New mechanisms leading to the intermittency in Shilnikov chaos: Randomization theory of the infinite-modal maps

Statistical properties of intermittent chaos generated by infinite-modal maps are investigated. Infinite-modal maps treated in this paper are related to the Shilnikov chaos, which appears in ordinary differential equations. First, we present that the infinite-modal maps generate strong intermittency with bursts like so-called "on-off intermittency". Furthermore, we develop a randomization theory of the infinite-modal maps based on the Weyl's theorem, to explain that the intermittent mechanism is generally described by a nonlinear multiplicative random process which is a generalization of the standard on-off intermittency. Second, two statistical properties are analytically derived; one is a stationary distribution, and the other is a laminar-duration distribution. Near the critical state, the stationary distribution is shown to be a log-normal distribution, and the laminar-duration distribution is analytically obtained as a function of a threshold. These theoretical results are successfully confirmed in the numerical examinations, and the previous results for the on-off intermittency are all explained systematically in these analytical formulae.