Multi-Photon Phosphor Feasibility Research

Statistical properties of dynamical chaos

“…This leads to the fact that the probability of switchings between the two states in one revolution about the fixed point is approximately equal to 1/2. This particular aspect of the dynamics ensures that the ACF of the x(t) and y(t) oscillations on the Lorenz attractor has the form defined by expression (14). However, the finite width of the peaks in the distribution and deviations from the probability 1/2 can lead to an ACF that decays to a certain finite, nonvanishing value.…”

“…Our studies of the statistical properties of the Lorenz attractor have demonstrated that the properties of the ACF are mainly defined by a random switching process and slightly depend on the rotation about the saddle-foci. The classical model of telegraph signal enables one to describe the behavior of ψ(τ ) for the Lorenz attractor by using expression (14). In particular, this expression closely approximates a linear decay of the ACF from 1.0 to 0.2 that allows us to estimate theoretically the correlation time.…”

“…In general, there is no stationary probability measure on nonhyperbolic chaotic attractors without noise, independent of the initial distribution. In this case the continuous limit transition D → 0 does not exist for the probability density of noisy nonhyperbolic systems [14]. Moreover, the probability characteristics of nonhyperbolic chaos are very sensitive to even the slightest changes of the system parameters [13,42,59,73].…”

“…A nonhyperbolic attractor is a maximal attractor of the dynamical system and encloses a countable set of regular and chaotic attracting subsets [11,12]. When δ-correlated Gaussian noise is added to the system, an invariant measure on such attractors exists too [14]. In the nonhyperbolic case the behavior of phase trajectories is significantly affected by noise [15,16,17,18] while it changes only slightly in systems with hyperbolic and nearly hyperbolic chaos [15,16,19,20].…”

“…When Gaussian noise sources are added to a system, basins of attraction of all the system's attractors can merge. As a result, a unified stationary probability density appears, independent of the initial state [14]. However, the statistical description of nonhyperbolic chaos encounters principal difficulties.…”