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  Why can any pair of master coordinates be used to calculate a nonlinear mode of a nonlinear dynamical system?

+ 4 like - 0 dislike

This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied.

In symplectic mechanics, consider a dynamical system with no rigid body movements (all eigenvalues are non-zero, ie $\lambda_i\neq 0$). The centre manifold states that the dynamics can be written in the following form: $$ \begin{bmatrix} \dot{\mathbf{x}}_\mathrm{s} \\ \dot{\mathbf{x}}_\mathrm{c} \\ \dot{\mathbf{x}}_\mathrm{u} \end{bmatrix} =\begin{bmatrix} [ A_\mathrm{s}] & & \\ & [A_\mathrm{c}] & \\& &[A_\mathrm{u}] \end{bmatrix}\begin{bmatrix} \mathbf{x_\mathrm{s}} \\ \mathbf{x_\mathrm{c}} \\ \mathbf{x_\mathrm{u}} \end{bmatrix} +\text{nonlinear terms}$$ and that there exists a centre manifold (also stable and unstable manifolds, but they are not of interest here) of dimension the number of lines of $A_\mathrm{u}$.

Because all eigenvalues corresponding to the centre manifold ($\mathrm{Re}(\lambda_i)=0$) are purely imaginary and non-zero (no rigid body movements), $A_\mathrm{c}$ is composed of $2\times2$ blocks on the diagonal.

Now, we are searching for periodic orbits, so we focus and the centre manifold and we name the coordinates of $\mathbf{x}_\mathrm{u}$ such that $\mathbf{x}_\mathrm{u}=[u_1,v_1,\dots,u_n,v_n]^T$ (even dimension, as explained above).

In order to calculate the nonlinear normal mode $i$, many people write that any pair $(u_i,v_i)$ can be chosen as master coordinates and that all other pair $(u_j,v_j)$, $i\neq j$, can be written as: $$\left\lbrace \begin{aligned} u_j &=f_j(u_i,v_i) \\ v_j&=g_j(u_i,v_i)\end{aligned}\right.$$

Question: What ensures that choosing a pair of master coordinates is always possible? If not, could you give a counterexample?

This post imported from StackExchange Physics at 2014-08-17 11:31 (UCT), posted by SE-user anderstood
asked Aug 16, 2014 in Theoretical Physics by anderstood (20 points) [ no revision ]

I believe these are often only local coordinates, and there may be a singular set of problems (of measure zero) where such a choice is impossible. But I don't have a specific counterexample.

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