Quasi-periodic Regimes Research Articles

Transition to turbulence in toroidal pipes

AbstractIncompressible flow in toroidal pipes of circular cross-section was investigated by three-dimensional, time-dependent numerical simulations using a finite volume method. The computational domain included a whole torus and was discretized by up to ${\ensuremath{\sim} }11. 4\ensuremath{\times} 1{0}^{6} $ nodes. Two curvatures $\delta $ (radius of the cross-section/radius of the torus), namely 0.3 and 0.1, were examined; a streamwise forcing term was imposed, and its magnitude was made to vary so that the bulk Reynolds number ranged between ${\ensuremath{\sim} }3500$ and ${\ensuremath{\sim} }14\hspace{0.167em} 700$. The results were processed by different techniques in order to confirm the spatio-temporal structure of the flow. Consecutive transitions between different flow regimes were found, from stationary to periodic, quasi-periodic and chaotic flow. At low Reynolds number, stationary flow was predicted, exhibiting a symmetric couple of Dean vortex rings and a strong shift of the streamwise velocity maximum towards the outer wall. For $\delta = 0. 3$, between $\mathit{Re}= 4556$ and $\mathit{Re}= 4605$ a first transition occurred from stationary to periodic flow, associated with a supercritical Hopf bifurcation and giving rise to a travelling wave which took the form of a varicose streamwise modulation of the Dean vortex ring intensity. A further transition, associated with a secondary Hopf bifurcation, occurred between $\mathit{Re}= 5042$ and $\mathit{Re}= 5270$ and led to a quasi-periodic flow characterized by two independent fundamental frequencies associated with distinct travelling waves, the first affecting mainly the Dean vortex rings and similar to that observed in purely periodic flow, the second localized mainly in the secondary flow boundary layers and manifesting itself as an array of oblique vortices produced at the edge of the Dean vortex regions. Both the periodic and the quasi-periodic regimes were characterized by an instantaneous anti-symmetry of the oscillatory components with respect to the equatorial midplane of the torus. For $\delta = 0. 1$, between $\mathit{Re}= 5139$ and $\mathit{Re}= 5208$ a direct (‘hard’) transition from steady to quasi-periodic flow occurred. Hysteresis was also observed: starting from a quasi-periodic solution and letting the Reynolds number decrease, both quasi-periodic and periodic stable solutions were obtained at Reynolds numbers below the critical value. A further decrease in $\mathit{Re}$ led to steady-state solutions. This behaviour suggests the existence of a subcritical Hopf bifurcation followed by a secondary Hopf bifurcation. The resulting periodic and quasi-periodic flows were similar to those observed for the higher curvature, but the travelling modes were now instantaneously symmetric with respect to the equatorial midplane of the torus. Also, the further transition from quasi-periodic to chaotic flow occurred with different modalities for the two curvatures. For $\delta = 0. 3$, a centrifugal instability of the main flow in the outer region occurred abruptly between $\mathit{Re}= 7850$ and $\mathit{Re}= 8160$, while a further increase of $\mathit{Re}$ up to 13 180 did not cause any relevant change in the distribution and intensity of the fluctuations. For $\delta = 0. 1$ the transition to chaotic flow was gradual in the range $\mathit{Re}= 6280$ to 8160 and affected mainly the inner region; only a further increase of $\mathit{Re}$ to 14 700 caused fluctuations to appear also in the outer region.

Quasi-periodic synchronisation of self-modulation oscillations in a ring chip laser by an external periodic signal

The synchronisation of periodic self-modulation oscillations in a ring Nd:YAG chip laser under an external periodic signal modulating the pump power has been experimentally investigated. A new quasi-periodic regime of synchronisation of self-modulation oscillations is found. The characteristic features of the behaviour of spectral and temporal structures of synchronised quasi-periodic oscillations with a change in the external signal frequency are studied.

Desynchronization transitions in nonlinearly coupled phase oscillators

We consider the nonlinear extension of the Kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter. A bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed. We demonstrate that for small ensembles it is typically mediated by stable cluster states, that disappear with creation of heteroclinic cycles, while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs.

Defects in spatiotemporal diagrams and their relations to phase coherence and lack of observability

Spatiotemporal systems are commonly investigated in terms of spatiotemporal diagrams and, most often, the analysis is limited to the first instabilities. Due to the lack of a Takens-like theorem for spatiotemporal systems, the resulting dynamics is almost never interpreted using phase portraits reconstructed from one variable locally recorded. This work is an attempt to make an explicit link between reconstructed phase portraits and spatiotemporal diagrams. Defects distributions are interpreted in terms of a lack of phase coherence. The lack of a simple structure--as a torus characterized by a closed curve for Poincaré section when a quasiperiodic regime is identified--is tentatively interpreted in terms of observability. A first link is thus made between the defects distribution and the nature of the underlying dynamics.

