[SciPy-user] Eigenvalues of the Floquet matrix

[Nils Wagner]

On Sat, 14 Mar 2009 16:47:07 -0400
  Rob Clewley <rob.clewley at gmail.com> wrote:
> On Sat, Mar 14, 2009 at 4:21 PM, Nils Wagner
> <nwagner at iam.uni-stuttgart.de> wrote:
>> 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)
> 
>> How can I improve the numerical results concerning the
>> symmetry ?
> 
> I can't see straight away what's causing the numerical 
>inaccuracy.
> 
>> BTW, the example is taken from a recent paper by 
>>Seyranian.
> 
> If you send me Seyranian's paper I might be able to work 
>it out. Could
> you do that?
> 
> -Rob
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Rob,

I have added another test. Consider the dynamical systems

\dot{X} = A(t) X          (1)

\dot{Y} = -A^T(t) Y       (2)


X^T Y = I          (3),

where I denotes the identity matrix.

Differentiating (3) yields

\dot{X}^T Y + X^T \dot{Y} = 0   (4)

Inserting (1),(2) in (4) confirms (3)

Here is the product of X^T Y

[[ 0.99682762  0.00427521 -0.00417167  0.00414188]
  [ 0.00212284  0.99717958  0.00329442 -0.00330161]
  [ 0.00122548 -0.00120682  0.98516109  0.01544185]
  [-0.00108133  0.00108144  0.01406736  0.98537091]]

Again, I cannot reproduce the theory. The off-diagonal
elements should be zero...

Nils


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