EXPLOSION OF STRANGE ATTRACTORS EXHIBITED BY DUFFING'S EQUATION

Various fascinating phenomena occur in nonlinear systems.' They are discussed generally by making use of steady solutions of :I diRerentia1 equation. The equation is usually derived from a real system by neglecting small uncertain factors-hence, it is deterministic. I f a single solution of the equation is asymptotically stable and its basin is large compared with random noise. the corresponding phenomenon turns out to be deterministic. But if a bundle of solutions containing infinitely many unstable periodic solutions is asymptotically orbitally stable. a chaotic phenomenon appears, which results from the small uncertain lactors in the real system. That is. the representative point of the physical state wanders chaoticnllq i n the bundle of solutions. Because of this characteristic. we have called the phenomenon a "chaotically transitional process." '* We have long been studying this subject in connection with forced oscillatory phenomena in nonlinear electrical and electronic circuits. As analytical solutions of differential equations cannot be expected in these problems, we have been relying on analog and digital computers. Thus, both the global structure of solutions and the long-term movement of a representative point have been examined. So the question arises, Are computer solutions valid'? However, we have not entered into details of this problem, though we have found consistency between the analog and digital results. Therefore. our results may lack mathematical rigor; nevertheless, they will have an important influence on many researchers in various fields. This paper also describes the chaotically transitional processes exhibited by Duffing's equation. Special attention is directed toward the transition of the processes and the explosion of strange attractors is clarified.