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Bubbling and On-Off Intermittency in Bailout Embeddings

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Bubbling and On-Off Intermittency in Bailout Embeddings PHYSICAL REVIEW E 68, 016217 ͑2003͒ Bubbling and on-off intermittency in bailout embeddings Julyan H. E. Cartwright,1,* Marcelo O. Magnasco,2,† Oreste Piro,3,‡ and Idan Tuval3,§ 1Laboratorio de Estudios Cristalogra´ficos, CSIC, E-18071 Granada, Spain 2Abdus Salam International Centre for Theoretical Physics, I-34014 Trieste, Italy 3Institut Mediterrani d’Estudis Avanc¸ats, CSIC–UIB, E-07071 Palma de Mallorca, Spain ͑Received 14 February 2003; published 21 July 2003͒ We establish and investigate the conceptual connection between the dynamics of the bailout embedding of a Hamiltonian system and the dynamical regimes associated with the occurrence of bubbling and blowout bifurcations. The roles of the invariant manifold and the dynamics restricted to it, required in bubbling and blowout bifurcating systems, are played in the bailout embedding by the embedded Hamiltonian dynamical system. The Hamiltonian nature of the dynamics is precisely the distinctive feature of this instance of a bubbling or blowout bifurcation. The detachment of the embedding trajectories from the original ones can thus be thought of as transient on-off intermittency, and noise-induced avoidance of some regions of the embedded phase space can be recognized as Hamiltonian bubbling. DOI: 10.1103/PhysRevE.68.016217 PACS number͑s͒: 05.45.Gg I. INTRODUCTION ture of the dynamics away from the invariant manifold. In the foregoing, we have supposed that both the re- Symmetry is a fundamental concept in science, both at the stricted and the unrestricted dynamics are dissipative. But we level of physical laws and when applied to specific applica- might envision a case in which the global dissipative dynam- tions. In dynamics, the presence of symmetries may imply ics have an invariant manifold on which the restricted dy- namics is conservative; described, for example, by a Hamil- the existence of lower-dimensional submanifolds of the full tonian flow or a volume-preserving ͑Liouvillian͒ map. The phase space, which are dynamically invariant, because sym- orbits on the invariant manifold then exhibit, instead of at- metric states must evolve into symmetric states. The motion tractors, the typical Hamiltonian phase-space structure char- restricted to these invariant manifolds is often chaotic. This acterized by the coexistence of chaotic regions and occurs, for instance, in synchronized chaotic oscillators and Kolmogorov-Arnold-Moser ͑KAM͒ tori. However, since the in extended systems with spatial symmetry. The existence of full dynamics are dissipative, a part or even the whole family chaotic attractors in the restricted dynamics can have inter- of the restricted Hamiltonian orbits may be global attractors. esting consequences for the behavior of the full system ͓1,2͔. Furthermore, a parameter can now control the transversal An interesting question is whether the attractors on invari- stability of the invariant manifold in the same way as in ant manifolds are also attractors of the unrestricted dynam- blowout bifurcations; but now governing which part of the ics. Consider the case in which a parameter controls the dy- Hamiltonian dynamics is globally attracting, and which is namics transversal to the invariant manifold while leaving not. unaffected the dynamics restricted to the invariant manifold The aim of this work is to show that these dynamics are ͓2͔. Below a critical value of this parameter, the manifold is observed in a physical system of finite-sized particles driven transversally stable and any attractor within it is also a global by an incompressible fluid flow ͓7,8͔, and have been har- attractor. When the parameter exceeds the critical value, nessed to work in the different context of the study of chaotic some invariant set embedded in the attractor becomes trans- dynamical systems in the technique of bailout embedding, ͓ ͔ versally unstable. Although most trajectories initially close to first introduced by us in Ref. 9 . In Sec. II we recall the the attractor in the invariant manifold may still remain close necessary background on bailout embeddings. In Sec. III we consider bubbling and blowout bifurcations and in Sec. IV to it, there is now a small set of points, in any neighborhood we relate bailout embeddings with blowout bifurcations. of the attractor, which diverges from the invariant manifold. Section V follows with a discussion of the addition of noise Increasing the parameter still further, the full attractor be- to a bailout embedding, and the occurrence of bubbling. We comes transversally unstable. These two scenarios, named conclude in Sec. VI. bubbling and blowout bifurcations, may lead either to the so-called riddled basins ͓3͔ or to a strong temporal