To integrate the given equations of motion and obtain the time series over N periods of oscillation, along with the corresponding spectrum for the displacement $X(\tau)$, we can modify your code to avoid using the asym_ODE function. Instead, we'll directly define the system of differential equations within the MATLAB script.

Here's a full version of your MATLAB code that numerically integrates the equations and computes the frequency spectrum using ode45:

matlabCopy code
clear variables
set(0, 'DefaultAxesFontName', 'Courier New', 'DefaultAxesFontSize', 11)

% Parameters
epsilon = 0.5;
X0 = 1; U0 = 0;   % Initial conditions
init = [X0, U0];   % Initial state vector

% Define ODE system
ode_system = @(tau, Y, epsilon) [Y(2); 
                                -Y(1) + epsilon * Y(1)^3];

% Set ODE solver options
options = odeset('RelTol', 1e-6, 'AbsTol', 1e-6);

% Compute period T (using integral approximation)
E = (1/2) - epsilon/3;
coeffs = [(epsilon/3), -1/2, 0, E];
sols = roots(coeffs);
sols = sort(sols, 'ascend');
Xc = sols(1);
integrand = @(y) 1./sqrt((1-y.^2) - (2*epsilon/3)*(1 - y.^3));
T = 2 * integral(integrand, Xc, 1);
p = 2 * pi / T;

% Set simulation time parameters
N = 8;  % Number of periods
Tmax = N * T; 
n = 2^13; % Number of points in time domain
dt = Tmax / n; 
t = 0:dt:Tmax-dt; % Time vector

% Frequency domain
domega = 2 * pi / Tmax;
omega_max = n * domega;
omega = (0:n-1) * domega;
omega(n/2+1:n) = omega(n/2+1) - omega_max;

% Solve the ODE using ode45
[t, soln] = ode45(@(tau, Y) ode_system(tau, Y, epsilon), t, init, options, epsilon);

% Extract position and velocity from the solution
X = soln(:, 1); 
U = soln(:, 2);

% Compute the FFT of the displacement X
spec = fft(X);

% Plot the time series for position X and velocity U
figure(1)
set(gcf,'Name','Position and velocity time series','NumberTitle','off')

subplot(2,1,1)
plot(t/T, X, 'b'); 
axis([0 N -1.5 1.5])
grid on; 
box on; 
xlabel('Time \tau/T'); 
ylabel('X(\tau)')
title(['\epsilon = ', num2str(epsilon)])

subplot(2,1,2)
plot(t/T, U, 'r'); 
axis([0 N -1.5 1.5])
grid on; 
box on; 
xlabel('Time \tau/T'); 
ylabel('U(\tau)')

% Compute the spectrum and plot it
s = find(omega >= 0);
figure(2)
set(gcf,'Name','Fast Fourier Transform','NumberTitle','off')
semilogy(omega(s)/p, abs(spec(s)), 'm');
grid on;
box on;
axis([0 10 1e-4 1e4]); 
xticks(0:1:10);
xlabel('Frequency \omega/p'); 
ylabel('Spectrum |FFT[X(\tau)]|')
title(['\epsilon = ', num2str(epsilon)])

Explanation:

1. ODE System Definition: We define the system of equations in the function ode_system as: $\frac{d^2 X}{d\tau^2} + X = \epsilon X^3$ which becomes a system of two first-order ODEs: $\frac{dX}{d\tau} = U$ $\frac{dU}{d\tau} = -X + \epsilon X^3$

2. Time and Frequency Domain Setup: We calculate the period $T$ of the oscillator using the integral, and we define the time vector t for N periods. The frequency domain is also computed using the FFT parameters.

3. ODE Solver: We use ode45 to numerically integrate the system. The time evolution of both $X(\tau)$ and $U(\tau)$ is computed.

4. Plotting:

   • The first figure plots the position $X(\tau)$ and velocity $U(\tau)$ as a function of time over $N$ periods.
   • The second figure shows the frequency spectrum of the displacement $X(\tau)$ obtained via FFT.

Adjustments:

• You can adjust the epsilon parameter, the number of periods N, or the number of time steps n to change the resolution or range of the simulation.
• The frequency spectrum is plotted using a logarithmic scale (semilogy) to visualize the amplitude of frequencies.

Would you like further explanations on any of the steps or modifications to the code?