## The Bubbling Transition

### Shankar C. Venkataramani, Brian R. Hunt, and Edward Ott

#### Phys. Rev. E 54 (1996), 1346-1360.

Recently, physically important examples of dynamical systems that have a chaotic attractor embedded in an invariant submanifold have been pointed out, and the unusual dynamical consequences of this situation have been studied. As a parameter $\epsilon$ of the system is increased, a periodic orbit embedded in the attractor on the invariant manifold can become unstable for perturbations transverse to the invariant manifold. This bifurcation is called the bubbling transition, and it can lead to the occurrence of a recently discovered, new kind of basin of attraction, called a riddled basin. In this paper we study the effects of noise and asymmetry on the bubbling transition. We find that, in the presence of noise or asymmetry, the attractor is replaced either by a chaotic transient or an intermittently bursting time evolution, and we derive scaling relations, valid near the bubbling transition, for the characteristic time (i.e., the average chaotic transient lifetime or the average interburst time interval) as a function of the strength of the asymmetry and the variance of the additive noise. We also present numerical evidence for the predicted scalings.

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