The double scroll bifurcations

Abstract This paper describes the dynamics of the simplest physical system known to date whose chaotic dynamics and rich bifurcation phenomena have been observed not only in the laboratory, but also reconfirmed by extensive computer simulation of its associated mathematical model: a 3rd‐order autonomous ordinary differential equation. The physical system is a 5‐element electrical circuit whose only non‐linearity is a non‐linear resistor characterized by a 3‐segment piecewise‐linear voltage‐current characteristic. Despite the simplicity of the circuit, however, it is imbued with an extremely rich variety of bifurcation phenomena. By changing the capacitance values, many phenomena, including Hopf bifurcation, period‐doubling cascades, Rössler's spiral‐type and screw‐type attractors, 4 periodic windows, ‘double‐scroll’ attractor, 1 boundary crisis, 5 Shilnikov‐type phenomenon, 6 etc. have been observed experimentally and confirmed by computer simulation. Other attractors and periodic windows have also been observed by varying the conductance values. In addition, Rössler's spiral‐type and screw‐type attractors have been observed from the same circuit, where the non‐linear resistor has only one break point, i.e. it is described by a 2‐ segment piecewise‐linear v ‐ i characteristic. This means that extremely complicated non‐periodic (chaotic) waveforms can arise in the simplest third‐order uncoupled electrical circuit in which all elements except one (a resistor ) are linear and passive, and in which the constitutive relation of the non‐linear resistor is made of the simplest conceivable non‐linearity, namely 2 straight‐line segments.