We consider a system of coupled two oscillators with external force. At first we introduce the symmetrical property of the system. When the external force is not applied, the two oscillators are synchronized at the opposite phase. We obtain a bifurcation diagram of periodic solutions in the coupled system when the single oscillator has a stable anti-phase solution. We find that the synchronized oscillations eventually become in-phase when the amplitude of the external force is increased.

This paper clarifies the bifurcation structure of the chaotic attractor in an interrupted circuit with switching delay from theoretical and experimental view points. First, we introduce the circuit model and its dynamics. Next, we define the return map in order to investigate the bifurcation structure of the chaotic attractor. Finally, we discuss the dynamical effect of switching delay in the existence region of the chaotic attractor compared with that of a circuit with ideal switching.

In this research, we propose an effective stability analysis method to impacting systems with periodically moving borders (periodic borders). First, we describe an n-dimensional impacting system with periodic borders. Subsequently, we present an algorithm based on a stability analysis method using the monodromy matrix for calculating stability of the waveform. This approach requires the state-transition matrix be related to the impact phenomenon, which is known as the saltation matrix. In an earlier study, the expression for the saltation matrix was derived assuming a static border (fixed border). In this research, we derive an expression for the saltation matrix for a periodic border. We confirm the performance of the proposed method, which is also applicable to systems with fixed borders, by applying it to an impacting system with a periodic border. Using this approach, we analyze the bifurcation of an impacting system with a periodic border by computing the evolution of the stable and unstable periodic waveform. We demonstrate a discontinuous change of the periodic points, which occurs when a periodic point collides with a border, in the one-parameter bifurcation diagram.

In this article, we propose a square wave generator whose switching threshold values are switched by external inputs. This circuit is designed to simulate the synchronized luminescence of coupled fireflies. We investigate the behavior of the solutions in two coupled oscillators. The dynamics are demonstrated by a linear autonomous equation piecewisely, therefore, a one-dimensional return map is derived. We also prove the existence of stable in-phase synchronization in the coupled oscillator by using the return map, and we show the existence of regions of periodic solutions within a parameter space. Some theoretical results are confirmed by laboratory measurements.