Results for

1. Multibody Simulation Onramp
2. Analyzing Results in Simulink
3. Battery Pack Modeling
4. Introduction to Motor Control
5. Signal Processing Techniques for Streaming Signals
6. Core Signal Processing Techniques in MATLAB (learning path – includes the four short courses listed below)

• Signal Generation and Resampling
• Spectral Analysis Techniques
• Time-Frequency Analysis
• Filter Design and Analysis Methods

• 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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