@acer That is very useful. Thanks

@janhardo You have done an amazing job. Thank you for your answers

@C_R i have played with some more attractors. I'm addicted to Maple :)

Download Attractors_2.mwDownload Attractors_2.mw

@C_R 

Download Attractors.mw

@janhardo but with different parameters and in the interval [ 0 800] the results are the following

restart;
unprotect(gamma);
 Digits := 15
 gamma := 0.318;
alpha := -1;
beta := 1;
delta := 0.1;
omega := 1.4;

sys := {diff(v(t), t) = -delta*v(t) - alpha*x(t) - beta*x(t)^3 + gamma*cos(omega*t), diff(x(t), t) = v(t)}:
 


ics := {v(0) = 0, x(0) = 0};
sol := dsolve(sys union ics, {v(t), x(t)}, numeric, range = 0 .. 800, output = listprocedure, maxfun = 0, abserr = 0.1e-17, relerr = 0.1e-12);
                  ics := {v(0) = 0, x(0) = 0}

 


X := subs(sol, x(t));
Y := subs(sol, v(t));


plot('[X(t), Y(t)]', t = 0 .. 800, numpoints = 500, title = "Trajectory", color = ["#40e0d0", "SteelBlue"]):

with(plots);
phaseplot := odeplot(sol, [x(t), v(t)], 0 .. 800, numpoints = 10000, color = red, thickness = 2, axes = boxed, gridlines, title = "Phase-space Diagram");

but in Matlab using the command ode45 the result is

@janhardo I have not thought about this approach. Very cool! However, we get the same result. Probably ode45 from MATLAB has some other options as a command.

@acer I tried the interval [270 800] now it is closer to the desirable plot but not identical

@acer When I use 

maxfun = 0, abserr = 0.1e-17, relerr = 0.1e-12

as @Preben Alsholm showed above the plot in the interval [789 900] is the same but in the interval [0 800] is different. Also when I run it, this message appears [Length of output exceeds limit of 1000000]

restart;
unprotect(gamma);
 Digits := 15
 gamma := 0.318;
alpha := -1;
beta := 1;
delta := 0.1;
omega := 1.4;

sys := {diff(v(t), t) = -delta*v(t) - alpha*x(t) - beta*x(t)^3 + gamma*cos(omega*t), diff(x(t), t) = v(t)}:
 

ics := {v(0) = 0, x(0) = 0};
sol := dsolve(sys union ics, {v(t), x(t)}, numeric, range = 0 .. 800, output = listprocedure, maxfun = 0, abserr = 0.1e-17, relerr = 0.1e-12);
                  ics := {v(0) = 0, x(0) = 0}

X := subs(sol, x(t));
Y := subs(sol, v(t));

plot('[X(t), Y(t)]', t = 0 .. 800, numpoints = 500, title = "Trajectory", color = ["#40e0d0", "SteelBlue"]);

with(plots);
phaseplot := odeplot(sol, [x(t), v(t)], 0 .. 800, numpoints = 10000, color = red, thickness = 2, axes = boxed, gridlines, title = "Phase-space Diagram");

phaseplot := odeplot(sol, [x(t), v(t)], 789 .. 800, numpoints = 3500, color = blue, thickness = 1, axes = boxed, gridlines, title = "Phase-space Diagram")

Hello @acer! I just used the command ode45 without specific options and I plotted the graphs as you can see below in my MATLAB's code. 

% Define parameters
gamma = 0.318;
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 = [0 800];

% 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 Portrait');
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 Portrait - Tail of the Solution');
grid on;
 

Hello @Preben Alsholm. Your animations are amazing. However, as @Carl Love mentioned below the second plot below shows only positive values for x(t) whereas the corresponding plot in the paper shows only negative values for x(t). How could I find the right way to create this plot?

 @Carl Love, I found the   Sample Project  sample_project_0.pdf , so now I am sure the initial conditions are correct. However, my question is that when I run it in Matlab the solutions come out similar but when I run it in Maple they are the ones you posted. Why are there such big differences? Especially with the negative solutions

@acer  Thank you for your answer. I was told that Ι must first register for a membership account so the Maple Ambassador Program Manager can add me to the Ambassador Portal: https://www.mapleprimes.com/register/ . However, I can't find the portal to see if I am allowed access

Hello  @Carl Love. I had excactly the same results, for this reason I tried different initial conditions, but still the results were not like in the document

@janhardo I wish I could give you a like

@Scot Gould  My goal in this exercise was to plot the equilibrium points on the phase plane. However, I appreciate your advice. It is the first time I will use Maple for my lessons.