Consider a linear time-invariant system \[ \dot x(t) = A x(t), \ \ x \in \mathbb{R}^n, \tag{1} \] where \(A\) is a constant \(n \times n\) matrix with real coefficients. Let \(\kappa(t)\) be the first curvature of the trajectory \(x(t)\) of system (1). The main results of the paper are the following:
1. If there exists a measurable set \(E_1 \subseteq \mathbb{R}^n\) whose Lebesgue measure is positive, such that for all initial values \(x(0) \in E_1\) the limit \[ \lim\limits_{t \to +\infty} \kappa(t)\tag{2} \] is not zero or it does not exist, then the zero solution of system (1) is stable.
2. If the matrix \(A\) is invertible and there exists a measurable set \(E_2 \subseteq \mathbb{R}^n\) whose Lebesgue measure is positive, such that for all initial values \(x(0) \in E_2\) the limit (2) is \(+\infty\), then the zero solution of (1) is asymptotically stable.