Can Ode 45 solve piecewise function in the iteration？

In the iterative process of Ode45, the variable values change with time. The relationship between variable values corresponds to different system of differential equations. Whether it is possible to select the corresponding system of differential equations to continue the iterative calculation according to the relationship of the variable values.

For example:

The initial value is (x0, y0). At first time, x0>y0, so Ode45 selects the system (1) of differential equations for iterative calculation. Then the variable values x and y change with time. When x<=y, the system (2) of differential equations is selected to continue the iterative calculation. And so on.

Code:

t0=0;
tf=10;
x10=8;
x20=5;
x0=[x10,x20];
[t, x1] = ode45(@fun,[t0:tf], x0);
x2=[t,x1];
plot(x1(:, 1),x1(:, 2));

function f=fun(t,x)
f=zeros(2,1);
f(1)=-(x(1)-x(2))/x(1);      % system(1),x(1)>x(2)
f(2)=(x(1)-x(2))/x(1)+1;     % system(1),x(1)>x(2)
% f(1)=(x(2)-x(1))/x(2);     % system(2),x(1)<=x(2)
% f(2)=1-(x(2)-x(1))/x(2);   % system(2),x(1)<=x(2)
end

The initial value is x0=[x10, x20]=[8, 5], where x10 > x20, so Ode45 selects the system (1) of differential equations for iterative calculation. When t=3s, x(1)=7.6491, x(2)=8.3509, where x(1)<=x(2). Then I want the system (2) of differential equations is selected to continue the iterative calculation by using [7.6491, 8.3509] as a new initial value.

Thank you.

[darova]

Do you have a code or something?

[Cola]

@darova Thanks， I update the question with an example code.

[Cola]

I don't know if these codes are right.

Code1:

function f=fun(t,x)
f=zeros(2,1);

f(1)=(-(x(1)-x(2))/x(1))*(x(1)-x(2)>0)+((x(2)-x(1))/x(2))*(x(1)-x(2)<=0);     
f(2)=((x(1)-x(2))/x(1)+1)*(x(1)-x(2)>0)+(1-(x(2)-x(1))/x(2))*(x(1)-x(2)<=0);    

end

Code2:

function f=fun(t,x)
f=zeros(2,1);

if x(1)>x(2)
f(1)=-(x(1)-x(2))/x(1);      % system(1),x(1)>x(2)
f(2)=(x(1)-x(2))/x(1)+1;     % system(1),x(1)>x(2)
else
f(1)=(x(2)-x(1))/x(2);     % system(2),x(1)<=x(2)
f(2)=1-(x(2)-x(1))/x(2);   % system(2),x(1)<=x(2)
end

end

Answer 1

[Bjorn Gustavsson]

Your best approach is to use the events-handling procedure to interupt and restart the ode-integration whenever x and y passes the other. See the help and documentation for odeset and look at the code to ballode for an example of how to handle events.

In your case you might be able to proceed straight-away with the integration provided all f_1, f_2, g_1, g_2 and all their derivatives to some degree match, but I'm not even sure of that.

HTH

[Cola]

Thank you for helps.