Quasi-periodic regime of self-modulation oscillations with a periodic low-frequency envelope in a ring chip laser

The dynamics of a ring chip Nd:YAG laser radiation in an external static magnetic field inducing an optical nonreciprocity of the resonator is studied experimentally. The amplitude and frequency nonreciprocities of the ring cavity induced by this magnetic field varied with moving the magnet with respect to the active element of the chip laser. A previously unknown quasi-periodic self-modulation lasing regime in which the self-modulation intensity oscillations of counterpropagating waves have an antiphase low-frequency envelope is found. The temporal and spectral characteristics of radiation are studied and the conditions of the appearance of this lasing regime are determined.

C-type period-doubling transition in nephron autoregulation

The functional units of the kidney, called nephrons, utilize mechanisms that allow the individual nephron to regulate the incoming blood flow in response to fluctuations in the arterial pressure. This regulation tends to be unstable and to generate self-sustained oscillations, period-doubling bifurcations, mode-locking and other nonlinear dynamic phenomena in the tubular pressures and flows. Using a simplified nephron model, the paper examines how the regulatory mechanisms react to an external periodic variation in arterial pressure near a region of resonance with one of the internally generated mode-locked cycles. We show how the stable and unstable resonance cycles generated in this response undergo interconnected cascades of period-doubling bifurcations and how each period doubling leads to the formation of a new pair of saddle-node bifurcation curves along the edges of the resonance zone. We also show how period doubling of the resonance cycles is accompanied by a torus-doubling process in the quasiperiodic regime that exists outside of the resonance zone.

Spectroscopic determination of kinetic parameters for frequency sweeping Alfvén eigenmodes

A method for analyzing fundamental kinetic plasma parameters, such as linear drive and external damping rate, based on experimental observations of chirping Alfvén eigenmodes, is presented. The method, which relies on new semiempirical laws for nonlinear chirping characteristics, consists of fitting procedures between the so-called Berk–Breizman model and the experiment in a quasiperiodic chirping regime. This approach is applied to the toroidicity induced Alfvén eigenmode (TAE) on JT-60 Upgrade (JT-60U) [N. Oyama et al., Nucl. Fusion 49, 104007 (2009)], which yields an estimation of the kinetic parameters and suggests the existence of TAEs far from marginal stability. Two collision models are considered, and it is shown that dynamical friction and velocity-space diffusion are essential to reproduce nonlinear features observed in experiments. The results are validated by recovering measured growth and decay of perturbation amplitude and by estimating collision frequencies from experimental equilibrium data.

Dynamic regimes of the dual-wavelength vertical external cavity surface-emitting laser

The stability of the stationary state and periodic oscillations of the radiation intensity of the dual-wavelength vertical external cavity surface-emitting laser is analyzed. It is demonstrated that the stationary state is always stable for the laser with independent active areas that contain quantum dots of different depths. A map of dynamic regimes with the values of parameters that correspond to the cw lasing, periodic oscillations of intensity, and complicated dynamics (quasi-periodic or chaotic regimes) is constructed for the laser with coupled active areas. The dynamics of the laser radiation in the quasi-periodic regime is numerically simulated. It is demonstrated that relatively slow variations in the amplitude with characteristic times of several or several tens of microseconds are typical of the short laser pulses that are formed at time intervals of tens of picoseconds. The period of the pulse train is approximately equal to the round-trip time of the external laser cavity.

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Kármán equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincaré maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.

COMPLEX DYNAMICS AND CHAOS IN A HYBRID SYSTEM MODELING A CONTROLLED REVERSE FLOW REACTOR

In this work some complex behaviors of a controlled reverse flow reactor is presented. The control system introduces discrete events making the model an infinite dimensional hybrid system. The study is conducted through continuation techniques and brute force numerical simulations. Together with standard bifurcations like pitchfork, saddle-node and Neimark–Sacker, varying the set-point parameter of the controller, several novel aspects are singled out: an unusual sequence of period-adding bifurcation phenomena, a new route to chaos and the coexistence of Zeno states with quasi-periodic and chaotic regimes. The period-adding phenomena dictate the transition between symmetric and asymmetric multiperiodic regimes and a simple rule for the occurrence of symmetry breaking and recovery is found. The new route to chaos is a transition from a quasi-periodic regime to chaos due to the presence of Zeno phenomena, typical of hybrid systems. The chaos is characterized by Zeno-like oscillations.

