In this paper, we examine oscillatory modes generated by the Hopf bifurcations of equilibrium points except for the origin in a system of coupled four oscillators. (The bifurcation analyses of the origin for many coupled oscillators were already done.) The Hopf bifurcations of the equilibrium points with strong symmetrical property and the generated oscillatory modes are classified. We observe four-phase, in-phase and a pair of anti-phase synchronized states. Even in a system of four coupled oscillators, we discover the existence of a stable three-phase oscillation. By the numerical bifurcation analysis of generated periodic oscillations we find out successive period-doubling bifurcations as the route to chaos and show some of them are symmetry-breaking bifurcations. As a result of the symmetry-breaking period-doubling bifurcations, a periodic solution with complete synchronization becomes a chaotic solution with phase synchronization.

We developed the computer calligraphy, that is, a computer formation of brush-written Kanji characters using calligraphic knowledge. The style of brush handwriting depends mainly on the way of using a writing brush. Brush writing skills include the direction of brush at the beginning, curvature and turning the brush, the brush-up at the termination point in a stroke. We make up the calligraphic knowledge base according to the above mentioned brush writing skills. For simulating real brush movement, we represent the brush contact form that is the brush shape on the writing plane as a brush-touch. The system can control the size and direction of this brush-touch during the brush simulation. The system simulates the real brush writing to move the brush-touch along the skeleton letter shape in the standard database. We get the brush written Kanji from the locus of the brush-touch movement. We can extend this system to the new on line training system for brush writing using the simulation of brushtouch movement modified by the pressure, speed and rotation of the writing brush, and the skeleton letters written by a learner from the tablet. This system is also useful for students learning how to write Japanese letters beautifully with brush.