  In this paper, we establish the Stroock-Varadhan type support theorems for
stochastic differential equations (SDEs) under Lyapunov conditions, which
significantly improve the existing results in the literature where the
coefficients of the SDEs are required to be globally Lipschitz and of linear
growth. Our conditions are very mild to include many important models, e.g.
Threshold Ornstein-Ulenbeck process, Stochastic SIR model, Stochastic
Lotka-Volterra systems, Stochastic Duffing-van der Pol oscillator model, which
have polynomial the coefficients. To obtain the support theorem, we prove a new
Wong-Zakai approximation problem, which is of independent interest.
