Taming a Chaotic Dripping Faucet via a Global Bifurcation

We carried out a fluid dynamical simulation for a forced dripping faucet system using a new algorithm that was recently developed. The simulation shows that periodic external forcing induces transitions from chaotic to periodic motion and vice versa. We further constructed an improved mass-sprig model for the same system on the basis of data obtained from the fluid dynamical computations. A detailed analysis of this simple model demonstrated that the periodic motion is realized after a homoclinic bifurcation, although a stable periodic orbit is generated via a Hopf bifurcation which occurs just after a saddle node one. Chaos occurs widely in engineering and natural systems. In the past few years practical implementations of controlling chaos have been studied with great interest [1-6]. Such control schemes can be divided broadly into two categories, feedback control and nonfeedback control. The pioneering work on controlling chaos by Ott, Grebogi, and Yorke (OGY method)[2] is a representative example of feedback control systems. The OGY method aims to stabilize one of many unstable periodic orbits embedded in the chaotic attractor through small time-dependent perturbations in an accessible system parameter. Although the OGY method is very general, it is difficult for some high-speed systems to implement the control procedure. In the nonfeedback control systems, on the other hand, the applied perturbation is independent of the system’s state. So far, controlling chaos by applying a suitable weak periodic perturbation, which is sometimes called taming chaos [3], has been studied in many chaotic dynamical systems ([4, 5] and references therein). Recently, analyzing a constrained system in which a one-dimensional Poincaré map is derived, Tamura et al. [4] showed that taming chaos occurs by a saddle node bifurcation. However, many issues in taming chaos are still not well understood, such as phase effect [6], phase diagram structures [5], and bifurcation structures for higher-dimensional dynamical systems [5]. ∗E-mail: k [email protected]