Numerical integration "backwards"
14 Ansichten (letzte 30 Tage)
Ältere Kommentare anzeigen
Hello everyone,
I have a system of nonlinear equations on the form
x1' = f1(x1,x2,x3)
x2' = f2(x1,x2,x3)
x3' = f3(x1,x2,x3)
Let the flow F(x,t) be the solution to the system, that is, F(x0,t) as a function of t is an integral curve that goes through x0 at t=0.
I need the value of the inverse flow to some particular values of x0 and t0, that is, the value of F^-1(x0,t0) = F(x0,-t0).
Wen I use Matlab's odeXX functions to integrate F() numerically, I get well-behaved integral curves. Now, the problem is, when I try to use them to integrate F^-1() by reversing the time span parameter (or by exchanging the signs of the right hand sides of the differential equations), then apparently a numerical unstability occurs, and the integral curve grows extremely fast.
Does any one know how I can get the curve for F^-1(x0,t0), for particular t0 and x0?
Code used so far:
%code to compute the solution for F(x0,t0):
[t1,flow] = ode15s(@f1f2f3,[0 t0],x0);
%code to compute the solution for F^-1(x0,t0)=F(x0,-t0):
[t2,flowinv] = ode15s(@f1f2f3,[t0 0],x0);
1 Kommentar
Akzeptierte Antwort
Andrew Newell
am 3 Apr. 2011
You seem to be talking about equations for some physical phenomenon like fluid flow. In general, you can't integrate physical equations backwards. You can only do it if you don't have dissipative terms like friction. The numerical blowup is probably telling you that you have such a term. This is not really a MATLAB issue.
EDIT: You could get an intuitive feel for whether it is feasible or not by plotting a phase portrait of the system, i.e., choose a grid of reasonable starting values, calculate the flow, and then plot all the curves together. Do you have attractors?
7 Kommentare
Andrew Newell
am 8 Apr. 2011
Yes. The phase portrait is just a picture of flow lines while the Poincaré map is a less intuitive plot that looks at intersections of trajectories in some traversal section in phase space. Sorry, I can't really explain it in a comment. The point is that if the orbits are periodic they'll keep intersecting at the same points, but if it's chaotic you'll get a much more complex pattern.
Weitere Antworten (0)
Siehe auch
Kategorien
Mehr zu Numerical Integration and Differential Equations finden Sie in Help Center und File Exchange
Produkte
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!