Exact stationary solutions to Fokker–Planck–Kolmogorov equation for oscillators using a new splitting technique and a new class of stochastically equivalent systems

• We provide new classes of exact solutions to reduced FPK equation for oscillators.
• A new splitting technique is established.
• Obtained exact pdfs exhibit a rich structure (bistability, limit cycle, hybrid).
• New classes of stochastic equivalent systems have been reported.

Finding exact stationary probabilistic solutions to nonlinear oscillators under additive and/or multiplicative white-noise excitation has been an interesting and difficult problem arising in theoretical and applied stochastic dynamics. As a rule, such results are obtained by solving the corresponding stationary or reduced, as it is commonly called, Fokker–Plank–Kolmogorov (rFPK) equation. In the present work, the rFPK equation is solved by using a novel splitting technique, under which it is replaced by a Pfaffian system, which is always solvable, and a scalar equation, which is solved under some restrictions, relating the coefficients of the oscillator with the intensities of excitation noises. Using this technique, stationary solutions are obtained to two classes of stochastic oscillators, generalizing the results of Wang and Zhang [9] and Dimentberg [19]. Finally, the present splitting method is applied to establish a new equivalence class between stochastic oscillators and stochastic differential equations.