New universal bifurcation scenario in one-dimensional trimodal maps

By means of star products and high precision numerical calculation, an abnormal phenomenon is found in period-p-tupling bifurcation processes in one-dimensional trimodal maps. A route of transition to chaos, presented by a right-associative non-normal star product, breaks the Feigenbaum's metric universality, namely, the conventional Feigenbaum's successive rates exhibit a strong divergence. To overcome the divergence, an approximate scheme of accelerating convergence is proposed; and the Feigenbaum scenario is included as a special case in the new bifurcation scenario. It will provide access to understanding non-normal star products and their corresponding renormalization.     

The dynamics of random interfaces in phase transitions

A review is given of recent developments involving the dynamics of random interfaces formed in the evolution of metastable and unstable systems. Topics which are discussed include interface growth and nonequilibrium dynamical scaling. The possibility of there being dynamical universality classes in first-order phase transitions is also discussed. There are a large number of systems of experimental interest which include binary alloys, binary fluids, and polymer mixtures. Other systems studied by computer simulation include the kinetic Ising, Potts, andZ _N models.Work supported by NSF grant No. DMR-8013700.     

Universal power spectra for the reverse bifurcation sequence

Many dynamical systems exhibit forward and reverse period-doubling bifurcation sequences, the latter being intrinsically noisy. Feigenbaum has predicted the amplitude of sharp spectral components in the forward sequence from universality arguments. In the same spirit we derive the approximate form of the broad band features in the reverse sequence. Our results give a power-law behavior of the integrated noise spectrum similar to that recently reported by Huberman and Zisook.     

Dynamical analysis of fractional-order Rössler and modified Lorenz systems

This Letter is devoted to the dynamical analysis of fractional-order systems, namely the Rössler and a modified Lorenz system. The work here described compares the dynamical regimes of such fractional-order systems to that of the corresponding standard systems. It turns out that most of the chaotic attractors are topologically equivalent to those found in the original integer-order systems, although in some particular (and apparently rare) cases unusual bifurcation patterns and attractors are found.     

Collective oscillation in a hamiltonian system

Oscillation of macroscopic variables is discovered in a metastable state of the Hamiltonian system of the mean-field model. The duration of the oscillation is divergent with the system size. This long-lasting periodic or quasiperiodic collective motion appears through Hopf bifurcation, which is a typical route in low-dimensional dissipative dynamical systems. The origin of the oscillation is explained, with a self-consistent analysis of the distribution function, as the self-organization of a self-excited swing state through the mean field. The universality of the phenomena is discussed.     

Analysis of critical and post-critical behaviour of non-linear dynamical systems by the normal form method,Part I: Normalization formulae

A method, based on normal form theory, is presented to study the dynamical behaviour of a system in the neighbourhood of a nearly critical equilibrium state associated with a bifurcation condition. Explicit formulae for the normalization procedure are derived. These formulae can be numerically programmed, avoiding usual complicated algebraic calculations and making the method effectively applicable for n-dimensional systems. Rather general bifurcations can be included: e.g., non-linear flutter (Hopf bifurcation), divergence and internal resonance.     

Degenerate Hopf bifurcation in the presence of symmetry

We investigate the bifurcation of time-periodic states from a stationary state destabilized by the undamping of a set of modes associated with a degenerate pair of complex-conjugate frequencies. This problem is of particular interest for bifurcations in driven systems with symmetry whose order-parameter dimension n is even and n ≥ 4. For this case of a degenerate Hopf bifurcation a star of symmetry-equivalent limit cycles bifurcates in analogy to the star of symmetry-related domains arising at a symmetry-breaking phase transition in equilibrium systems. We illustrate this fact by analyzing a concrete example with n = 4. Within the framework of an amplitude expansion, we explicitly construct the time-periodic states and discuss their stability. In particular, it is shown that fairly general conclusions for the bifurcation behaviour can be drawn on the sole basis of the knowledge of the order-parameter symmetry.     

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. This class includes all the standard constructions of star products. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance properties and give necessary and sufficient conditions for them to have a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.     

Geometric Analysis of Bifurcation and Symmetry Breaking in a Gross—Pitaevskii Equation

Gross–Pitaevskii and nonlinear Hartree equations are equations of nonlinear Schrödinger type that play an important role in the theory of Bose–Einstein condensation. Recent results of Aschbacher et al.(3) demonstrate, for a class of 3-dimensional models, that for large boson number (squared L ²norm), $N$ , the ground state does not have the symmetry properties of the ground state at small $N$ . We present a detailed global study of the symmetry breaking bifurcation for a 1-dimensional model Gross–Pitaevskii equation, in which the external potential (boson trap) is an attractive double-well, consisting of two attractive Dirac delta functions concentrated at distinct points. Using dynamical systems methods, we present a geometric analysis of the symmetry breaking bifurcation of an asymmetric ground state and the exchange of dynamical stability from the symmetric branch to the asymmetric branch at the bifurcation point.     

