Bubbling transition
| Peter Ashwin (2006), Scholarpedia, 1(8):1725. | doi:10.4249/scholarpedia.1725 | revision #193783 [link to/cite this article] |
The bubbling transition is a bifurcation in certain nonlinear dynamical systems where a change in parameter results in a qualitative change in the way the attractor responds to noise and/or other dynamical perturbations. Although there is little change to the structure of a chaotic attractor itself, its basin of attraction changes. Before a bubbling transition takes place there is a chaotic attractor that is asymptotically stable. After the transition the attractor is only an attractor in the weaker sense of being a Milnor (measure) attractor; see Ashwin et al. (1994) and Venkataramani et al. (1996ab) for a description and more details. The qualitative manifestation of the bubbling transition in a physical system is the emergence of intermittent bursts of chaotic trajectories away from the previously constrained attractor.
<review> You need a simple phase-space cartoon here illustrating the bifurcation, possibly showing one unstable orbit corresponding to the Lyapunov exponent crossing zero. Even better, put an animation here. </review>
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Bubbling from Invariant Manifolds
A bubbling transition gives a change in the way the system responds to small perturbations as follows:
- Before the transition, the dynamics remains in a small neighbourhood of the attractor.
- After the transition, the dynamics can make finite size excursions away from the attractor, even if the perturbation is of an arbitrarily small amplitude. These excursions away from the attractor are intermittent and short-lived; hence the name `bubbling'.
Such transitions is not generic except in cases where the dynamics is constrained to have an invariant manifold and these are often caused by the presence of symmetries in the system. Coupled map or coupled oscillator systems are typical examples where symmetry might play a prominent role in the dynamics. Before the transition, the dynamics in directions transverse to the invariant manifold are stable. However, as the system goes through the bubbling transition, an invariant subset (such as an unstable periodic orbit) looses stability in one of the transverse directions. This transversely unstable invariant subset and its pre-images result in a collection of tonguelike channels punctuating on the chaotic attractor (see illustration).File:Bubbling.jpg
With small perturbations, these structures provide an escape route for trajectories embedded within the chaotic attractor. The observed effect is called bubbling. With more and more invariant subsets within the chaotic attractor becoming transversely unstable, it would come to a point when the chaotic attractor itself becomes transverse unstable and this results in a Blowout Bifurcation.
<review>
"per-images" in caption is a bit unclear
I think there are several results on the different scalings in this transition. Could these be mentioned and briefly explained. In particular, what are the universal features of the bubbling transition
It is implied that the blowout bifurcation first happens at the unstable periodic orbits. This should be justified by a reference to several papers by Hunt.
Reference should be made to normal hyperbolicity as well. Note that an article on the subject will be written by Neil Fenichel. Please discuss the connection and provide a link to that article.
</review>
An Example from Synchronization
Consider a system of two coupled systems of ODEs of the form \[ \begin{matrix} \dot{x}_1 = f(x_1) + K(x_2-x_1) \\ \dot{x}_2 = f(x_2) + K(x_1-x_2) \end{matrix} \] for \(x_i\in\R^n\) and \(K\) a scalar. This system has an invariant synchronized subspace \[ N=\{(x,x)~:~x\in \R^n\} \] on which the dynamics is governed by the single equation \(\dot{x}=f(x)\). Suppose typical initial conditions for this are asymptotic to a chaotic attractor \(A\) with a positive Liapunov exponent for typical trajectories. We can examine the linearized transverse stability by setting \(v=x_2-x_1\). The transverse Liapunov exponents are then the rates \(\lambda_{\perp}\) of exponential growth of solutions \(v(t)\) of the variational equation \( \dot{v}=Df(x(t))v-2Kv. \) These depend on the trajectory \(x(t)\). One obtains bubbling when typical \(x(t)\) have all transverse Liapunov exponents negative; \(\lambda_{\perp}<0\) while for some exceptional trajectories \(x(t)\) within the attractor (usually unstable periodic orbits) we have \(\lambda_{\perp}>0\). The bubbling transition is the first point at which one can find a non-negative transverse Liapunov exponent for an invariant subset (such as an UPO) within the attractor. For the two coupled chaotic oscillators, \(K\) sufficiently positive will cause all transverse Liapunov exponents to become negative, meaning there is a bubbling transition on decreasing \(K\) through some critical value.
Example: Coupled Rossler Oscillators
Suppose we have two linearly coupled Rössler systems
\[
\begin{matrix}
\dot{x}_1 & = & -y_1-z_1+K(x_2-x_1), \\
\dot{y}_1 & = & x_1+ay_1+K(y_2-y_1), \\
\dot{z}_1 & = & b+z_1(x_1-c)+K(z_2-z_1)
\end{matrix}
\]
with similar equtions for \(x_2,y_2,z_2\) on interchanging the subscripts. Using \(a=b=0.2\), \(c=6.6\)
and \(K=0.045\) in the absence of noise (left below) the difference \(dx=x_2-x_2\) as a function of \(t\) settles down to a synchronized state whereas for noise of amplitude \(10^{-6}\) (right below) there are
occasional large amplitude bursts, indicative of bubbling. Note that the system is chaotic in both cases, though this is not visible in the difference \(dx\). The bursts coincide with the trajectory remaining close to some unstable dynamics in the chaotic attractor for long enough.
Increasing \(K\) a small amount beyond this value gives a bubbling transition beyond which low noise gives only a low amplitude response.
<review> One is left with the impression that bubbling only appears in the case of diffusively coupled identical chaotic systems. Is this the only place where it has been studied. References and a brief explanation of other examples would suffice. </review>
Types of Bubbling Transition
The bubbling transition is associated with a bifurcation of some unstable dynamics within the invariant manifold \(N\); for example an unstable periodic orbit within \(N\) typically transversely bifurcates to create new invariant sets near \(N\). The response of the system to noise depends on the nature of this bifurcation. If the bifurcation is subcritical the attractor will start to have a riddled basin of attraction meaning that small noisy perturbations will eventually drive the system away from \(N\); this is called a riddling bifurcation (also called the hard bubbling transition). If the bifurcation is supercritical then addition of noise results in deviations away from \(N\) that always decay back to \(N\); this is the usual bubbling bifucation (also called the soft bubbling transition).<review>This section can be expanded a bit more with more details.</review>
References
- Ashwin P., Buescu J., Stewart I. (1994) Bubbling of attractors and synchronization of chaotic oscillators. Physics Letters A, 193:126-139
- Venkataramani S.C., Hunt B.R., Ott E., Gauthier D.J., Bienfang J.S. (1996a) Transitions to bubbling of chaotic systems. Physical Review Letters 77:5361-5364
- Venkataramani S.C., Hunt B.R., Ott E. (1996b) Bubbling transition. Physical Review E 54:1346-1360
See Also
Basin of Attraction Blowout Bifurcation, Chaos, Intermittency, Chaotic Oscillators, Crises, Riddled Basin of Attraction, Normal Hyperbolicity
