Exact Stationary Solutions to a Class of Non-linear Stochastic Oscillators. Establishing New Benchmark Cases for Testing Numerical Solution Schemes

The development of numerical solution schemes for stochastic oscillators under additive and/or parametric white noise excitation also calls for the determination of benchmark cases, i.e. oscillators with known characteristics and response probability density functions (pdfs). In the present paper, the stationary response pdfs for a new class of oscillators generalizing the results of Wang & Zhang (Nonlinearity 13(2000), pp. 907-920) are obtained by solving the corresponding reduced Fokker-Planck-Kolmogorov (rFPK) equation using a new splitting technique. To the best of our knowledge, this class of oscillators has not been presented before. The obtained response stationary pdfs exhibit complicated behavior, including bistabilities, limit-cycles, oscillations, and hybrid (combination of the previous three) patterns, making the studied oscillators an interesting class of benchmark cases.