Mechanisms for the hard bubbling transition in symmetrically coupled chaotic systems (original) (raw)
Related papers
Global effect of transverse bifurcations in coupled chaotic systems
JOURNAL-KOREAN PHYSICAL SOCIETY, 2003
We investigate the global effect of transverse bifurcations in symmetrically coupled onedimensional maps. A transition from strong to weak synchronization occurs via a first transverse bifurcation of a periodic saddle embedded in a synchronous chaotic attractor (SCA). For the case of a supercritical transverse bifurcation, a soft bubbling transition occurs. On the other hand, a subcritical transverse bifurcation leads to a hard transition. The global effect of such subcritical hard bifurcations are found to depend on whether they may or may not induce a "contact" between the SCA and its basin boundary. For the case of a "contact" bifurcation, an absorbing area, surrounding the SCA and acting as a bounded trapping vessel, disappears; hence, basin riddling occurs. However, for the case of a "non-contact" bifurcation, such an absorbing area is preserved; hence, hard bubbling takes place. Through a detailed numerical analysis, we give explicit examples for all kinds of transverse bifurcations leading to bubbling and riddling.
Transitions to Bubbling of Chaotic Systems
Physical Review Letters, 1996
Certain dynamical systems exhibit a phenomenon called bubbling, whereby small perturbations induce intermittent bursting. In this Letter we show that, as a parameter is varied through a critical value, the transition to bubbling can be "hard" (the bursts appear abruptly with large amplitude) or "soft" (the maximum burst amplitude increases continuously from zero), and that the presence or absence of symmetry in the unperturbed system has a fundamental effect on these transitions. These results are confirmed by numerical and physical experiments. [S0031-9007(96)01852-2] PACS numbers: 05.45. + b, 47.20.Ky,
Loss of Chaos Synchronization through the Sequence of Bifurcations of Saddle Periodic Orbits
Physical Review Letters, 1997
In the work we investigate the bifurcational mechanism of the loss of stability of the synchronous chaotic regime in coupled identical systems. We show that loss of synchronization is a result of the sequence of soft bifurcations of saddle periodic orbits which induce the bubbling and riddling transitions in the system. A bifurcation of a saddle periodic orbit embedded in the chaotic attractor determines the bubbling transition. The phenomenon of riddled basins occurs through a bifurcation of a periodic orbit located outside the symmetric subspace.
Symmetry-Conserving and Breaking Blow-Out Bifurcations in Coupled Chaotic Systems
2001
We consider blow-out bifurcations of synchronous chaotic attractors on invariant subspaces in coupled chaotic systems with symmetries. Through a blow-out bifurcation, the synchronous chaotic attractor becomes unstable with respect to perturbations transverse to the invariant subspace, and then a new asynchronous chaotic attractor may appear. However, the system symmetry may be preserved or violated when such a transition from synchronous to asynchronous chaotic motion occurs. Here we investigate the underlying mechanism for the symmetry preservation and violation. It is thus found that the shape of a minimal invariant absorbing area controlling the global dynamics and acting as a trapping bounded vessel determines whether the symmetry is conserved or not. For the case of a symmetric absorbing area, a symmetry-conserving blow-out bifurcation occurs while in the case of an asymmetric absorbing area, a symmetry-breaking blow-out bifurcation takes place.
