Chaos from a time-delayed Chua´s circuit

By replacing the parallel LC “resonator” in Chua´s circuit by a lossless transmission line that is terminated by a short circuit, we obtain a “time-delayed Chua´s circuit”, whose time evolution is described by a pair of linear partial differential equations with a nonlinear boundary condition. If we neglect the capacitance across the Chua´s diode, which is described by a nonsymmetric piecewise-linear V_R-I_R characteristic, the resulting idealized “time-delayed” Chua´s circuit is described exactly by a scalar nonlinear difference equation with continuous time, which makes it possible to characterize its associated nonlinear dynamics and spatial chaotic phenomena. From a mathematical view point, circuits described by ordinary differential equations can generate only temporal chaos, whereas the time-delayed Chua´s circuit can generate spatial temporal chaos. Except for stepwise periodic oscillations, the typical solutions of the idealized time-delayed Chua´s circuit consist of either weak turbulence or strong turbulence, which are examples of “ideal” (or “dry”) turbulence. In both cases, we can observe infinite processes of spatial-temporal coherent structure formations. Under weak turbulence, the graphs of the solution tend to limit sets that are fractals with a Hausdorff dimension between 1 and 2 and is therefore larger than the topological dimension (of sets). Under strong turbulence, the “limit” oscillations are oscillations whose amplitudes are random functions. This means that the attractor of the idealized time-delayed Chua´s circuit already contains random functions, and spatial self-stochasticity phenomenon can be observed