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Landscape is one of the key notions in literature on biological processes andphysics of complex systems with both deterministic and stochastic dynamics. Thelarge deviation theory (LDT) provides a possible mathematical basis for thescientists' intuition. In terms of Freidlin-Wentzell's LDT, we discussexplicitly two issues in singularly perturbed stationary diffusion processesarisen from nonlinear differential equations: (1) For a process whosecorresponding ordinary differential equation has a stable limit cycle, thestationary solution exhibits a clear separation of time scales: an exponentialterms and an algebraic prefactor. The large deviation rate function attains itsminimum zero on the entire stable limit cycle, while the leading term of theprefactor is inversely proportional to the velocity of the non-uniform periodicoscillation on the cycle. (2) For dynamics with multiple stable fixed pointsand saddles, there is in general a breakdown of detailed balance among thecorresponding attractors. Two landscapes, a local and a global, arise in LDT,and a Markov jumping process with cycle flux emerges in the low-noise limit. Alocal landscape is pertinent to the transition rates between neighboring stablefixed points; and the global landscape defines a nonequilibrium steady state.There would be nondifferentiable points in the latter for a stationary dynamicswith cycle flux. LDT serving as the mathematical foundation for emergentlandscapes deserves further investigations.

Landscapes of Non-gradient Dynamics Without Detailed Balance: Stable Limit Cycles and Multiple Attractors