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. For an example application to turbulent flow, see Szezech et al. (2011).
Contents |
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).
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. Both blowout and bubbling transitions occur typically at low order stable periodic orbits; see for example work of Hunt and Ott (1996).
Note that the invariant manifold must lose Normal Hyperbolicity as a precursor to a bubbling bifurcation. Because of this, bubbling bifurcations are not expected in systems other than those where the invariant manifold is forced by a symmetry or other constraint on the dynamics.
An Example from Synchronization
Bubbling arises in many situations; for example see the references below, but to illustrate the phenomenon with a simple example, 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 Lyapunov 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.
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 bifurcation (also called the soft bubbling transition). Zimin et al. (2003) have examined a number of bifurcation scenarios leading to bubbling transitions.
Note that this is a noise-induced transverse instability, and not the same as the 'bubbling' of Bier and Bountis (1984) who discuss truncated and reversed period doubling cascades.
References
- Ashwin P., Buescu J., Stewart I. (1994) Bubbling of attractors and synchronization of chaotic oscillators. Physics Letters A, 193:126-139
- Bier M., Bountis, T.C. (1984) Remerging Feigenbaum trees in dynamical systems, Physics Letters A 104:239-244
- Hunt B., Ott E. (1996) Optimal periodic orbits of chaotic systems,Phys. Rev. Lett. 76:2254-2257
- Szezech J.D., Lopes S.R., Caldas I.L., Viana R.L. (2011) Blowout bifurcation and spatial mode excitation in the bubbling transition to turbulence, Physica A 390:365-373* 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
- Zimin A., Hunt B., Ott E. (2003) Bifurcation scenarios for the bubbling transition, Phys. Rev. E 67:016204
Internal references
- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Edward Ott (2006) Basin of attraction. Scholarpedia, 1(8):1701.
- Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
- Edward Ott (2006) Crises. Scholarpedia, 1(10):1700.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Christophe Letellier and Otto E. Rossler (2006) Rossler attractor. Scholarpedia, 1(10):1721.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
External links
See Also
Basin of Attraction Blowout Bifurcation, Chaos, Intermittency, Chaotic Oscillators, Crises, Riddled Basin of Attraction, Normal Hyperbolicity
