Duffing equation:Transition to Chaos
The Original Equation is the following:
Let . This implies that
Then we eewrite it as a System of First-Order Equations
Using the substitution for , the second-order equation can be transformed into the following system of first-order equations:
Exploring the Effect of .
% Define parameters
gamma = 0.338;
alpha = -1;
beta = 1;
delta = 0.1;
omega = 1.4;
% Define the system of equations
odeSystem = @(t, y) [y(2);
-delta*y(2) – alpha*y(1) – beta*y(1)^3 + gamma*cos(omega*t)];
% Initial conditions
y0 = [0; 0]; % x(0) = 0, v(0) = 0
% Time span
tspan = [0 2000];
% Solve the system
[t, y] = ode45(odeSystem, tspan, y0);
% Plot the results
figure;
plot(t, y(:, 1));
xlabel(‘Time’);
ylabel(‘x(t)’);
title(‘Solution of the nonlinear system’);
grid on;
% Plot the phase portrait
figure;
plot(y(:, 1), y(:, 2));
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
% Define the tail (e.g., last 10% of the time interval)
tail_start = floor(0.9 * length(t)); % Starting index for the tail
tail_end = length(t); % Ending index for the tail
% Plot the tail of the solution
figure;
plot(y(tail_start:tail_end, 1), y(tail_start:tail_end, 2), ‘r’, ‘LineWidth’, 1.5);
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
ax = gca;
chart = ax.Children(1);
datatip(chart,0.5581,-0.1126);
Then I used but the results were not that I expected for
My code gives me the following. Any suggestion?
% Define parameters
gamma = 0.35;
alpha = -1;
beta = 1;
delta = 0.1;
omega = 1.4;
% Define the system of equations
odeSystem = @(t, y) [y(2);
-delta*y(2) – alpha*y(1) – beta*y(1)^3 + gamma*cos(omega*t)];
% Initial conditions
y0 = [0; 0]; % x(0) = 0, v(0) = 0
% Time span
tspan = [0 3000];
% Solve the system
[t, y] = ode45(odeSystem, tspan, y0);
% Plot the phase portrait
figure;
plot(y(:, 1), y(:, 2));
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
% Define the tail (e.g., last 10% of the time interval)
tail_start = floor(0.9 * length(t)); % Starting index for the tail
tail_end = length(t); % Ending index for the tail
% Plot the tail of the solution
figure;
plot(y(tail_start:tail_end, 1), y(tail_start:tail_end, 2), ‘r’, ‘LineWidth’, 1.5);
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.35$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
ax = gca;
chart = ax.Children(1);
datatip(chart,0.5581,-0.1126);The Original Equation is the following:
Let . This implies that
Then we eewrite it as a System of First-Order Equations
Using the substitution for , the second-order equation can be transformed into the following system of first-order equations:
Exploring the Effect of .
% Define parameters
gamma = 0.338;
alpha = -1;
beta = 1;
delta = 0.1;
omega = 1.4;
% Define the system of equations
odeSystem = @(t, y) [y(2);
-delta*y(2) – alpha*y(1) – beta*y(1)^3 + gamma*cos(omega*t)];
% Initial conditions
y0 = [0; 0]; % x(0) = 0, v(0) = 0
% Time span
tspan = [0 2000];
% Solve the system
[t, y] = ode45(odeSystem, tspan, y0);
% Plot the results
figure;
plot(t, y(:, 1));
xlabel(‘Time’);
ylabel(‘x(t)’);
title(‘Solution of the nonlinear system’);
grid on;
% Plot the phase portrait
figure;
plot(y(:, 1), y(:, 2));
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
% Define the tail (e.g., last 10% of the time interval)
tail_start = floor(0.9 * length(t)); % Starting index for the tail
tail_end = length(t); % Ending index for the tail
% Plot the tail of the solution
figure;
plot(y(tail_start:tail_end, 1), y(tail_start:tail_end, 2), ‘r’, ‘LineWidth’, 1.5);
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
ax = gca;
chart = ax.Children(1);
datatip(chart,0.5581,-0.1126);
Then I used but the results were not that I expected for
My code gives me the following. Any suggestion?
