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A relaxation oscillator incorporating nanoscale niobium dioxide memristors that exhibit both a current- and a temperature-controlled negative differential resistance produces chaotic dynamics that aid biomimetic computing.At present, machine learning systems use simplified neuron models that lack the rich nonlinear phenomena observed in biological systems, which display spatio-temporal cooperative dynamics. There is evidence that neurons operate in a regime called the edge of chaos1 that may be central to complexity, learning efficiency, adaptability and analogue (non-Boolean) computation in brains2,3,4,5,6,7. Neural networks have exhibited enhanced computational complexity when operated at the edge of chaos2, and networks of chaotic elements have been proposed for solving combinatorial or global optimization problems8. Thus, a source of controllable chaotic behaviour that can be incorporated into a neural-inspired circuit may be an essential component of future computational systems. Such chaotic elements have been simulated using elaborate transistor circuits that simulate known equations of chaos9,10,11,12, but an experimental realization of chaotic dynamics from a single scalable electronic device has been lacking5,6,13. Here we describe niobium dioxide (NbO2) Mott memristors each less than 100 nanometres across that exhibit both a nonlinear-transport-driven current-controlled negative differential resistance and a Mott-transition-driven temperature-controlled negative differential resistance. Mott materials have a temperature-dependent metal-insulator transition that acts as an electronic switch, which introduces a history-dependent resistance into the device. We incorporate these memristors into a relaxation oscillator14 and observe a tunable range of periodic and chaotic self-oscillations15. We show that the nonlinear current transport coupled with thermal fluctuations at the nanoscale generates chaotic oscillations. Such memristors could be useful in certain types of neural-inspired computation by introducing a pseudo-random signal that prevents global synchronization and could also assist in finding a global minimum during a constrained search. We specifically demonstrate that incorporating such memristors into the hardware of a Hopfield computing network can greatly improve the efficiency and accuracy of converging to a solution for computationally difficult problems.