论文标题
随机锥线性线性系统会稳定吗？
Will Random Cone-wise Linear Systems Be Stable?
论文作者
论文摘要
我们考虑了一个简单的模型，用于围绕尖端样平衡的多维锥线性线性动力学。我们假设本地线性进化是$ \ mathbf {v}^\ prime = \ mathbb {a} \ mathbf {v} $或$ \ mathbb {b} \ mathbf {v} $（with $ \ mathbb {a} $，$ \ mathbb { n $矩阵）取决于$ \ mathbf {v} $的第一个组件的符号。我们与随机扩散持久性问题建立了牢固的联系。当$ n \ to \ infty $时，我们发现lyapounov指数是非自我平等的，即，根据时间和初始条件，可以观察到同一系统的明显稳定性和明显的不稳定。还讨论了有限的$ n $效果，并导致锥体捕获现象。
We consider a simple model for multidimensional cone-wise linear dynamics around cusp-like 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 a rotationally invariant ensemble of $N \times N$ 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 Lyapounov 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.
