IDEAL TURBULENCE IN AN IDEALIZED TIME-DELAYED CHUA’S CIRCUIT

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Abstract
By replacing the parallel LC “resonator” in Chua’s circuit by a lossless transmission line, 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, described by a nonsymmetric piecewiselinear vR–iR 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 viewpoint, circuits described by ordinary differential equations can generate only temporal chaos, while the time-delayed Chua’s circuit can generate spatiotem poral 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 spatiotemporal coherent structure formations. Under weak turbulence, the graphs of the solution tend to limit sets which are fractals with a Hausdorff dimension between 1 and 3, 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.

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