Effects of noise on the phase dynamics of nonlinear oscillators

Various properties of human rhythmic movements have been successfully modeled using nonlinear oscillators. However, despite some extensions towards stochastical differential equations, these models do not comprise different statistical features that can be explained by nondynamical statistics. For instance, one observes certain lag one serial correlation functions for consecutive periods during periodic motion. This work aims at an extension of dynamical descriptions in terms of stochastically forced nonlinear oscillators such as $\ddot{\xi }+{\omega }_0^2\xi =n(\xi ,\dot{\xi })+q(\xi ,\dot{\xi })\Psi \mathrm{}\left(t\right)\mathrm{}$, where the nonlinear function $n(\xi ,\dot{\xi })$ generates a limit cycle and $\Psi \mathrm{}\left(t\right)\mathrm{}$ denotes colored noise that is multiplied via $q(\xi ,\dot{\xi })$. Nonlinear self-excited systems have been frequently investigated, particularly emphasizing stability properties and amplitude evolution. Thus, one can focus on the effects of noise on the frequency or phase dynamics that can be analyzed by use of time-dependent Fokker-Planck equations. It can be shown that noise multiplied via polynoms of arbitrary finite order cannot generate the desired period correlation but predominantly results in phase diffusion. The system is extended in terms of forced oscillators in order to find a minimal model producing the required error correction.