Dependence of magnetic field generation by thermal convection on the rotation rate: A case study

Dependence of magnetic field generation on the rotation rate is explored by direct numerical simulation of magnetohydrodynamic convective attractors in a plane layer of conducting fluid with square periodicity cells for the Taylor number varied from zero to 2000, for which the convective fluid motion halts (other parameters of the system are fixed). We observe 5 types of hydrodynamic (amagnetic) attractors: two families of two-dimensional (i.e. depending on two spatial variables) rolls parallel to sides of periodicity boxes of different widths and parallel to the diagonal, travelling waves and three-dimensional “wavy” rolls. All types of attractors, except for one family of rolls, are capable of kinematic magnetic field generation. We have found 21 distinct nonlinear convective MHD attractors (13 steady states and 8 periodic regimes) and identified bifurcations in which they emerge. In addition, we have observed a family of periodic, two-frequency quasiperiodic and chaotic regimes, as well as an incomplete Feigenbaum period doubling sequence of bifurcations of a torus followed by a chaotic regime and subsequently by a torus with 1/3 of the cascade frequency. The system is highly symmetric. We have found two novel global bifurcations reminiscent of the SNIC bifurcation, which are only possible in the presence of symmetries. The universally accepted paradigm, whereby an increase of the rotation rate below a certain level is beneficial for magnetic field generation, while a further increase inhibits it (and halts the motion of fluid on continuing the increase), remains unaltered, but we demonstrate that this “large-scale” picture lacks many significant details.

Experimental study of targeted energy transfer from an acoustic system to a nonlinear membrane absorber

Abstract This paper deals with the application of the concept of targeted energy transfer to the field of acoustics, providing a new approach to passive sound control in the low frequency domain, where no efficient dissipative mechanism exists. The targeted energy transfer, also called energy pumping, is a phenomenon that we observe by combining a pure nonlinear oscillator with a linear primary system. It corresponds to an almost irreversible transfer of vibration energy from the linear system to the auxiliary nonlinear one, where the energy is finally dissipated. In this study, an experimental set-up has been developed using the air inside a tube as the acoustic linear system, a thin circular visco-elastic membrane as an essentially cubic oscillator and the air inside a box as a weak coupling between those two elements. In this paper, which mainly deals with experimental results, it is shown that several regimes exist under sinusoidal forcing, corresponding to the different nonlinear normal modes of the system. One of these regimes is the quasi-periodic energy pumping regime. The targeted energy transfer phenomenon is also visible on the free oscillations of the system. Indeed, above an initial excitation threshold, the sound extinction in the tube follows a quasi-linear decrease that is much faster than the usual exponential one. During this linear decrease, the energy of the acoustic medium is irreversibly transferred to the membrane and then damped into this element called nonlinear energy sink. We present also the frequency responses of the system which shows a clipping of the original resonance peak of the acoustic medium and we finally demonstrate the ability of the nonlinear absorber to operate in a large frequency band, tuning itself to any linear system.

Stochastic effects during the action of the pump noise on bistable self-modulation oscillations in a solid-state ring laser

Nonlinear radiation dynamics of a solid-state ring laser is studied in the region of laser parameters corresponding to the parametric resonance between the self-modulation and relaxation oscillations. Bistable regions are found in which, apart from the self-modulation regime of the first kind, a stable quasi-periodic self-modulation regime exists. Temporal and spectral emission parameters of counterpropagating waves are considered in the bistable self-modulation generation regimes. The effect of noise on the bistable self-modulation oscillations is studied. It is shown that during the interaction of noise, spectral peaks split at relaxation and self-modulation frequencies.

Synchronization of modulated traveling baroclinic waves in a periodically forced, rotating fluid annulus

Frequency entrainment and nonlinear synchronization are commonly observed between simple oscillatory systems, but their occurrence and behavior in continuum fluid systems are much less well understood. Motivated by possible applications to geophysical fluid systems, such as in atmospheric circulation and climate dynamics, we have carried out an experimental study of the interaction of fully developed baroclinic instability in a differentially heated, rotating fluid annulus with an externally imposed periodic modulation of the thermal boundary conditions. In quasiperiodic and chaotic amplitude-modulated traveling wave regimes, the results demonstrate a strong interaction between the natural periodic modulation of the wave amplitude and the externally imposed forcing. This leads to partial or complete phase synchronization. Synchronization effects were observed even with very weak amplitudes of forcing, and were found with both 1:1 and 1:2 frequency ratios between forcing and natural oscillations.

