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 stoch
astic interpretation different from the traditional Itos and Stratonovichs, 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.
