Periodic Initial Conditions for IVP

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Angus

[ebon kelp]

but then I'm not sure how to continue

Should I find a T such that x(0 + T) = x(T) = x(0) ?

or should I compute the constants (c1, c2, c3) by computing the inverse matrix composed of the eigenvectors multiplied by the initial solution

with V being the matrix of the (independent) eigenvectors then compute $C = V^{-1}z$

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Angus

[smoky pine]

Do you have a given period?

If you do, I would maybe suggest finding 3 initial conditions based on the periods

x(0) = 0
x(T) = 0
x(2T) = 0

I guess in a sense, that is 3 equations with 3 unknowns

[ebon kelp]

Nope, all that is given is $$\dot x = Ax, x(0) = z$$, A is given but I want to know how to approach such a question in general and not just the answer to this one

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Angus

[ebon kelp]

A is 3x3 and z is in R^3

[smoky pine]

Well you've got to get equations from somewhere right

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Angus

[ebon kelp]

with eigenvalues 2, sqrt(2)i, -2sqrt(2)i and

and the 2nd and 3rd eigenvalues are purely imaginary which makes them rotations

so my question is moreso how can I justify setting the constant of the eigenvector of the first eigenvalue (2) to zero