2); 
dydt(3) = b + y(3)*(y(1) - c); 
end 

Output plot from “smooth_rossler_script.m”:

Hope it helps!