A three-dimensional non-autonomous conservative chaotic system based on cross-phase-coupled oscillators: Theoretical analysis and electronic realization

This study concerns a conservative, non-autonomous chaotic system composed of three ordinary differential equations, each of which solely comprises a sine function of time and cross-coupling terms. While devoid of an attractive invariant set, the system is not Hamiltonian. It was recently introduced as an abstraction of a chaotic oscillator circuit based on CMOS inverter rings. Moreover, it has aspects of structural resemblance with particular configurations of the Kuramoto model, as well as coupled Josephson junctions, and can be intended to represent a high-order interaction. Numerical simulations, based on reformulations as an equivalent autonomous system, reveal two distinct chaotic regimes under the assumption of homogeneous or heterogeneous harmonic excitation. Under homogeneous excitation, the trajectories remain bounded and are organized into a chaotic sea interspersed with quasi-periodic islands. In the heterogeneous case, labyrinth chaos is generated, characterized by aperiodic wandering akin to a perturbed Wiener process and reminiscent of the behavior of the Thomas system. The trajectory spread underlying the transition between bounded and unbounded behaviors is controlled by the cross-coupling strength, and, in the presence of growing dissipation, the orbits are increasingly constrained. A realization based on transfer functions and voltage-controlled oscillators is also presented, and was simulated both at block level and in terms of a realistic electronic circuit. A minimalist experimental implementation using commonplace 555-type timer integrated circuits was constructed and found to reproduce salient aspects of the dynamics. Besides the distinguishing structural features of the equation system and its dynamics, the resulting circuit appears well suited for future applications in chaos engineering and the construction of networks of coupled units.