Quasiperiodic Transition Research Articles

Coupled instabilities drive quasiperiodic order-disorder transitions in Faraday waves.

We present an experimental study of quasiperiodic transitions between a highly ordered square-lattice pattern and a disordered, defect-riddled state, in a circular Faraday system. We show that the transition is driven initially by a long-wave amplitude modulation instability, which excites the oscillatory transition phase instability, leading to the formation of dislocations in the Faraday lattice. The appearance of dislocations dampens amplitude modulations, which prevents further defects from being created and allows the system to relax back to its ordered state. The process then repeats itself in a quasiperiodic manner. Our experiments reveal an unexpected mechanism for temporal quasiperiodicity that results from a coupling between two distinct instabilities on the route to chaos.

Route to chaos in whispering gallery mode coupled opto-mechanical systems

In opto-mechanical systems, the nonlinearity caused by radiation pressure can lead to various abundant dynamical phenomena such as chaos. Chaos is an important branch of nonlinear dynamics, and researchers focus on understanding the transitions from order to chaos in different systems. In this paper, we investigate the chaotic dynamics in a system consisting of two evanescently coupled identical cavity opto-mechanical subsystems, where the optical fields are in whispering gallery modes. To thoroughly analyze the transition from order to chaos in our system, we utilize the bifurcation diagrams, the Lyapunov exponents, and phase space trajectories to characterize the system properties. It is found that the coupling strength between the two opto-mechanical subsystems plays a crucial role in determining the systemic dynamic behaviors. There are two routes to chaos in our system i.e. the period-doubling transition and the quasiperiodic transition. These routes correspond to strong coupling and weak coupling between the two opto-mechanical subsystems, respectively. Furthermore, the results show that the synchronization between the oscillations in the two opto-mechanical subsystems can occur under strong coupling. In this situation, the dynamic behaviors of the two opto-mechanical subsystems are exactly identical and the manipulation of the coupling strength is equivalent to the tuning of the frequency detuning between the cavity fields and their corresponding driving fields. Consequently, the coupled system behaves as a single opto-mechanical system, enabling a period-doubling transition to chaos through increasing the coupling strength. In the case of weak coupling, the dynamics of the two opto-mechanical subsystems are no longer synchronized, and the coupled system dynamic behaviors unfold in an eight-dimensional phase space. The limit cycles experience the Hopf bifurcation, resulting in the emergence of a toric attractor. Within a certain range of parameters, i.e. appropriate frequency detunings, the two-dimensional torus becomes unstable as coupling strength increases, leading to a quasiperiodic transition into chaos in our coupled opto-mechanical system.

Glycolytic Wave Patterns in a Simple Reaction‐diffusion System with Inhomogeneous Influx: Dynamic Transitions

An inhomogeneous profile of chemostatted species generates a rich variety of patterns in glycolytic waves depicted in a Selkov reaction-diffusion framework here. A key role played by diffusion amplitude and symmetry in the chemostatted species profile in dictating the fate of local spatial dynamics involving periodic, quasiperiodic, and chaotic patterns and transitions among them are investigated systematically. More importantly, various dynamic transitions, including wave propagation direction changes, are illustrated in interesting situations. Besides numerical results, our analytical formulation of the amplitude equation connecting complex Ginzburg-Landau and Lambda-omega representation shed light on the phase dynamics of the system. This systematic study of the glycolytic reaction-diffusion wave is in line with previous experimental results in open spatial reactor and will provide a knowledge about the dynamics that shape and control biological information processing and related phenomena.

Quasiperiodic many-body localization transition in dimension d>1

The nature of the many-body localization (MBL) transition and even the existence of the MBL phase in random many-body quantum systems have been actively debated in recent years. In spatial dimension $d>1$, there is some consensus that the MBL phase is unstable to rare thermal inclusions that can lead to an avalanche that thermalizes the whole system. In this note, we explore the possibility of MBL in quasiperiodic systems in dimension $d>1$. We argue that (i) the MBL phase is stable at strong enough quasiperiodic modulations for $d = 2$, and (ii) the possibility of an avalanche strongly constrains the finite-size scaling behavior of the MBL transition. We present a suggestive construction that MBL is unstable for $d \geq 3$.

