On a coupled relaxation oscillator

An analytic investigation is presented for a piecewise-linear version of a two-coupled relaxation oscillator. The circuit dynamics are described by a four-dimensional piecewise-linear differential equation containing two small parameters (ε₁, ε₂). In the case of (ε₁, ε₂ )→0, the phase space of the system degenerates into four overlapping halfplanes connected by a transitional condition. Then the Poincare return map is derived rigorously as a one-dimensional homeomorphism of the circle. The mapping is shown to account for the fact that the actual circuit exhibits various complicated synchronous phenomena and asynchronous phenomena