Systems with a complex dynamics like glasses or models of biological evolution are often pictured in terms of a complex landscape, with a large number of possible collective states. We show on the example of a stochastic spin model with nonreciprocal and heterogeneous interactions how the complex landscape picture can be generalized far from equilibrium, where collective states may become time-dependent and exhibit, e.g., spontaneous oscillations, often hidden by the presence of disorder. We identify relevant observables, like the density of entropy production rate, to unveil the spontaneous collective time dependence, and we determine a configurational entropy which counts the number of oscillating collective states when this number grows exponentially with system size.

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• Complex systems
• Entropy production
• Disordered systems