MATLAB: Finding frequency and domain of equation using ode45

Question

dear all

i use following code to find answer of the following equation :

u ̈+u+u^3=0

function dydt= vdp1(t,u)
dydt=[u(2);-u(1)-((u(1))^3)];

clc
clear all
for a=0.1:0.1:0.3
[t,y]=ode45(@vdp1,[0 60],[0 a]);
hold on
plot(t,y(:,1)) 
end

is there any way to find frequency and domain of this equation ? i know ode 45 gives nonuniform answer but can i use interpolation and if it is imposible i really appreciate if someone can help me finde the frequance and domain of this equation

Answer 1

Try this:

vdp1 = @(t,u) [u(2); -u(1)-((u(1))^3)];
tspan = linspace(0, 60, 240);
a=0.1:0.1:0.3;
for k = 1:numel(a)
    [t,y{k}]=ode45(@vdp1,tspan,[0 a(k)]);
end
ym = cell2mat(y);
figure
plot(t,ym(:,1:2:size(ym,2)))
grid
xlabel('Time')
lgdstrc = sprintfc('a = %.1f',a);
title('Time Domain Plot')
legend(lgdstrc)
Ts = mean(diff(tspan));                             % Sampling Interval
Fs = 1/Ts;                                          % Sampling Frequency
Fn = Fs/2;                                          % Nyquist Frequency
L = numel(tspan);                                   % Signal Length
FTvdp1 = fft(ym(:,1:2:size(ym,2)))/L;               % Fourier Trasnsform
Fv = linspace(0, 1, fix(L/2)+1)*Fn;                 % Frequency Vector
Iv = 1:numel(Fv);                                   % Index Vector
figure
plot(Fv, abs(FTvdp1(Iv,:)))
grid
xlabel('Frequency')
title('Frequency Domain Plot')
legend(lgdstrc)
xlim([0 1.2])

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