Bubbling bifurcation: Loss of synchronization and shadowing breakdown in complex systems
Physica D: Nonlinear Phenomena, 2005
Complex dynamical systems with many degrees of freedom may exhibit a wealth of collective phenomena related to highdimensional chaos. This paper focuses on a lattice of coupled logistic maps to investigate the relationship between the loss of chaos synchronization and the onset of shadowing breakdown via unstable dimension variability in complex systems. In the neighborhood of the critical transition to strongly non-hyperbolic behavior, the system undergoes on-off intermittency with respect to the synchronization manifold. This has been confirmed by numerical diagnostics of synchronization and non-hyperbolic behavior, the latter using the statistical properties of finite-time Lyapunov exponents. (R. Viana). interrelated in a nontrivial manner; (ii) they can exhibit both ordered and random behaviors; and (iii) they display a hierarchy of structures over a wide range of lengths [1]. Spatially extended systems built from coupled chaotic maps or flows typically belong to the category of complex systems, for different parts of the lattice can exhibit different dynamics, say, regular and chaotic, forming structures where coherent and incoherent behavior coexist . There are many quantitative ways to characterize the complexity of a given system, more effectively being a mix of sundry 0167-2780/$ -see front matter
Mechanism for the riddling transition in coupled chaotic systems
Physical Review E, 2001
We investigate the loss of chaos synchronization in coupled chaotic systems without symmetry from the point of view of bifurcations of unstable periodic orbits embedded in the synchronous chaotic attractor ͑SCA͒. A mechanism for direct transition to global riddling through a transcritical contact bifurcation between a periodic saddle embedded in the SCA and a repeller on the boundary of its basin of attraction is thus found. Note that this bifurcation mechanism is different from that in coupled chaotic systems with symmetry. After such a riddling transition, the basin becomes globally riddled with a dense set of repelling tongues leading to divergent orbits. This riddled basin is also characterized by divergence and uncertainty exponents, and thus typical power-law scaling is found.
Journal of Mathematical Sciences, 2009
This paper introduces a new 2D piecewise smooth discrete-time chaotic mapping with rarely observed phenomenon – the occurrence of the same chaotic attractor via different and distinguishable routes to chaos: period doubling and border-collision bifurcations as typical futures. This phenomenon is justified by the location of system equilibria of the proposed mapping, and the possible bifurcation types in smooth dissipative systems.
Destruction of Chaotic Attractors in Coupled Chaotic Systems
Journal- Korean Physical Society
We investigate the sudden destruction of hyperchaotic attractors in symmetrically coupled onedimensional maps. An asynchronous hyperchaotic attractor may appear through a blow-out bifurcation, where the synchronous chaotic attractor on the invariant synchronization line becomes unstable with respect to perturbations transverse to the synchronization line. It is found that such a hyperchaotic attractor may be broken up suddenly through stabilization of a periodic saddle embedded in the hyperchaotic attractor via a reverse subcritical pitchfork or period-doubling bifurcation. After break up of the hyperchaotic attractor, the asymptotic state transforms from the hyperchaotic state to a periodic state. Note that this sudden destruction of the hyperchaotic attractor occurs without any collision with its basin boundary, in constrast to the boundary crisis and is robust under a small perturbation of parameter mismatching.
Role of invariant and minimal absorbing areas in chaos synchronization
Physical Review E, 1998
In this paper the method of critical curves, a tool for the study of the global dynamical properties of two-dimensional noninvertible maps, is applied to the study of chaos synchronization and related phenomena of riddling, blowout, and on-off intermittency. A general procedure is suggested in order to obtain the boundary of a particular two-dimensional compact trapping region, called absorbing area, containing the onedimensional chaotic set on which synchronized dynamics occur. The main purpose of the paper is to show that only invariant and minimal absorbing areas are useful to characterize the global dynamical behavior of the dynamical system when a Milnor attractor with locally riddled basin or a chaotic saddle exists, and may strongly influence the effects of riddling and blowout bifurcations. Some examples are given for a system of two coupled logistic maps, and some practical methods and numerical tricks are suggested in order to ascertain the properties of invariance and minimality of an absorbing area. Some general considerations are given concerning the transition from locally riddled to globally riddled basins, and the role of the absorbing area in the occurrence of such transition is discussed.
Bubbling of attractors and synchronisation of chaotic oscillators
Physics Letters A, 1994
We present a system of two coupled identical chaotic electronic circuits that exhibit a blowout bifurcation resulting in loss of stability of the synchronised state. We introduce the concept of bubbling of an attractor, a new type of intermittency that is triggered by low levels of noise, and demonstrate numerical and experimental examples of this behaviour. In particular we observe bubbling near the synchronised state of two coupled chaotic oscillators. We give a theoretical description of the behaviour associated with locally riddled basins, emphasising the role of invariant measures. In general these are non-unique for a given chaotic attractor, which gives rise to a spectrum of Lyapunov exponents. The behaviour of the attractor depends on the whole spectrum. In particular, bubbling is associated with the loss of stability of an attractor in a dynamically invariant subspace, and is typical in such systems.