Self-organized partially synchronous dynamics in populations of nonlinearly coupled oscillators

We analyze a minimal model of a population of identical oscillators with a nonlinear coupling—a generalization of the popular Kuramoto model. In addition to well-known for the Kuramoto model regimes of full synchrony, full asynchrony, and integrable neutral quasiperiodic states, ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamics: self-organized bunch states and self-organized quasiperiodic dynamics. The analysis based on the Watanabe–Strogatz ansatz allows us to describe the self-organized bunch states in any finite ensemble as a set of equilibria, and the self-organized quasiperiodicity as a two-frequency quasiperiodic regime. An analytic solution in the thermodynamic limit of infinitely many oscillators is also discussed.

The death of quasi-periodic regimes in a system of dissipatively coupled van der Pol oscillators under pulsed drive action

We have studied a system of coupled van der Pol oscillators under pulsed drive action. In the case where the coupled oscillators exhibit the death of oscillations, the introduction of a pulsed drive leads to the appearance of islands corresponding to quasi-periodic regimes in the drive period-amplitude plane, which are also dying with an increase in the frequency detuning.

Simulation of flow around a row of square cylinders

In this paper, the low-Reynolds number (Re= 80) flow around a row of nine square cylinders placed normal to the oncoming flow is investigated using the lattice-Boltzmann method. The effects of the cylinder spacing on the flow are studied for spacing to diameter ratios of 0.3 to 12. No significant interaction between the wakes is observed with spacings greater than six times the diameter. At smaller spacings, the flow regimes as revealed by vorticity field and drag coefficient signal are: synchronized, quasi-periodic and chaotic. These regimes are shown to result from the interaction between primary (vortex shedding) and secondary (cylinder interaction) frequencies; the strength of the latter frequency in turn depends on the cylinder spacing. The secondary frequency is also related to transition between narrow and wide wakes behind a cylinder.The mean drag coefficient and Strouhal number are found to increase rapidly with a decrease in spacing; correlations of these parameters with spacing are proposed. The Strouhal number based on gap velocity becomes approximately constant for a large range of spacings, highlighting the significance of gap velocity for this class of flows. It is also possible to analyse the vortex pattern in the synchronized and quasi-periodic regimes with the help of vorticity dynamics. These results, most of which have been obtained for the first time, are of fundamental significance.

Stabilizing the Rössler system by external pulses on a runaway trajectory

The action of external pulses on the Rossler system prior to the onset of a saddle-node type bifurcation on a runaway trajectory is considered. It is shown that a pulsed drive can lead to the appearance of stable periodic and quasi-periodic regimes in the nonautonomous system.

Breeding and predictability in the baroclinic rotating annulus using a perfect model

Abstract. We present results from a computational study of predictability in fully-developed baroclinically unstable laboratory flows. This behaviour is studied in the Met Office/Oxford Rotating Annulus Laboratory Simulation – a model of the classic rotating annulus laboratory experiment with differentially heated cylindrical sidewalls, which is firmly established as an insightful laboratory analogue for certain kinds of atmospheric dynamical behaviour. This work is the first study of "predictability of the first kind" in the annulus experiment. We devise an ensemble prediction scheme using the breeding method to study the predictability of the annulus in the perfect model scenario. This scenario allows one simulation to be defined as the true state, against which all forecasts are measured. We present results from forecasts over a range of quasi-periodic and chaotic annulus flow regimes. A number of statistical and meteorological techniques are used to compare the predictability of these flows: bred vector growth rate and dimension, error variance, "spaghetti plots", probability forecasts, Brier score, and the Kolmogorov-Smirnov test. These techniques gauge both the predictability of the flow and the performance of the ensemble relative to a forecast using a climatological distribution. It is found that in the perfect model scenario, the two quasi-periodic regimes examined may be indefinitely predictable. The two chaotic regimes (structural vacillation and period doubled amplitude vacillation) show a loss of predictability on a timescale of hundreds to thousands of seconds (65–280 annulus rotation periods, or 1–3 Lyapunov times).

High-speed videocamera investigation of the wave structure development on an unstable cavity boundary

The formation of wave structures on an unstable (in the Rayleigh-Taylor criterion) boundary of a plane cavity is studied using high-speed video filming. It is shown that at the head of the cavity a quasiperiodic wave regime develops, with a mean wavelength similar to that obtained from a linear stability analysis for the cavity boundary. Observations of the initial-regime breakdown show that its scenarios are similar to the development of subharmonic instability in the one-mode regime. The waves are classified with respect to the wave development rate. It is shown that the large-wave amplitude growth law is on average closely approximated by a quadratic parabola, with the total gas entrainment from the cavity being proportional to the square of the cavity length. The existence of a scaling effect is detected, which in the case considered reduces mainly to a dependence of the gas entrainment coefficient on the Weber number. It is shown that with decrease in the Weber number the gas entrainment coefficient may significantly increase.