bursting II. BAILOUT EMBEDDINGS termed on-off intermittency ͓1,4–6͔, depending on the na- Given an arbitrary dynamical system x˙ ϭ f (x), we define a bailout embedding as one of the form * ϳ URL: http://lec.ugr.es/ julyan; Electronic address: d [email protected] ͓vϪ f ͑x͔͒ϭϪk͑x͓͒vϪ f ͑x͔͒, †URL: http://www.ictp.trieste.it/ϳmagnasco; Electronic address: dt [email protected] dx ͑1͒ ‡ ϳ ϭv. URL: http://www.imedea.uib.es/ piro; Electronic address: dt [email protected] § URL: http://www.imedea.uib.es/ϳidan; Electronic address: This is a dynamical system defined in the space of variables [email protected] x and v. Notice, though, that the original dynamics, satisfy- 1063-651X/2003/68͑1͒/016217͑7͒/$20.0068 016217-1 ©2003 The American Physical Society CARTWRIGHT et al. PHYSICAL REVIEW E 68, 016217 ͑2003͒ ing x˙ ϭ f (x)ϭv, also solves Eqs. ͑1͒. This implies that v k x ϩ ϭx ϩ sin͑2␲y ͒ϵ f ͑x ,y ͒ ͑mod1͒, ϭ f (x) defines an invariant submanifold for the bailout em- n 1 n 2␲ n x n n bedding. However, we can design an embedding to satisfy ͑3͒ the condition that k(x)Ͻ0 on a set of unwanted orbits of the y ϩ ϭy ϩx ϩ ϵ f ͑x ,y ͒ ͑mod1͒. original dynamics while k(x)Ͼ0 on the others. We thus n 1 n n 1 y n n force the unwanted orbits to be unstable in the larger dynam- Parameter k, controlling the nonlinearity, takes the standard ics even though they are stable in the original system. In this map from the integrable limit at kϭ0 to a highly chaotic way, trajectories in the embedding tend to detach or bail out regime when kӷ1. At intermediate k this map shows a mix- from the unwanted set of the original system, bouncing ture of quasiperiodic trajectories on the KAM tori together around in the larger space until eventually reaching a stable with chaotic ones, depending on where we set the initial region with k(x)Ͼ0, where they may converge back onto conditions. As the value of k is increased, the region domi- trajectories of the original dynamical system. In other words, nated by chaotic trajectories pervades most of the phase by means of a bailout embedding we can create a larger space except for increasingly small islands of KAM quasi- version of the dynamics in which specific sets of orbits are periodicity. Notice that the only factor that decides whether removed from the asymptotic set, while preserving the dy- we are in one of these islands or in the surrounding chaotic namics of another set of orbits, the wanted one, as attractors sea is the initial condition of the trajectory. Therefore, in of the enlarged dynamical system. order to locate one of these islands, it would be necessary to There is a remarkable example in nature where a bailout make the initial condition scan the whole phase space while embedding describes the dynamics of a physical system. The watching for quasiperiodicity in the resulting dynamics. In a motion of finite-sized neutrally buoyant spherical particles bailout embedding, on the other hand, the KAM trajectories suspended in an incompressible fluid flow embeds the La- of the embedded system can be transformed into effective ˙ϭ grangian dynamics of a perfect tracer satisfying x u(x), global attractors in the extended phase space. The search for where u(x) is the Eulerian velocity field describing the flow KAM islands then becomes a matter of simply iterating the ͓ ͔ ͓ ͔ 10 . In the full model of the drag on a sphere 7 , this is a bailout map forwards until its trajectories converge to those highly nontrivial embedding. For example, the contribution of the embedded one. to the drag, known as a Basset-Boussinesq force, contains The bailout embedding of the standard map is obtained by the difference between the fluid and particle accelerations simply replacing f in Eq. ͑2͒ with the definitions from Eq. integrated over the whole trajectory. However, in the limit ͑3͒. Accordingly, K(x) in Eq. ͑2͒ becomes where the particles are sufficiently small and the Basset- Boussinesq term can be disregarded, the dynamics reduce to 1 k cos͑2␲y ͒ a bailout embedding of the form given in Eq. ͑1͒ with u(x) ͑ ͒ϭ Ϫ␥ͩ n ͪ ͑ ͒ K x e ␲ ͒ϩ . 4 ϭϪ ␥ϩ ␥ 1 k cos͑2 y n 1 playing the role of f and k(x) ( “ f ), where turns out to be the Stokes coefficient ͓8͔. In this case, it appears that the embedding ͑the finite-sized particle dynamics͒ de- These replacements then lead to the coupled second-order taches from the original dynamics ͑the passive scalar one͒ iterative system near saddle points and other unstable regions of the basic ϭ ϩ ͒ volume-preserving flow, converging back only in the KAM xnϩ1 un f x͑xn ,y n , islands that act now as attractors of the embedding. This ϭ ϩ ͒ result generalizes immediately to any Hamiltonian or y nϩ1 vn f y͑xn ,y n , divergence-free flow using a similar form of k(x), and can ͑5͒ be used as a method to target small KAM islands in these Ϫ␥ u ϩ ϭe u ϩk cos͑2␲y ͒v , types of systems ͓9͔.
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