How to define event function in ode45 solver as the slop of variable achieve certain value?

Hi! I want to solve the Van Der Pol equations using ode45 with event function as its stopping criteria. I would like to set the stopping criteria as the slope of is small enough:

where is the change of the value of at the nearest time step and its previous time step.

I am not sure can event function access the value of both at the current time and the previous time, since it seems that the events function receives both the current time and the current state vector.

In this case, how to revise the code? Thanks a lot for any suggestion!

Here is the code

function main

options = odeset('Events',@event_function);

initial_cond = [2;0;0];

[t,y] = ode45(@vanderpol,[0 20],initial_cond,options);

plot(t,y(:,1),'-o',t,y(:,2),'-o',t,y(:,3),'-o');

title('Solution of van der Pol Equation (\mu = 1) with ODE45');

xlabel('Time t');

ylabel('Solution y');

legend('y_1','y_2','z');

function dydt = vanderpol(t,y)

dydt = [y(2); (1-y(1)^2)*y(2)-y(1); y(1)^2+y(2)^2];

function [value,isterminal,direction] = event_function(t,y)

value = y(1)-0.1; % when value = 0, an event is triggered

isterminal = 1; % terminate after the first event

direction = 0; % get all the zeros

Answer 1

[Torsten]

Maybe someone could answer if he/she knew what z is in your code. Is it y(1), y(2) or y(3) ?

Anyhow: The slopes of your variables is given by

dydt = [y(2); (1-y(1)^2)*y(2)-y(1); y(1)^2+y(2)^2];

So choose the one you need to specify your event.

[H_W]

My apology! z is . Thus I just need to write value = y(1)^2+y(2)^2 to solve the problem. Thank you!

But I am still confused since this is because the slope is exactly given by current states

If I ask for previous states, can event function access it? For example, I want to specify the stopping criteria as on the previous state equals to 1.

[Torsten]

The events function only refers to current states, but you could save the value of the last call. See

https://www.mathworks.com/help/matlab/ref/persistent.html

for details.

[H_W]

This should be really helpful to my problem! Thanks a lot for the link!