Results for

• Plotting the Potential Energy and Identifying Extrema

% Define the range of x values

x = linspace(-2, 2, 1000);

% Define the potential function V(x)

V = -x.^2 / 2 + x.^4 / 4;

% Plot the potential function

figure;

plot(x, V, 'LineWidth', 2);

hold on;

% Mark the minima at x = ±1

plot([-1, 1], [-1/4, -1/4], 'ro', 'MarkerSize', 5, 'MarkerFaceColor', 'g');

% Add LaTeX title and labels

title('Duffing Potential Energy: $$V(x) = -\frac{x^2}{2} + \frac{x^4}{4}$$', 'Interpreter', 'latex');

xlabel('$$x$$', 'Interpreter', 'latex');

ylabel('$$V(x)$$','Interpreter', 'latex');

grid on;

hold off;

• Solving and Plotting the Duffing Equation

% Define the system of ODEs for the non-chaotic Duffing equation

duffing_ode = @(t, X) [X(2);

X(1) - X(1).^3];

% Time span for the simulation

tspan = [0 10];

% Initial conditions [x(0), v(0)]

initial_conditions = [1; 1];

% Solve the ODE using ode45

[t, X] = ode45(duffing_ode, tspan, initial_conditions);

% Extract displacement (x) and velocity (v)

x = X(:, 1);

v = X(:, 2);

% Plot both x(t) and v(t) in the same figure

figure;

plot(t, x, 'b-', 'LineWidth', 2); % Plot x(t) with blue line

hold on;

plot(t, v, 'r--', 'LineWidth', 2); % Plot v(t) with red dashed line

% Add title, labels, and legend

title(' Component curve solutions to $$\ddot{x}-x+x^3=0$$','Interpreter', 'latex');

xlabel('t','Interpreter', 'latex');

ylabel('$$x(t) $$ and $$v(t) $$','Interpreter', 'latex');

legend('$$x(t)$$', ' $$v(t)$$','Interpreter', 'latex');

grid on;

hold off;

% Phase portrait with nullclines, equilibria, and vector field

figure;

hold on;

% Plot phase portrait

plot(x, v,'r', 'LineWidth', 2);

% Plot equilibrium points

plot([0 1 -1], [0 0 0], 'ro', 'MarkerSize', 5, 'MarkerFaceColor', 'g');

% Create a grid of points for the vector field

[x_vals, v_vals] = meshgrid(linspace(-2, 2, 20), linspace(-1, 1, 20));

% Compute the vector field components

dxdt = v_vals;

dvdt = x_vals - x_vals.^3;

% Plot the vector field

quiver(x_vals, v_vals, dxdt, dvdt, 'b');

% Set axis limits to [-1, 1]

xlim([-1.7 1.7]);

ylim([-1 1]);

% Labels and title

title('Phase-Plane solutions to $$\ddot{x}-x+x^3=0$$','Interpreter', 'latex');

xlabel('$$ (x)$$','Interpreter', 'latex');

ylabel('$$v(v)$$','Interpreter', 'latex');

grid on;

hold off;

% Lorenz Attractor Parameters

sigma = 10;

beta = 8/3;

rho = 28;

% Lorenz system of differential equations

f = @(t, a) [-sigma*a(1) + sigma*a(2);

rho*a(1) - a(2) - a(1)*a(3);

-beta*a(3) + a(1)*a(2)];

% Time span

tspan = [0 100];

% Initial conditions

a0 = [1 1 1];

% Solve the system using ode45

[t, a] = ode45(f, tspan, a0);

% Plot using scatter3 with time-based color mapping

figure;

scatter3(a(:,1), a(:,2), a(:,3), 5, t, 'filled'); % 5 is the marker size

title('Lorenz Attractor');

xlabel('x(t)');

ylabel('y(t)');

zlabel('z(t)');

grid on;

colorbar; % Add a colorbar to indicate the time mapping

view(3); % Set the view to 3D

% Define the parameters

a = 2.07;

b = 1.79;

% Define the system of differential equations

dynamics = @(t, X) [ ...

X(2) + a * X(1) * X(2) + X(1) * X(3); % dx/dt

1 - b * X(1)^2 + X(2) * X(3); % dy/dt

X(1) - X(1)^2 - X(2)^2 % dz/dt

];

% Initial conditions

X0 = [0.63; 0.47; -0.54];

% Time span

tspan = [0 100];

% Solve the system using ode45

[t, X] = ode45(dynamics, tspan, X0);

% Plot the results with color gradient

figure;

colormap(jet); % Set the colormap

c = linspace(1, 10, length(t)); % Color data based on time

% Create a 3D line plot with color based on time

for i = 1:length(t)-1

plot3(X(i:i+1,1), X(i:i+1,2), X(i:i+1,3), 'Color', [0 0.5 0.9]*c(i)/10, 'LineWidth', 1.5);

hold on;

end

% Set plot properties

title('Sprott Attractor');

xlabel('x(t)');

ylabel('y(t)');

zlabel('z(t)');

grid on;

colorbar; % Add a colorbar to indicate the time mapping

view(3); % Set the view to 3D

hold off;

% Define the parameters

a = 0.2;

b = 0.2;

c = 5.7;

% Define the system of differential equations

dynamics = @(t, X) [ ...

-(X(2) + X(3)); % dx/dt

X(1) + a * X(2); % dy/dt

b + X(3) * (X(1) - c) % dz/dt

];

% Initial conditions

X0 = [10.0; 0.00; 10.0];

% Time span

tspan = [0 100];

% Solve the system using ode45

[t, X] = ode45(dynamics, tspan, X0);

% Plot the results

figure;

scatter3(X(:,1), X(:,2), X(:,3), 5, t, 'filled');

title('Rössler Attractor');

xlabel('x(t)');

ylabel('y(t)');

zlabel('z(t)');

grid on;

colorbar; % Add a colorbar to indicate the time mapping

view(3); % Set the view to 3D

%% Parameters for Rabinovich-Fabrikant Attractor

alpha = 0.14;

gamma = 0.10;

dt = 0.01;

num_steps = 5000;

% Initial conditions

x0 = -1;

y0 = 0;

z0 = 0.5;

% Preallocate arrays for performance

x = zeros(1, num_steps);

y = zeros(1, num_steps);

z = zeros(1, num_steps);

% Set initial values

x(1) = x0;

y(1) = y0;

z(1) = z0;

% Generate the attractor

for i = 1:num_steps-1

x(i+1) = x(i) + dt * (y(i)*(z(i) - 1 + x(i)^2) + gamma*x(i));

y(i+1) = y(i) + dt * (x(i)*(3*z(i) + 1 - x(i)^2) + gamma*y(i));

z(i+1) = z(i) + dt * (-2*z(i)*(alpha + x(i)*y(i)));

end

% Create a time vector for color mapping

t = linspace(0, 100, num_steps);

% Plot using scatter3

figure;

scatter3(x, y, z, 5, t, 'filled'); % 5 is the marker size

title('Rabinovich-Fabrikant Attractor');

xlabel('x(t)');

ylabel('y(t)');

zlabel('z(t)');

grid on;

colorbar; % Add a colorbar to indicate the time mapping

view(3); % Set the view to 3D

1. Strange Attractors
2. Attractor

1. Create and save a "template file" with the desired label and then call copyfile to make a copy of that file and then write your results to the new copy, or
2. If using Windows you can create and/or open the file using ActiveX and then apply the desired label from your MATLAB program's code.

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