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Time Period of Limit Cycle numerically

  1. Aug 2, 2013 #1
    1. The problem statement, all variables and given/known data

    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 >>)

    2. Relevant 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

    3. 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 doesnt 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 ?
     

    Attached Files:

    Last edited: Aug 2, 2013
  2. jcsd
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