= Solution
Follow a perturbation through one complete circuit. The chain rule gives
$$
\delta x_k=M\delta x_0+O(\|\delta x_0\|^2),\qquad M=Df(x_{k-1})\cdots Df(x_1)Df(x_0).
$$
After $m$ circuits the linear perturbation is $M^m\delta x_0$. With distinct <eigenvalues>, the <eigenvectors> span over the complex numbers, and each component is multiplied by $\lambda_j^m$. Thus the <periodic orbit> is linearly asymptotically stable exactly when
$$
\boxed{|\lambda_j(M)|<1\quad\text{for every }j.}
$$
These are the orbit's discrete <Floquet multipliers>. Any modulus greater than one causes instability; a unit modulus prevents decay in the linear system, and nonlinear stability then needs further analysis. The finitely many intermediate <derivative> factors do not affect convergence to zero between complete circuits.
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