[SciPy-user] Eigenvalues of the Floquet matrix

Rob Clewley rob.clewley@gmail....
Sun Mar 15 12:56:00 CDT 2009

Hi Nils,

> The eigenvalues \lambda of the Floquet matrix are called multipliers.
> Multipliers of system (1) possess symmetry: If \lambda is the
> multiplier, then 1/\lambda is also a multiplier.

I am still not certain what the problem with your code is
specifically, but I can verify that when I put your model and
calculations into PyDSTool I get symmetric eigenvalues.

> The computation of a stability chart is a time-consuming task even for low dimensional systems.

The integrations with PyDSTool are an order of magnitude faster or
more even at the higher tolerances, so that should help a lot.

> How can I improve the numerical results concerning the symmetry ?

It could be just that the VODE integrator is not having its absolute
and relative tolerances set small enough (I'm not sure what these are
by default for VODE but I'm using values for Radau down near 1e-12). I
am also integrating with smaller maximum step size (0.001).

For the first pair of (omega, delta) = (16.2, 0.0128205128205)
parameter values that your code detects as giving a stable orbit, my
code does not agree. Your final point on the last of the four computed
orbits ("ic=3" in your loop) is

q1: 0.04280477
q1dot: 0.35978081
q2: 0.33468969
q2dot: 0.48781428

whereas mine is

q1:  0.0404874798281
q1dot:  0.350985516948
q2:  0.336574584737
q2dot:  0.504365520376

which would appear to suggest a source of error during integration
between our codes. My eigenvalues of F are

[ 0.96241252+0.j          0.00799704+0.99996802j  0.00799704-0.99996802j
1.03905548+0.j        ]

which are indistinguishable, as a set, from their reciprocals. I've
attached my PyDSTool script in case changing the VODE tolerances don't
get you the accuracy you need. Please let me know if you get VODE to
be more agreeable, I'm curious.

-Rob
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