An Invariant Manifold is a manifold embedded in a phase space with the property that it is invariant under the flow, i.e., orbits that start out in the Manifold remain in it. The phase space of many dynamical systems have embedded in them, invariant manifolds whose dimensions are smaller than the dimensions of the entire phase space. This situation occurs naturally in systems with symmetry, where the symmetric states form an invariant manifold. We investigate the bifurcations for such systems, in which the dynamics restricted to the invariant manifold has a chaotic attractor, as we vary a generic parameter p of the system.
Fig. 5 is a schematic representation of a dynamical system that has an embedded chaotic attractor.
The invariant manifold is attracting in the transverse direction if all the periodic orbits embedded in the chaotic attractor are attracting in the transverse direction. As the parameter p is varied, a periodic orbit can become unstable in the transverse direction, while the chaotic attractor is still transversely attracting on average. This bifurcation is called the bubbling transition and it leads to the formation of a riddled basin of attraction for the attractor in the invariant manifold. A basin of attraction is riddled if it has positive measure, but is nowhere dense, that is, there is a dense set of holes corresponding to the basins of other attractors.
In the presence of noise or a small asymmetry, the attractor is replaced by a chaotic transient or an intermittently bursting time evolution. We derive scaling relations for the characteristic time as a function of the asymmetry amplitude or the noise strength and test these predictions by numerical simulations.
As the parameter p is changed, the chaotic attractor can become transversely unstable on average. This transformation is called the blowout bifurcation and it can lead to an extreme form of intermittent bursting called on-off intermittency. We obtain results for the power spectrum of an on-off intermittent time series and the fractal dimension of its level set using a stochastic differential equation model, and test the results by numerical simulation.
