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Threshold-linear networks consist of simple units interacting in the presence of a threshold nonlinearity. Competitive threshold-linear networks have long been known to exhibit multistability, where the activity of the network settles into one of potentially many steady states. In this work, we find conditions that guarantee the absence of steady states, while maintaining bounded activity. These conditions lead us to define a combinatorial family of competitive threshold-linear networks, parametrized by a simple directed graph. By exploring this family, we discover that threshold-linear networks are capable of displaying a surprisingly rich variety of nonlinear dynamics, including limit cycles, quasi-periodic attractors, and chaos. In particular, several types of nonlinear behaviors can co-exist in the same network. Our mathematical results also enable us to engineer networks with multiple dynamic patterns. Taken together, these theoretical and computational findings suggest that threshold-linear networks may be a valuable tool for understanding the relationship between network connectivity and emergent dynamics.