Universal Algorithm of the Nonequilibrium Dynamics of Chaotic Systems

Scenarios of transition of nonlinear dynamical systems to chaos are considered based on the study of the behavior of nonlinear dynamical systems represented as feedback systems and analysis of universality and scale invariance (fractal) properties of threshold dynamical structures at points of unstable equilibrium of these systems.     

Critical fluctuation universality in chemically oscillatory systems: A soluble master equation

Multiple time scale arguments are used to show that near a Hopf bifurcation to a chemical oscillation the dynamics of the system reduces to that of a classic soluble limit cycle system. A birth and death master equation is then introduced and the spectrum of the resulting transition operator is shown to be complex. Exact solutions of the master equation are obtained both for the steady and (for a rather general class of systems) excited states. Thus a simple basis of universality of critical properties in chemical oscillations is provided.Research supported in part by a grant from the National Science Foundation.     

Experimental bifurcation diagram of a circuit-implemented neuron model

An experimental bifurcation diagram of a circuit implementing an approximation of the Hindmarsh-Rose (HR) neuron model is presented. Measured asymptotic time series of circuit voltages are automatically classified through an ad hoc algorithm. The resulting two-dimensional experimental bifurcation diagram evidences a good match with respect to the numerical results available for both the approximated and original HR model. Moreover, the experimentally obtained current-frequency curve is very similar to that of the original model. The obtained results are both a proof of concept of a quite general method developed in the last few years for the approximation and implementation of nonlinear dynamical systems and a first step towards the realisation in silico of HR neuron networks with tunable parameters.     

Bifurcation phenomena in two-dimensional piecewise smooth discontinuous maps

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, there exist a large number of switching dynamical systems that have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. In this paper we present a systematic approach to the problem of analyzing the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We first motivate the analysis by considering the bifurcations occurring in a familiar physical system-the static VAR compensator used in electrical power systems-and then proceed to formulate the theory needed to explain the bifurcation behavior of such systems. We then integrate the observed bifurcation phenomenology of the compensator with the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps.     

Optically induced phase transition of excitons in coupled quantum dots

The weak classical light excitations in many semiconductor quantum dots have been chosen as important solid- state quantum systems for processing quantum information and implementing quantum computing. For strong classical light we predict theoretically a novel phase transition as a function of magnitude of this classical light from the deformed to the normal phases in resonance ease, and the essential features of criticality such as the scaling behaviour, critical exponent and universality are also present in this paper.     

Discontinuity-induced bifurcations of equilibria in piecewise-smooth and impacting dynamical systems

A rich variety of dynamical scenarios has been shown to occur when a fixed point of a non-smooth map undergoes a border-collision. This paper concerns a closely related class of discontinuity-induced bifurcations, those involving equilibria of n-dimensional piecewise-smooth flows. Specifically, transitions are studied which occur when a boundary equilibrium, that is one lying within a discontinuity manifold, is perturbed. It is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios. Conditions to classify among the possible simplest scenarios are given for piecewise-smooth continuous, Filippov and impacting systems. Also, we investigate the possible birth of other attractors (e.g. limit cycles) at a boundary-equilibrium bifurcation.     

Hyperchaos--chaos--Hyperchaos Transition in a Class of On--Off Intermittent Systems Driven by a Family of Generalized Lorenz Systems

Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems＇ dynamics. After the bifurcation the phenomenon of on-off intermittency appears. It is observed that the systems undergo a symmetric hyperchaos-chaos-hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.     

The VKS experiment: turbulent dynamical dynamos

The VKS experiment studies dynamo action in the flow generated inside a cylinder filled with liquid sodium by the rotation of coaxial impellers (the von Kármán geometry). We report observations related to the self-generation of a stationary dynamo when the flow forcing is symmetric, i.e. when the impellers rotate in opposite directions at equal angular velocities. The bifurcation is found to be supercritical, with a neutral mode whose geometry is predominantly axisymmetric. We then report the different dynamical dynamo regimes observed when the flow forcing is asymmetric, including magnetic field reversals. We finally show that these dynamics display characteristic features of low dimensional dynamical systems despite the high degree of turbulence in the flow. To cite this article: VKS Collaboration, C. R. Physique 9 (2008).     

Comparison of the influences of nodal dynamics on network synchronized regions

A new piecewise linear unified chaotic (PLUC) system is firstly presented, and then its fundamental dynamical behaviors are analyzed. This modified chaotic system, as well as the unified chaotic (UC) one, is taken as network nodal oscillators for investigating the difference of influences of nodal dynamics on the bifurcation of network synchronized regions. It is found that beyond the greatly similar bifurcation modes between PLUC and UC networks, the synchronized regions in PLUC networks are far narrower at almost each parameter a than those in UC networks for most of inner coupling matrices, indicating the PLUC node makes the network more difficult to synchronization. Our numerical investigations show that this phenomenon is closely related with nodal dynamical properties, such as the boundary of attractors, the largest Lyapunov exponent and Lyapunov dimension.