## Plotting Phase Portrait of Duffing Equation

I study a paper that describes a stationary problem where the function satisfies the boundary conditions and is governed by a modified Laplace equation with a non-linear term. Here’s a breakdown of the equations and the conditions they represent:

1. Equation Details:

– The first equation:

This is a partial differential equation (PDE) where denotes the Laplacian in dimensions. The equation includes a linear term proportional to and a non-linear term proportional to . The parameter seems to modulate the influence of the spatial derivatives, while and scale the linear and non-linear terms, respectively.

Boundary conditions:

These conditions specify that $ Phi $ is zero at the boundaries of the domain.

2. Special Case (Non-coupled Particles, đź™‚

In the limit $ k = 0 $, the spatial derivative terms disappear, simplifying the equation to:

Here, seems to represent the state of a system described by the Duffing equation, which is a well-known model for non-linear oscillators with a cubic nonlinearity. The term introduces damping into the system, representing energy loss.

Here’s the Matlab code for simulating and visualizing the dynamics of the Duffing equation:

% Parameters

omega_d_squared = 1;

beta = 1;

delta = 0.5;

% Duffing equations

duffingEquations = @(t, y) [y(2); omega_d_squared * y(1) – beta * y(1)^3 – delta * y(2)];

% Initial conditions and time span

initial_conditions = [

-1.5, -1.5;

0.01, 0;

-0.01, 0;

1.5, 1.5

];

time_span = [0, 50];

% Colors for the plots

colors = [

0, 0, 1; % Blue

1, 0, 0; % Red

1, 0, 0; % Red

0, 1, 0 % Green

];

% Line styles

line_styles = {‘–‘, ‘-‘, ‘-‘, ‘-‘};

% Plot the solutions

figure;

hold on;

box on; % Add box around the plot

for i = 1:size(initial_conditions, 1)

[t, sol] = ode45(duffingEquations, time_span, initial_conditions(i, :));

plot(sol(:, 1), sol(:, 2), ‘Color’, colors(i, :), ‘LineStyle’, line_styles{i}, ‘LineWidth’, 1);

end

% Customize the plot

axis([-2 2 -2 2]);

grid on;

grid minor;

set(gca, ‘XTick’, -2:0.5:2, ‘YTick’, -2:0.5:2, ‘GridLineStyle’, ‘:’);

set(gca, ‘Box’, ‘on’);

% Add arrows

for i = 1:size(initial_conditions, 1)

[t, sol] = ode45(duffingEquations, time_span, initial_conditions(i, :));

% Adding arrows at specific points

for j = 1:5:length(sol)-1

quiver(sol(j, 1), sol(j, 2), sol(j+1, 1) – sol(j, 1), sol(j+1, 2) – sol(j, 2), ‘AutoScale’, ‘off’, ‘Color’, colors(i, :), ‘LineWidth’, 1);

end

end

% Labels and title

xlabel(‘$U_n$’, ‘Interpreter’, ‘latex’);

ylabel(‘$U_n^prime$’, ‘Interpreter’, ‘latex’);

title(‘Duffing Oscillator Phase Portrait’);

hold off;

My question is the following, how could I those arrows on the trajectoriesI study a paper that describes a stationary problem where the function satisfies the boundary conditions and is governed by a modified Laplace equation with a non-linear term. Here’s a breakdown of the equations and the conditions they represent:

1. Equation Details:

– The first equation:

This is a partial differential equation (PDE) where denotes the Laplacian in dimensions. The equation includes a linear term proportional to and a non-linear term proportional to . The parameter seems to modulate the influence of the spatial derivatives, while and scale the linear and non-linear terms, respectively.

Boundary conditions:

These conditions specify that $ Phi $ is zero at the boundaries of the domain.

2. Special Case (Non-coupled Particles, đź™‚

In the limit $ k = 0 $, the spatial derivative terms disappear, simplifying the equation to:

Here, seems to represent the state of a system described by the Duffing equation, which is a well-known model for non-linear oscillators with a cubic nonlinearity. The term introduces damping into the system, representing energy loss.

