Stable torus and its bifurcation phenomena in a simple three-dimensional autonomous circuit

It is well known that a stable torus is observed as a result after the system meets the super-critical Neimark–Sacker bifurcation for a limit cycle. Although tori are easily observed in two-dimensional and periodically forced dynamical systems, there is a few papers about stable tori in three-dimensional autonomous systems. Besides, as physical circuit implementations, such circuits contain very special active elements, or have difficulty in realizing. In this paper, we show a very simple circuit of three-dimensional autonomous system, an extended Bonhöffer–van der Pol (BVP) oscillator, which is demonstrating a stable torus. Firstly we explain a discovery of the torus in a computer simulation of the model equation. To implement it as a circuitry, we design a new nonlinear resistor. Although this contains an FET and an op-amp, it is simpler than any other nonlinear resistors proposed in previous papers. We confirm that this BVP oscillator can generate a stable torus in a real circuitry. We thoroughly investigate the bifurcation phenomena of various limit cycles and tori in this circuit, i.e., a super-critical Neimark–Sacker and tangent bifurcations of limit cycles are concretely obtained, furthermore, phase locking and chaos regions are clarified in a bifurcation diagram.