I have the following definition and theorem are taken from [1], which I like as text book for ODE theory. However, I think that there is a problem below.
Definition 8.1.2. Let $C$ be a critical point for the system $X^{\prime}=F(X)$. The point $C$ is said to be
(i) stable if, for given $\varepsilon>0$, there exists a $\delta>0$ such that whenever $\|X(0)-C\|<\delta$, $\|X(t)-C\|<\varepsilon$ for all $t>0$;
(ii) asymptotically stable if there exists a $\gamma>0$ such that whenever $\|X(0)-C\|<\gamma$, $\lim_{t\to\infty}\|X(t)-C\|=0$;
(iii) strictly stable if it is stable and asymptotically stable;
(iv) unstable if it is not stable.
Theorem 8.2.1. Let $X^{\prime}=AX$ be a linear autonomous system with $n\times n$ real nonsingular constant coefficient matrix $A$. Let the critical point be at the origin in $\mathbb{R}^{n}$. Then the critical point is
(i) strictly stable if the real parts of the eigenvalues of $A$ are negative;
(ii) stable if $A$ has at least one pair of pure imaginary eigenvalues of
multiplicity one;
(iii) unstable otherwise.
Consider the system $\dfrac{{\mathrm d}}{{\mathrm d}t}\left(\begin{array}{c}x\\ y\\ z\end{array}\right) =\left(\begin{array}{rrr}0&-1&0\\1&0&0\\0&0&1\end{array}\right)\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$, which has the eigenvalues $\pm{\mathrm i}$ and $1$. This system has the general solution $\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}{c}c_{1}\cos(t)-c_{2}\sin(t)\\ c_{1}\sin(t)+c_{2}\cos(t)\\ c_{3}{\mathrm e}^{t}\end{array}\right)$, where $c_{1},c_{2},c_{3}$ are arbitrary constants. Hence, $\left(\begin{array}{c}0\\0\\0\end{array}\right)$ is surely unstable due to the definition since for $c_{3}\neq0$ the norm of the solution diverges to infinity as $t\to\infty$. However, the theorem says that it is stable. Am I reading the theorem wrong?
References
[1] T. Myint-U, Ordinary Differential Equations, Elsevier North-Holland Publishing, New York 1978.
Yes, (ii) is completely wrong. It's also possible for the equilibrium to be unstable even though all eigenvalues are $\pm i$, e.g. for $$A = \pmatrix{0 & -1 & 1 & 0\cr 1 & 0 & 0 & 1\cr 0 & 0 & 0 & -1\cr 0 & 0 & 1 & 0\cr}$$ which has eigenvalues $\pm i$ and solutions such as $$ \pmatrix{t \cos(t)\cr t \sin(t)\cr \cos(t)\cr \sin(t)\cr}$$ What is true is that a linear system is stable if the imaginary eigenvalues all have equal geometric and algebraic multiplicities and there are no eigenvalues of positive real part.