A matrix contraction process

作者：Wilkinson Michael^*; Grant John

来源：Journal of Physics A-Mathematical and Theoretical, 2018, 51(10): 105002.

DOI：10.1088/1751-8121/aaaa57

• random matrices
• stochastic process
• chaotic dynamical system

We consider a stochastic process in which independent identically distributed random matrices are multiplied and where the Lyapunov exponent of the product is positive. We continue multiplying the random matrices as long as the norm, epsilon, of the product is less than unity. If the norm is greater than unity we reset the matrix to a multiple of the identity and then continue the multiplication. We address the problem of determining the probability density function of the norm, P-epsilon. We argue that, in the limit as epsilon -> 0, P-epsilon similar to (ln(1/epsilon))(mu)epsilon(gamma), where mu and gamma are two real parameters.
Our motivation for analysing this matrix contraction process is that it serves as a model for describing the fine-structure of strange attractors, where a dense concentration of trajectories results from the differential of the flow being contracting in some region. We exhibit a matrix-product model for the differential of the flow in a random velocity field, and show that there is a phase transition, with the parameter mu changing abruptly from mu = 0 to mu = -3/2 as a parameter of the flow field model is varied.

• 出版日期2018-3-9