% Define parameters
gamma = 0.35;
alpha = -1;
beta = 1;
delta = 0.1;
omega = 1.4;
% Define the system of equations
odeSystem = @(t, y) [y(2);
-delta*y(2) – alpha*y(1) – beta*y(1)^3 + gamma*cos(omega*t)];
% Initial conditions
y0 = [0; 0]; % x(0) = 0, v(0) = 0
% Time span
tspan = [0 3000];
% Solve the system
[t, y] = ode45(odeSystem, tspan, y0);
% Plot the phase portrait
figure;
plot(y(:, 1), y(:, 2));
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
% Define the tail (e.g., last 10% of the time interval)
tail_start = floor(0.9 * length(t)); % Starting index for the tail
tail_end = length(t); % Ending index for the tail
% Plot the tail of the solution
figure;
plot(y(tail_start:tail_end, 1), y(tail_start:tail_end, 2), ‘r’, ‘LineWidth’, 1.5);
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.35$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
ax = gca;
chart = ax.Children(1);
datatip(chart,0.5581,-0.1126); The Original Equation is the following:
Let . This implies that
Then we eewrite it as a System of First-Order Equations
Using the substitution for , the second-order equation can be transformed into the following system of first-order equations:
Exploring the Effect of .
% Define parameters
gamma = 0.338;
alpha = -1;
beta = 1;
delta = 0.1;
omega = 1.4;
% Define the system of equations
odeSystem = @(t, y) [y(2);
-delta*y(2) – alpha*y(1) – beta*y(1)^3 + gamma*cos(omega*t)];
% Initial conditions
y0 = [0; 0]; % x(0) = 0, v(0) = 0
% Time span
tspan = [0 2000];
% Solve the system
[t, y] = ode45(odeSystem, tspan, y0);
% Plot the results
figure;
plot(t, y(:, 1));
xlabel(‘Time’);
ylabel(‘x(t)’);
title(‘Solution of the nonlinear system’);
grid on;
% Plot the phase portrait
figure;
plot(y(:, 1), y(:, 2));
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
% Define the tail (e.g., last 10% of the time interval)
tail_start = floor(0.9 * length(t)); % Starting index for the tail
tail_end = length(t); % Ending index for the tail
% Plot the tail of the solution
figure;
plot(y(tail_start:tail_end, 1), y(tail_start:tail_end, 2), ‘r’, ‘LineWidth’, 1.5);
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
ax = gca;
chart = ax.Children(1);
datatip(chart,0.5581,-0.1126);
Then I used but the results were not that I expected for
My code gives me the following. Any suggestion?
% Define parameters
gamma = 0.35;
alpha = -1;
beta = 1;
delta = 0.1;
omega = 1.4;
% Define the system of equations
odeSystem = @(t, y) [y(2);
-delta*y(2) – alpha*y(1) – beta*y(1)^3 + gamma*cos(omega*t)];
% Initial conditions
y0 = [0; 0]; % x(0) = 0, v(0) = 0
% Time span
tspan = [0 3000];
% Solve the system
[t, y] = ode45(odeSystem, tspan, y0);
% Plot the phase portrait
figure;
plot(y(:, 1), y(:, 2));
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.318$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
% Define the tail (e.g., last 10% of the time interval)
tail_start = floor(0.9 * length(t)); % Starting index for the tail
tail_end = length(t); % Ending index for the tail
% Plot the tail of the solution
figure;
plot(y(tail_start:tail_end, 1), y(tail_start:tail_end, 2), ‘r’, ‘LineWidth’, 1.5);
xlabel(‘x(t)’);
ylabel(‘v(t)’);
title(‘Phase-Plane $$ddot{x}+delta dot{x}+alpha x+beta x^3=0$$ for $$gamma=0.35$$, $$alpha=-1$$,$$beta=1$$,$$delta=0.1$$,$$omega=1.4$$’,’Interpreter’, ‘latex’);
grid on;
ax = gca;
chart = ax.Children(1);
datatip(chart,0.5581,-0.1126); ode45, duffing equation, plot, differential equations, plotting MATLAB Answers — New Questions
