Time Period of Limit Cycle numerically

[leothorn]

Homework Statement

Consider the system of equations : x' = x-y-x^3 and y'=x+y-y^3
a) Draw a phase plot ( Done numericlly program listed here in matlab)
b) Prove analytically that at-least one stable cycle exists ( Used Poincare Bendixon theorem to prove done)
c) Compute the period of the cycle numerically ( No idea about this >>)

Homework Equations

Limitcycle.m

clc;
clear all;
options=[];
timeperiod=[];
phasepoints=[];
xnew =0.01;
ynew =0.01;
% x_start= -0.9:0.1:0.9;
% xstart = 0.9:-0.1:-0.9;
% x_start= [x_start,xstart];
% y_start= -0.9:0.1:0.9;
% ystart= -0.9:0.1:0.9;
% y_start=[y_start,ystart];
%for u=1:size(x_start,2)
[t y]=ode23('cyclefunc',[0 50],[xnew ynew],options);
timeperiod=[timeperiod;t];
phasepoints=[phasepoints;y];
%end
n=0.015;
points=y;
figure(2)
plot(phasepoints(:,1),phasepoints(:,2));
xlabel('X-axis');
ylabel('Y-axis');
% plot3(t,y(:,1),y(:,2));
% xlabel('Time');
% ylabel('Y-axis');
% zlabel('X-axis');
% startingNode=y(4,:);


% filename: cyclefunc.m
function dydt = f(t,y,flag)
dydt = [y(1)-y(2)-y(1)^3;
y(1)+y(2)-y(2)^3];
plot(y(1),y(2));
xlabel('X-axis');
ylabel('Y-axis');
drawnow;
hold on;
axis([-1.3 1.3 -1.3 1.3]);

These programs generate the limit cycle numerically and it is correct

The Attempt at a Solution

Now the attempt is the above program itself. I scoured the internet to search for an algorithm for generating time period numerically . I saw some programs like pplane which do it for you ( I doesn't show how it calculate though)
Any one can tell me a primitive algorithm for calculating this ?? Or if my current MATLAB program can be modified to calculate the same ?

 

[mighty2000]

Hello, great job on drawing the phase plot and proving the existence of a stable cycle using Poincare Bendixon theorem. As for computing the period of the cycle numerically, there are a few ways to do this. One way is to use the built-in function "findpeaks" in MATLAB to find the peaks of the limit cycle, and then use the time values at these peaks to compute the period. Another way is to use the "ode23" function to solve the system of equations for a longer time period and then use interpolation to find the time values at which the system returns to its starting point. I would suggest looking into these methods and seeing if you can modify your current code to implement them. Good luck!