EXACT SOLUTIONS FOR STATIONARY RESPONSES OF SEVERAL CLASSES OF NONLINEAR SYSTEMS TO PARAMETRIC AND/OR EXTERNAL WHITE NOISE EXCITATIONS

The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.

Dependence of Nonlinear Optical Response of Anatase TiO2 on Shape and Excitation Intensity

Nonlinear optical （NLO） properties of anatase TiO2 with nanostructures of nanopaxticle （NP）, nanowire （NW） and annealed nanowire （NWA） are studied by open-aperture and closed-aperture Z-scan techniques with a fem- tosecond pulsed laser at wavelengths of 532 nm and 780 nm simultaneously. At 532 nm, when increasing excitation intensity, NLO absorption of TiO2 NPs transforms from saturable absorption to reverse-saturable absorption. However, NWs and NWAs exhibit the opposite change. At 780nm, all samples show reverse-saturable absorption, but have different sensitivities to excitation intensity. Due to the larger surface-to-volume ratio of NPs and less defects of NWAs by annealing, nonlinear optical absorption coet^icients follow the order NPs≥ NWs≥ NWAs. The results also show that these shape and annealing effects axe dominant at low excitation intensity, but do not exhibit at the high excitation intensity. The NLO refractive index of NPs shows a positive linear relationship with the excitation intensity, whereas NW and NWAs exhibit a negative linear relationship. The results could provide some foundational guidance to applications of anatase TiO2 in optoelectronic devices or other aspects.