Here’s the Matlab code for simulating and visualizing the dynamics of the Duffing equation:

% Parameters

omega_d_squared = 1;

beta = 1;

delta = 0.5;

% Duffing equations

duffingEquations = @(t, y) [y(2); omega_d_squared * y(1) – beta * y(1)^3 – delta * y(2)];

% Initial conditions and time span

initial_conditions = [

-1.5, -1.5;

0.01, 0;

-0.01, 0;

1.5, 1.5

];

time_span = [0, 50];

% Colors for the plots

colors = [

0, 0, 1; % Blue

1, 0, 0; % Red

1, 0, 0; % Red

0, 1, 0 % Green

];

% Line styles

line_styles = {‘–‘, ‘-‘, ‘-‘, ‘-‘};

% Plot the solutions

figure;

hold on;

box on; % Add box around the plot

for i = 1:size(initial_conditions, 1)

[t, sol] = ode45(duffingEquations, time_span, initial_conditions(i, :));

plot(sol(:, 1), sol(:, 2), ‘Color’, colors(i, :), ‘LineStyle’, line_styles{i}, ‘LineWidth’, 1);

end

% Customize the plot

axis([-2 2 -2 2]);

grid on;

grid minor;

set(gca, ‘XTick’, -2:0.5:2, ‘YTick’, -2:0.5:2, ‘GridLineStyle’, ‘:’);

set(gca, ‘Box’, ‘on’);

% Add arrows

for i = 1:size(initial_conditions, 1)

[t, sol] = ode45(duffingEquations, time_span, initial_conditions(i, :));

% Adding arrows at specific points

for j = 1:5:length(sol)-1

quiver(sol(j, 1), sol(j, 2), sol(j+1, 1) – sol(j, 1), sol(j+1, 2) – sol(j, 2), ‘AutoScale’, ‘off’, ‘Color’, colors(i, :), ‘LineWidth’, 1);

end

end

% Labels and title

xlabel(‘$U_n$’, ‘Interpreter’, ‘latex’);

ylabel(‘$U_n^prime$’, ‘Interpreter’, ‘latex’);

title(‘Duffing Oscillator Phase Portrait’);

hold off;

My question is the following, how could I those arrows on the trajectoriesÂ I study a paper that describes a stationary problem where the function satisfies the boundary conditions and is governed by a modified Laplace equation with a non-linear term. Here’s a breakdown of the equations and the conditions they represent:

1. Equation Details:

– The first equation:

This is a partial differential equation (PDE) where denotes the Laplacian in dimensions. The equation includes a linear term proportional to and a non-linear term proportional to . The parameter seems to modulate the influence of the spatial derivatives, while and scale the linear and non-linear terms, respectively.

Boundary conditions:

These conditions specify that $ Phi $ is zero at the boundaries of the domain.

2. Special Case (Non-coupled Particles, đź™‚

In the limit $ k = 0 $, the spatial derivative terms disappear, simplifying the equation to:

Here, seems to represent the state of a system described by the Duffing equation, which is a well-known model for non-linear oscillators with a cubic nonlinearity. The term introduces damping into the system, representing energy loss.

Here’s the Matlab code for simulating and visualizing the dynamics of the Duffing equation:

% Parameters

omega_d_squared = 1;

beta = 1;

delta = 0.5;

% Duffing equations

duffingEquations = @(t, y) [y(2); omega_d_squared * y(1) – beta * y(1)^3 – delta * y(2)];

% Initial conditions and time span

initial_conditions = [

-1.5, -1.5;

0.01, 0;

-0.01, 0;

1.5, 1.5

];

time_span = [0, 50];

% Colors for the plots

colors = [

0, 0, 1; % Blue

1, 0, 0; % Red

1, 0, 0; % Red

0, 1, 0 % Green

];

% Line styles

line_styles = {‘–‘, ‘-‘, ‘-‘, ‘-‘};

% Plot the solutions

figure;

hold on;

box on; % Add box around the plot

for i = 1:size(initial_conditions, 1)

[t, sol] = ode45(duffingEquations, time_span, initial_conditions(i, :));

plot(sol(:, 1), sol(:, 2), ‘Color’, colors(i, :), ‘LineStyle’, line_styles{i}, ‘LineWidth’, 1);

end

% Customize the plot

axis([-2 2 -2 2]);

grid on;

grid minor;

set(gca, ‘XTick’, -2:0.5:2, ‘YTick’, -2:0.5:2, ‘GridLineStyle’, ‘:’);

set(gca, ‘Box’, ‘on’);

% Add arrows

for i = 1:size(initial_conditions, 1)

[t, sol] = ode45(duffingEquations, time_span, initial_conditions(i, :));

% Adding arrows at specific points

for j = 1:5:length(sol)-1

quiver(sol(j, 1), sol(j, 2), sol(j+1, 1) – sol(j, 1), sol(j+1, 2) – sol(j, 2), ‘AutoScale’, ‘off’, ‘Color’, colors(i, :), ‘LineWidth’, 1);

end

end

% Labels and title

xlabel(‘$U_n$’, ‘Interpreter’, ‘latex’);

ylabel(‘$U_n^prime$’, ‘Interpreter’, ‘latex’);

title(‘Duffing Oscillator Phase Portrait’);

hold off;

My question is the following, how could I those arrows on the trajectoriesÂ matlab, differential equations, plotting, equation, graphÂ MATLAB Answers â€” New Questions

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