高维动力系统中的混沌
Chaos in high-dimensional dynamical systems
摘要：
对于一般的耗散动力系统，我们研究什么分数的解决方案表现出混沌行为取决于相空间的维数$d$。我们发现，具有随机系数和初始条件的二次和三次非线性的$d$全局耦合常微分方程系统，其轨迹混沌的概率普遍地从$d=3$的$\sim 10^{-5} - 10^{-4}$增加到$d\sim 50$的本质上1。在大$d$的极限下，动力系统的不变测度表现出依赖于非线性程度但不依赖于系数选择的普适标度，最大李雅普诺夫指数收敛于普适标度极限.使用统计参数，我们提供了所观察到的缩放和混沌的概率的分析解释。
英文摘要：
For general dissipative dynamical systems we study what fraction of solutions exhibit chaotic behavior depending on the dimensionality $d$ of the phase space. We find that a system of $d$ globally coupled ODE's with quadratic and cubic non-linearities with random coefficients and initial conditions, the probability of a trajectory to be chaotic increases universally from $\sim 10^{-5} - 10^{-4}$ for $d=3$ to essentially one for $d\sim 50$. In the limit of large $d$, the invariant measure of the dynamical systems exhibits universal scaling that depends on the degree of non-linearity but does not depend on the choice of coefficients, and the largest Lyapunov exponent converges to a universal scaling limit. Using statistical arguments, we provide analytical explanations for the observed scaling and for the probability of chaos.
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