A robust model for the origin of optical quasi-periodic variability in supersoft X-ray sources

Supersoft X-ray sources (SSSs) are known as possible progenitors of Type Ia supernovae. The quasi-periodic variability has been detected in the optical light curves of SSSs. However, the exact origin of such quasi-periodic observable features remains a mystery. In this paper, we aim to reproduce the observed optical quasi-periodic variability of SSSs by proposing a white dwarf (WD) accretion model with a periodic mass transfer caused by the irradiation of supersoft X-ray onto the companion star. Methods. Assuming that a periodic mass transfer from the companion star to the WD can be caused while the supersoft X-ray irradiates the companion star, we used MESA to simulate the WD accretion process and the subsequent WD evolution by adopting a periodic jagged accretion rate. Comparing our results to the optical light curves of a well-observed SSS RX J0513.9-6951, we find that our models can reproduce the quasi-periodic transition between the optical high and low states of RX J0513.9-6951 because the periodic accretion rate can lead to the WD photosphere expands and contracts periodically in our models. In addition, we find that the transitional periods of the SSSs in our models strongly depend on the mass of the accreting WDs. The more massive the WD mass is, the shorter the transitional period. Based on our results, we suggest that the periodic mass transfer caused by the irradiation of supersoft X-ray onto the companion star may be the origin of the observed optical quasi-periodic variability in SSSs. In addition, our results indicate that the observed optical transition period of a SSS may be useful for the rough estimate of the mass of an accreting WD.

Microbubble PhoXonic resonators: Chaos transition and transfer

We report the activation of optomechanical chaotic oscillations in microbubble resonators (MBRs) through a blue-side excitation of its optical resonances. We confirm the sequence of quasi-periodical oscillation, spectral continuum and aperiodic motion; as well as the transition to chaos without external feedback or modulation of the laser source. In particular, quasi periodic transitions and a spectral continuum are reported for MBRs with diameters up to 600 µm, whereas only an abrupt transition into a spectral continuum is observed for larger microbubbles.

Local integrals of motion and the quasiperiodic many-body localization transition

We study the many-body localization (MBL) transition for interacting fermions subject to quasiperiodic potentials by constructing the local integrals of motion (LIOMs) in the MBL phase as time-averaged local operators. We study numerically how these time-averaged operators evolve across the MBL transition. We find that the norm of such time-averaged operators drops discontinuously to zero across the transition; as we discuss, this implies that LIOMs abruptly become unstable at some critical localization length of order unity. We analyze the LIOMs using hydrodynamic projections and isolating the part of the operator that is associated with interactions. Equipped with these data we perform a finite-size scaling analysis of the quasiperiodic MBL transition. Our results suggest that the quasiperiodic MBL transition occurs at considerably stronger quasiperiodic modulations and has a larger correlation-length critical exponent than previous studies had found.

Dynamic interlinking between near- and far-field wakes behind a pitching–heaving airfoil

This study investigates the transitional wake dynamics of a simultaneously pitching– heaving airfoil in the low Reynolds number regime. The transition from Kármán to reverse Kármán wakes and the subsequent wake deflection are known to take place entirely in the periodic regime as the dynamic heave velocity $\kappa h$ (proportionally, the amplitude-based Strouhal number $St_A$ ) is gradually increased. However, further increase in $\kappa h$ may result in the loss of dynamic stability of the deflected vortex street and an eventual transition to chaos. This phenomenon has largely remained unexplored in the existing literature. The present study attempts to fill this gap and aims to establish a dynamic link between the near- and far-field wake transitions. It is shown that the deflected jet undergoes a switching of deflection direction in the far-field as the near-field encounters a quasi-periodic transition with the increase in $\kappa h$ . The quasi-periodic behaviour of the near-field is seen to get interspersed with intermittent aperiodic windows, resulting in a complete reversal of the deflection direction through flipping of the immediate vortex couple. Eventually, the wake topology becomes fully chaotic through a series of rapid aperiodic jet-switching. To the best of the authors’ knowledge, this is the first study that investigates the role of this aperiodic jet-switching in ushering in chaos in the wake of a pitch–heave flapping system. The mechanism of jet-switching and the role of leading-edge vortex are also revealed. Flow field dynamics during these changes and the underlying vortex interactions are analysed using nonlinear dynamical tools.

On the destabilization of a periodically driven three-dimensional torus

We report experimental evidence of the destabilization of a 3D torus obtained when a small subharmonic perturbation is added to a 2D torus characteristic of a driven relaxation oscillator. The Poincare sections indicate that the torus breakup is sensitive to the phase difference between the main driving frequency and its first subharmonic perturbing component. The observed transition confirms the Newhouse, Ruelle and Takens quasiperiodic transition to chaos on a 3D torus. Numerical results on a sinusoidally perturbed circle map mirror the experimental results and confirm the key role of the phase difference in the transition between distinct dynamical regimes.

