# [SciPy-user] Eigenvalues of the Floquet matrix

Nils Wagner nwagner@iam.uni-stuttgart...
Sat Mar 14 15:21:46 CDT 2009

Hi all,

I am interested in the stability of time periodic ODE's of
the form

\dot{y} = A(t) y,  A(t)=A(t+T)          (1)

I have used scipy.integrate.ode to compute the eigenvalues
of the Floquet matrix. See attachment for details.

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.

The system is stable iff all eigenvalues are inside
the unit circle.

However, I cannot reproduce the symmetry of the
numerical multipliers computed by scipy, e.g.

>>> evals
array([ -2.55239771e-02+0.j        ,
-2.39509330e-02+0.99921743j,
-2.39509330e-02-0.99921743j,  -3.85602951e+01+0.j
])
>>> 1./evals
array([ -3.91788472e+01-0.j        ,
-2.39746890e-02-1.00020852j,
-2.39746890e-02+1.00020852j,  -2.59334115e-02-0.j
])

How can I improve the numerical results concerning the
symmetry ?

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

How can one accelerate the process ?

Is it possible to parallelize the integration, I mean
each processor could solve (1) for a new set of
initial conditions ?

Any pointer would be appreciated.

BTW, the example is taken from a recent paper by
Seyranian.

Nils

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