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Question: are the eigenvalues of a dynamical system invariant under a change of variables?

More specifically, consider a dynamical system A defined on a manifold $M_A$ by the evolution function $\phi_A(t,x)$ for all $x \in M_A$. Let $g$ be a diffeomorphism from $M_A$ to a different manifold $M_B$ and define the dynamical system $B$ on the manifold $M_B$ through the evolution function $\phi_B(t,y) = g \circ \phi_A \circ g^{-1}$ for $y \in M_B$.

I can see that the fixed points of system A map one-to-one to the fixed points of system B. For instance if a fixed point is stable in A, the corresponding fixed point in B will also be stable. Are the eigenvalues of the fixed points also the same in the two systems?

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Let $X$ denote the vector field defined by $$X_p=\frac{d}{dt}\phi_A(t,p)\vert_{t=0}, \quad p\in M_A.$$ Essentially, your ODE corresponding to the dynamical system reads $\dot x(t)=X_{x(t)}$. So asking about the eigenvalues means looking at how $X$ behaves close to a fixed point $q$. Let $v\in T_qM_A$ be a tangent vector at $q$, and $Y$ a local vector field such that $Y_q=v$. Since $X_q=0$ at a fixed point, the Lie derivative of $X$ wrt. $Y$ at $q$ reads $$(\mathcal{L}_YX)_q=(Y^i\partial_iX^j-X^i\partial_iY^j)_q\partial_j=v^i(\partial_iX^j)_q\partial_j,$$ i.e. it does not depend on the behavior of $Y$ away from $q$. Therefore the map $L:v\mapsto(\mathcal{L}_YX)_q$ is well-defined as a linear endomorphism of $T_qM_A$. One may check that if $M_A=\mathbb{R}^n$, this endomorphism is represented by the matrix that appears when linearizing the ODE around the fixed point, i.e. when writing

$$\dot x=f(x)\approx L(x-q).$$

One could obtain the same endomorphism by using a covariant derivative wrt. some connection $\nabla$, and show that the covariant derivative of $X$ in $q$ does not depend on the choice of connection.

Since the above endomorphism is well-defined, it behaves naturally under diffeomorphisms, and its eigenvalues are well-defined as well (they do not depend on the choice of coordinates/basis of the tangent space at $q$).

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