Universality and quantum criticality in quasiperiodic spin chains

Quasiperiodic systems are aperiodic but deterministic, so their critical behavior differs from that of clean systems and disordered ones as well. Quasiperiodic criticality was previously understood only in the special limit where the couplings follow discrete quasiperiodic sequences. Here we consider generic quasiperiodic modulations; we find, remarkably, that for a wide class of spin chains, generic quasiperiodic modulations flow to discrete sequences under a real-space renormalization-group transformation. These discrete sequences are therefore fixed points of a functional renormalization group. This observation allows for an asymptotically exact treatment of the critical points. We use this approach to analyze the quasiperiodic Heisenberg, Ising, and Potts spin chains, as well as a phenomenological model for the quasiperiodic many-body localization transition.

Quasiperiodic dynamical quantum phase transitions in multiband topological insulators and connections with entanglement entropy and fidelity susceptibility

We investigate the Loschmidt amplitude and dynamical quantum phase transitions in multiband one dimensional topological insulators. For this purpose we introduce a new solvable multiband model based on the Su-Schrieffer-Heeger model, generalized to unit cells containing many atoms but with the same symmetry properties. Such models have a richer structure of dynamical quantum phase transitions than the simple two-band topological insulator models typically considered previously, with both quasiperiodic and aperiodic dynamical quantum phase transitions present. Moreover the aperiodic transitions can still occur for quenches within a single topological phase. We also investigate the boundary contributions from the presence of the topologically protected edge states of this model. Plateaus in the boundary return rate are related to the topology of the time evolving Hamiltonian, and hence to a dynamical bulk-boundary correspondence. We go on to consider the dynamics of the entanglement entropy generated after a quench, and its potential relation to the critical times of the dynamical quantum phase transitions. Finally, we investigate the fidelity susceptibility as an indicator of the topological phase transitions, and find a simple scaling law as a function of the number of bands of our multiband model which is found to be the same for both bulk and boundary fidelity susceptibilities.

Multiple Transient Transitions Behavior Analysis of a Double Memristor’s Hidden System and Its Circuit

In this paper, a low-dimensional hidden nonlinear system is constructed by replacing two linear resistors of a simple integrating circuit with two active memristors. The system was analyzed in detail by using the Lyapunov exponent, 0–1 Test, Poincare map, phase diagram, power spectral density diagram, time-domain waveform, and chaotic characteristic diagram. The results show that the system can oscillate by itself under zero initial conditions, and there are various transient transition behaviors, such as from chaos to limit cycle, chaos to quasi-periodic, quasi-periodic to another quasi-periodic transition, quasi-periodic to periodic transition. Besides, these transient processes themselves include 2, 3, 4, and 5 different states respectively, showing multiple transient transitions behavior, which are not reported in the related literature. It is also found that the initial states of these transition states have rich symmetrical attractors and the stable state is multistability. In addition, the global entropy analysis method is adopted to test the universal existence of transient behaviors, which demonstrates that the system has rich nonlinear characteristics. Finally, the memristive circuit verifies the system’s physical feasibility and enriches the application of memristors in circuit theory.

Weyl Semimetal to Metal Phase Transitions Driven by Quasiperiodic Potentials.

We explore the stability of three-dimensional Weyl and Dirac semimetals subject to quasiperiodic potentials. We present numerical evidence that the semimetal is stable for weak quasiperiodic potentials, despite being unstable for weak random potentials. As the quasiperiodic potential strength increases, the semimetal transitions to a metal, then to an "inverted" semimetal, and then finally to a metal again. The semimetal and metal are distinguished by the density of states at the Weyl point, as well as by level statistics, transport, and the momentum-space structure of eigenstates near the Weyl point. The critical properties of the transitions in quasiperiodic systems differ from those in random systems: we do not find a clear critical scaling regime in energy; instead, at the quasiperiodic transitions, the density of states appears to jump abruptly (and discontinuously to within our resolution).

Quasiperiodic Quantum Ising Transitions in 1D.

Unlike random potentials, quasiperiodic modulation can induce localization-delocalization transitions in one dimension. In this Letter, we analyze the implications of this for symmetry breaking in the quasiperiodically modulated quantum Ising chain. Although weak modulation is irrelevant, strong modulation induces new ferromagnetic and paramagnetic phases which are fully localized and gapless. The quasiperiodic potential and localized excitations lead to quantum criticality that is intermediate to that of the clean and randomly disordered models with exponents of ν=1^{+} (exact) and z≈1.9, Δ_{σ}≈0.16, and Δ_{γ}≈0.63 (up to logarithmic corrections). Technically, the clean Ising transition is destabilized by logarithmic wandering of the local reduced couplings. We conjecture that the wandering coefficient w controls the universality class of the quasiperiodic transition and show its stability to smooth perturbations that preserve the quasiperiodic structure of the model.

