MEMRISTORS on Edge of Chaos: Quo Vadis?

Never in the annals of science and engineering has there been a phenomenon so ubiquitous, a paradigm so universal, or a discipline so multi-disciplinary as that of chaos. Yet chaos represents only the tip of an awesome iceberg, for beneath it lies a much finer structure of immense complexity, a geometric labyrinth of endless convolutions, and a surreal landscape of enchanting beauty. The bedrock which anchors these local and global bifurcation terrains is the omnipresent nonlinearity that was once randomly linearized by the engineers and applied scientists of yore, thereby forfeiting their only chance to grapple with reality. We now know that bifurcation and chaos are rather mundane phenomena that could occur in many natural and man-made systems where nonlinearity is present. The robustness of such counterintuitive, if not bizarre, phenomena has not only overthrown the age-old dogma of a deterministic world but also unloosed the underpinning of many long-held diagnostic and design methodologies from engineering and the applied sciences, which were based on linear approximations. Even the normally benign subcritical Hopf bifurcation phenomena could lead to catastrophic consequences if a system imbued with such dynamics were inadvertently kicked out of its unsuspected failure boundary. On the other hand, recent laboratory observations suggesting that chaotic regimes may in fact represent the ground state of the central nervous system point to the intriguing possibility of exploiting and controlling chaos for future scientific and engineering applications. Be that as it may, the subject of bifurcation and chaos has already established itself as a fundamental discipline cutting across virtually all scientific fields. This discipline is truly multi-disciplinary and unique because it shares no territorial precedents. For what discipline could one conjure up an analogue situation where an electronic engineer could engage in an in-depth dialogue with a biologist on the dynamics of irregular heartbeat rhythms, with a mathematician on the dimension of the Cantor set, with an economist on the chaotic patterns of the stock market, with a psychologist on the brain wave series registered during dreams, with a social scientist on the catastrophe of prison riots, or even with a musician on the technique of playing a clarinet to induce period doubling? In this 30th anniversary issue, we focus on two, among many, exciting and flourishing research areas, namely, Formula: see text Chua , 1971 , 2018a , 2018b ; Strukov et al. , 2008 ; Tour Williams , 2008 and Formula: see text Chua , 2024 . The fundamental theory which gives birth to the powerful “Formula: see text” is the “Formula: see text” Chua , 2005 , which can be interpreted as the complement of the “Formula: see text”. But the highlight of this 30th anniversary exposition is a colorful panorama for resolving the Hodgkin–Huxley blunder of identifying the sodium and potassium ion channels erroneously as time-dependent resistances Chua et al. , 2012b . The climax of the panorama is our definitive solution of the heretofore unresolved Turing Instability and the Smale Paradox Chua , 2024 .