  We find exact mappings for a class of limit cycle systems with noise onto
quasi-symplectic dynamics, including a van der Pol type oscillator. A dual role
potential function is obtained as a component of the quasi-symplectic dynamics.
Based on a stochastic interpretation different from the traditional Ito's and
Stratonovich's, we show the corresponding steady state distribution is the
familiar Boltzmann-Gibbs type for arbitrary noise strength. The result provides
a new angle for understanding processes without detailed balance and can be
verified by experiments.
