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Strongly correlated phases exhibit collective carrier dynamics that if properly harnessed can enable novel functionalities and applications. In this article, we investigate the phenomenon of electrical oscillations in a prototypical MIT system, vanadium dioxide (VO2). We show that the key to such oscillatory behaviour is the ability to induce and stabilize a non-hysteretic and spontaneously reversible phase transition using a negative feedback mechanism. Further, we investigate the synchronization and coupling dynamics of such VO2 based relaxation oscillators and show, via experiment and simulation, that this coupled oscillator system exhibits rich non-linear dynamics including charge oscillations that are synchronized in both frequency and phase. Our approach of harnessing a non-hysteretic reversible phase transition region is applicable to other correlated systems exhibiting metal-insulator transitions and can be a potential candidate for oscillator based non-Boolean computing.

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While Boolean logic has been the backbone of digital information processing, there exist classes of computationally hard problems wherein this paradigm is fundamentally inefficient. Vertex coloring of graphs, belonging to the class of combinatorial optimization, represents one such problem. It is well studied for its applications in data sciences, life sciences, social sciences and technology, and hence, motivates alternate, more efficient non-Boolean pathways towards its solution. Here we demonstrate a coupled relaxation oscillator based dynamical system that exploits insulator-metal transition in Vanadium Dioxide (VO2) to efficiently solve vertex coloring of graphs. Pairwise coupled VO2 oscillator circuits have been analyzed before for basic computing operations, but using complex networks of VO2 oscillators, or any other oscillators, for more complex tasks have been challenging in theory as well as in experiments. The proposed VO2 oscillator network harnesses the natural analogue between optimization problems and energy minimization processes in highly parallel, interconnected dynamical systems to approximate optimal coloring of graphs. We further indicate a fundamental connection between spectral properties of linear dynamical systems and spectral algorithms for graph coloring. Our work not only elucidates a physics-based computing approach but also presents tantalizing opportunities for building customized analog co-processors for solving hard problems efficiently.