Undergrad How to study an ODE in matrix form in a Hilbert space?

[SeM]

Hello, I have derived the matrix form of one ODE, and found a complex matrix, whose phase portrait is a spiral source. The matrix indicates further that the ODE has diffeomorphic flow and requires stringent initial conditions. I have thought about including limits for the matrix, however the diffeomorphism already indicates that we deal with infinity unless some conditions are applied.

I am however unaware on how to study the ODE in a Hilbert space, or simply represent it in it. I have the matrix: [ 2, i4; i5, 6] . Is there any particular thing that I have missed here? I.e. The Poincare map?

Thanks!

 

[S.G. Janssens]

Quite a number of different terms appear in your post.

Firstly, although $\mathbb{C}^n$ is also a Hilbert space, to me the term "Hilbert space" suggests infinite dimensionality.

As far as I see, you have a $2 \times 2$ complex matrix $A$ corresponding to an autonomous linear system of two complex ODEs. The time evolution of an initial condition $\mathbf{z}_0 \in \mathbb{C}^2$ is given by the matrix exponential $e^{tA}$ acting on $\mathbf{z}_0$, which in turn can be computed from the eigenvalues and eigenvectors of $A$.

 

[SeM]

Thanks Krylov.

I have looked at your suggestion, and I have computed already the general solution:

\begin{equation}
X_1(t) = \alpha e^{\lambda_1 t} \binom{a}{b}+ \beta e^{\lambda_2 t} \binom{c}{d}
\end{equation}

Is this what you had in mind?

Thanks!

 

[S.G. Janssens]

The initial condition is z_0 = cos(phi)

I think the initial condition should have two components, i.e. $\mathbf{z}_0 = (z_{0,1}, z_{0,2})$ with $z_{0,1}$ and $z_{0,2}$ in $\mathbb{C}$.

was not aware of the time-dimension in this.

For definiteness, I understood that you like to solve
$$
\frac{d\mathbf{z}(t)}{dt} = A\mathbf{z}(t), \quad A =
\begin{bmatrix}
2& 4i\\
5i& 6\end{bmatrix}, \quad
\mathbf{z}(t) = \mathbf{z}_0,
$$
where "time" $t$ is thought of as the independent variable.

How does an example of the matrix exponential in the time dimension computed for the eigenvalues and vectors look like? In a book a I have "Diff Eqns, Dynamical Syst and Intro to Chaos", it gives a description of the exponential matrix. Is it this you refer to?

Yes, is this the book by Hirsch, Devaney and Smale? I believe that there this is indeed discussed, but for for matrices with real entries only. (This is relevant for most applications. Of course, there may still be complex eigenvalue and eigenvectors, but $A$ itself is assumed real.) You would need a treatment that discusses the similar - but not identical - case of complex matrices.

Perhaps it is in the 1974 edition by Hirsch and Smale (a lot of linear algebra was unfortunately removed in subsequent editions), but otherwise you will find it in almost every linear algebra text that does not restrict itself to vector spaces over $\mathbb{R}$.

 

[SeM]

I changed the reply because I realized I may have understood you right in the first place, however, what you wrote now is new to me.

You mention definiteness, which appears as a time-evolution behavior of the matrix components. I have initial conditions for the analytical solution, which is x_0 = cos(theta), however, in order to study this from a Hilbert space perspective, this condition would not apply as you write.

Does the general solution I wrote above answer the time evolution point you mentioned? In case not, the respective chapter by Hirsch, Devaney and Smale discussed some more terms, such as A -¹ and I in this context. How can this approach be simplified in the given case? (Unless the general solution does not answer that):

PS: I have used the exponential of the matrix by multiplying it with the eigenvectors. The exponential term is the eigenvalue. However, you included A in your exponential term. Is this a different story?

 

[SeM]

PS: I have computed the exponential of the matrix, which is:

[ e^2, e^4i; e^5i, e^6]

However, how can this be applied to say something about the matrix, which the phase-plane portrait hasn't already?

Thanks

 

[SeM]

Dear Krylov, can the divergence or convergence of the eigenvectors say something about the general solution? I can't find anyting on divergence and convergence of eigenvectors on the web that says that.

Thanks!