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In this paper we consider piecewise affine differential equations modelinggene networks. We work with arbitrary decay rates, and under a local hypothesisexpressed as an alignment condition of successive focal points. The interactiongraph of the system may be rather complex (multiple intricate loops of anysign, multiple thresholds...). Our main result is an alternative theoremshowing that, if a sequence of region is periodically visited by trajectories,then under our hypotheses, there exists either a unique stable periodicsolution, or the origin attracts all trajectories in this sequence of regions.This result extends greatly our previous work on a single negative feedbackloop. We give several examples and simulations illustrating different cases.

Limit cycles in piecewise-affine gene network models with multiple interaction loops