Re: AUTO07p and continuation of periodic solutions

On Mar 6, 4:56 am, Alois Steindl <Alois.Stei...@xxxxxxxxxxxx> wrote:
"Sven" <s.schm...@xxxxxxxxxxx> writes:
yes, I know what you mean (cylinder theorem). That's why I introduced
an unfolding parameter lambda that destroys this family
of periodic solutions for lambda != 0.

I would be afraid that this causes additional troubles. You would have
to design your unfolding such that it admits a unique periodic
solution for lambda=0. (If I understand your method correctly.)> I was able to continue a periodic orbit in the Lorenz system for
varying r and I can continue a family of periodic solutions
in a system with integral with the help of a detuning parameter. Now I
try to glue these two things together, but don't know how
because the obvious solution doesn't work.

The way I tried to do this is the following. I extracted the numerical
data for this orbits using a software that integrates the
equations of motion. Then I converted this file (using fcon) into a
form AUTO can work with (a solution file you can access with
load(e="r3b",c="r3b.1",s=solution file)).
I guess I haven't really understood how AUTO continues in more than
one free parameter since you can only specify the continuation
range for the principal parameter.

I would suggest to choose one of the following strategies:
a) Keep the parameter mu fixed and calculate all periodic solutions
(at least the connected family, you are interested in.)
b) For varying parameter mu calculate special periodic solutions, like
resonant ones.

Of course there could be interesting combinations of these methods:
Find an interesting solution with a) and continue it with method b.

BTW, I think that you know already that you can leave out one periodic
boundary condition, because that one is usually automatically
fulfilled (if you start on the proper branch).

Good luck
Alois

--
Alois Steindl, Tel.: +43 (1) 58801 / 32558
Inst. for Mechanics and Mechatronics Fax.: +43 (1) 58801 / 32598
Vienna University of Technology, A-1040 Wiedner Hauptstr. 8-10

Hi Alois,

thank you very much for your answer, I appreciate this very much!

The method with the detuning parameter is prety much standard, see
recent papers by Doedel et. al., especially
"Continuation of periodic orbits in conservative and Hamiltonian
systems", Physica D 181 (2003) 1-38 by
F.J. Munoz-Almaraz.

Your suggested approach sounds good and I pretty much did what you
said. But as I said, the connection of both continuations (in mu and
the detuning parameter) won't work. So I figured I use an even simpler
Hamiltonian system with natural parameter (lambda). I found one in
Guckenheimer&Holmes (page 45-47). For lambda>0 it has an elliptic
equilibrium at the origin so I detuned, AUTO found the Hopf
bifurcation and calculated the family of periodic orbits. I took one
of them and continued that one in three parameters (lambda, detuning,
period) and it worked!, although I dont't know how AUTO decides how to
vary the parameters.
So, what I think the problem is, is that if I try to feed AUTO with
the numerical data of one complete periodic solution, something goes
wrong because I tried this with the same system and it failed. Now I
try to find out what is going on here.
The other issue is that I want AUTO to continue the periodic orbits
for fixed energy. I'd like to use the user-supplied routine pvls to
add the energy as a parameter but I can't get it working with C. I
think it is broken in AUTO07P.

Kind regards,

Sven Schmidt

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