ode45 fail with pendulum

Hello. I have proof that in my machine the pendulum goes all the way around, contradicting reality. Any thoughts?

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MATLAB Version: 9.8.0.1323502 (R2020a)

MATLAB License Number: 40875211

Operating System: Microsoft Windows 8.1 Version 6.3 (Build 9600)

Java Version: Java 1.8.0_202-b08 with Oracle Corporation Java HotSpot(TM) 64-Bit Server VM mixed mode

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MATLAB Version 9.8 (R2020a)

Simulink Version 10.1 (R2020a)

Optimization Toolbox Version 8.5 (R2020a)

Statistics and Machine Learning Toolbox Version 11.7 (R2020a)

Symbolic Math Toolbox

function pendule2
[~,y]=ode45(@fun,0:0.05:40,[pi-0.1,0]);
figure(1)
close(1)
figure(1)
[n,~]=size(y);
t1=0:0.05:2*pi;
x1=cos(t1);
y1=1+sin(t1);
for k=1:n
    hold on
    axis equal
    axis([-1 1 0 2])
    plot(x1,y1,'k--')
    plot(sin(y(k,1)),1-cos(y(k,1)),...
        'bo','MarkerSize',5,'MarkerFaceColor','b')
    plot([0 sin(y(k,1))], [1 1-cos(y(k,1))],'b')
    pause(0.01)
    clf
end
hold on
axis equal
axis([-1 1 0 2])
plot(x1,y1,'k--')
plot(sin(y(n,1)),1-cos(y(n,1)),...
    'bo','MarkerSize',5,'MarkerFaceColor','b')
plot([0 sin(y(n,1))], [1 1-cos(y(n,1))],'b')
text(0.85,0.1,'end')
    function [dydt]=fun(~,y)
        dydt=[y(2); -sin(y(1))];
    end
end

Answer 1

[Mischa Kim]

Hi Bobby, this is the wonderful world of numerical (vs symbolic) computation. ode45 is a numerical integrator that approximates the actual solution. In essence this mean that with every integration step there will be a small error that adds up over time. You can fine-tune how well you would like to approximate the solution by setting tolerance levels. Try, e.g.

options = odeset('RelTol',1e-10);
[~,y]   = ode45(@fun,0:0.05:40,[pi-0.1,0],options);