Plotting phase space for 2nd order nonlinear ODE

Question

Fue Xiong 2022-5-9

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1715135-plotting-phase-space-for-2nd-order-nonlinear-ode

回答： Steven Lord 2022-5-9

Hello, I'm trying to figure out how to plot the phase space for a 2nd order nonlinear ODE.

Problem Statement

Given the 2nd order ODE, (d^2x/dt^2) - u*(1-x^2)*(dx/dt)+x=0, initial conditions are x(0) = 2 and x'(0) = 0. Find the solution if u = 1 and t is from 0-30 seconds. Plot time vs x and the phase space where x vs x'

MATLAB Code (what I have so far)

[t, y] = ode45(@ODE45SolvMain,[0 30],[2 0])

figure(1)

plot(t,y(:,1),t,y(:,2))

xlabel('Time t')

ylabel('Displacement')

title('Displacement vs Time')

legend('x_1','x_2')

---------------------------------------------------------

% Function Definition

function [x_primes] = ODE45SolvMain(t,x)

u = 1;

x_primes = [x(2); u*((1-x(1)^2))*x(2)-x(1)]

My solution so far

This figure is for the time vs x plot. I am unsure how to begin and plot the phase space for the this ODE.

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Answer 1

Star Strider 2022-5-9

1
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1715135-plotting-phase-space-for-2nd-order-nonlinear-ode#answer_960450

在 MATLAB Online 中打开

I am not certain what you want to plot for the phase plot.

This calculates and plots

as a funciton of

and also calculates the derivatives (as the ‘derivs’ matrix) in the event those are also to be plotted —

[t, y] = ode45(@ODE45SolvMain,[0 30],[2 0]);

figure(1)

plot(t,y(:,1),t,y(:,2))

xlabel('Time t')

ylabel('Displacement')

title('Displacement vs Time')

legend('x_1','x_2')

figure(2)

plot(y(:,1),y(:,2))

xlabel('x_1')

% xlabel(']Time t')

ylabel('x_2')

% ylabel('Displacement')

title('Phase')

% legend('x_1','x_2')

% Derivative Calculations

for k = 1:numel(t)

derivs(k,:) = ODE45SolvMain(t(k),y(k,:));

end

Derivs = table(t,derivs(:,1),derivs(:,2), 'VariableNames',{'t','dy1dt','dy2dt'})

Derivs = 341×3 table

t dy1dt dy2dt __________ ___________ _______ 0 0 -2 2.5119e-05 -5.0236e-05 -1.9998 5.0238e-05 -0.00010047 -1.9997 7.5357e-05 -0.0001507 -1.9995 0.00010048 -0.00020092 -1.9994 0.00022607 -0.00045199 -1.9986 0.00035166 -0.00070296 -1.9979 0.00047726 -0.00095383 -1.9971 0.00060285 -0.0012046 -1.9964 0.0012308 -0.0024571 -1.9926 0.0018588 -0.0037072 -1.9889 0.0024868 -0.004955 -1.9851 0.0031147 -0.0062005 -1.9814 0.0062546 -0.012392 -1.9628 0.0093945 -0.018526 -1.9443 0.012534 -0.024603 -1.9261

% Function Definition

function [x_primes] = ODE45SolvMain(t,x)

u = 1;

x_primes = [x(2); u*((1-x(1)^2))*x(2)-x(1)];

end

.