We consider a simple model for multidimensional conewise linear dynamics around cusplike equilibria. We assume that the local linear evolution is either ${\mathbf{v}}^{\prime }=\mathbb{A}\mathbf{v}$ or $\mathbb{B}\mathbf{v}$ (with $\mathbb{A}, \mathbb{B}$ independently drawn from a rotationally invariant ensemble of symmetric $N\times N$ matrices) depending on the sign of the first component of $\mathbf{v}$. We establish strong connections with the random diffusion persistence problem. When $N\to \infty$, we find that the Lyapunov exponent is non-self-averaging, i.e., one can observe apparent stability and apparent instability for the same system, depending on time and initial conditions. Finite $N$ effects are also discussed and lead to cone trapping phenomena.

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• Dynamical systems
• First passage problems
• Non-Markovian processes
• Random matrix theory