Chaotic Dynamics of a Josephson Junction with a Ratchet Potential and Current-Modulating Damping

The chaotic dynamics of a Josephson junction with a ratchet potential and current-modulating damping are studied. Under the first-order approximation, we construct the general solution of the first-order equation whose boundedness condition contains the famous Melnikov chaotic criterion. Based on the general solution, the incomputability and unpredictability of the system’s chaotic behavior are discussed. For the case beyond perturbation conditions, the evolution of stroboscopic Poincare sections shows that the system undergoes a quasi-periodic transition to chaos with an increasing intensity of the rf-current. Through a suitable feedback controlling strategy, the chaos can be effectively suppressed and the intensity of the controller can vary in a large range. It is also found that the current between the two separated superconductors increases monotonously in some specific parameter spaces.

Chaotic Dynamics of Inner Ear Hair Cells

Experimental records of active bundle motility are used to demonstrate the presence of a low-dimensional chaotic attractor in hair cell dynamics. Dimensionality tests from dynamic systems theory are applied to estimate the number of independent variables sufficient for modelling the hair cell response. Poincaré maps are constructed to observe a quasiperiodic transition from chaos to order with increasing amplitudes of mechanical forcing. The onset of this transition is accompanied by a reduction of Kolmogorov entropy in the system and an increase in transfer entropy between the stimulus and the hair bundle, indicative of signal detection. A simple theoretical model is used to describe the observed chaotic dynamics. The model exhibits an enhancement of sensitivity to weak stimuli when the system is poised in the chaotic regime. We propose that chaos may play a role in the hair cell’s ability to detect low-amplitude sounds.

Periodic to quasi-periodic graphene superlattice transition by Fermi velocity modulation

Graphene superlattices have attracted great research interest in the last years, since it is possible to manipulate the electronic properties of graphene in these structures. They can be periodic, quasi-periodic or disordered, each one presenting different electronic properties. In this work we investigate the possibility of transforming a periodic and quasi-periodic graphene superlattice one into another by a Fermi velocity modulation. We analyzed finite and infinite superlattices. We considered an one-dimensional superlattice and show that the transformation can be perfectly done for charge carriers moving in the direction of the superlattice. However, for oblique moving, some features can not be transformed by the Fermi velocity. The results obtained here are relevant for the development of novel graphene-based electronic devices.

The Dynamics of Truncated Black Hole Accretion Disks. I. Viscous Hydrodynamic Case

Abstract Truncated accretion disks are commonly invoked to explain the spectro-temporal variability in accreting black holes in both small systems, i.e., state transitions in galactic black hole binaries (GBHBs), and large systems, i.e., low-luminosity active galactic nuclei (LLAGNs). In the canonical truncated disk model of moderately low accretion rate systems, gas in the inner region of the accretion disk occupies a hot, radiatively inefficient phase, which leads to a geometrically thick disk, while the gas in the outer region occupies a cooler, radiatively efficient phase that resides in the standard geometrically thin disk. Observationally, there is strong empirical evidence to support this phenomenological model, but a detailed understanding of the dynamics of truncated disks is lacking. We present a well-resolved viscous, hydrodynamic simulation that uses an ad hoc cooling prescription to drive a thermal instability and, hence, produce the first sustained truncated accretion disk. With this simulation, we perform a study of the dynamics, angular momentum transport, and energetics of a truncated disk. We find that the time variability introduced by the quasi-periodic transition of gas from efficient cooling to inefficient cooling impacts the evolution of the simulated disk. A consequence of the thermal instability is that an outflow is launched from the hot/cold gas interface, which drives large, sub-Keplerian convective cells into the disk atmosphere. The convective cells introduce a viscous θ − ϕ stress that is less than the generic r − ϕ viscous stress component, but greatly influences the evolution of the disk. In the truncated disk, we find that the bulk of the accreted gas is in the hot phase.

Chaotic Behavior of Oscillatory Hair Cells

The inner ear is capable of detecting sounds that elicit motions below the stochastic noise levels. Hair cells are specialized sensory cells essential for the hearing process. They convert mechanical energy from incoming sound into currents, by opening and closing mechanically sensitive ion channels in response to the induced deflections. Hair cells of certain species are also known to oscillate without external stimulation. The role of these spontaneous oscillations is not understood, but they are believed to be a signature of an underlying active mechanism. As this active process constitutes one of the most important open topics in auditory research, a deeper understanding of spontaneous motility could have important implications on understanding the extreme sensitivity of hearing. Hair cells have been shown to flicker between the oscillatory and quiescent states; this phenomenon was modeled with dynamic feedback acting on an internal control parameter that determines the dynamic state of the cell. This simulation predicted a positive Lyapunov exponent for innate bundle motility, indicating the presence of chaos. We will present experimental measurements of spontaneous hair bundle oscillations, which were obtained from the sacculus of the American bullfrog. Using the delay-coordinate technique, the phase space of the oscillator was reconstructed, allowing for estimation of the number of state variables required for modeling the system. Using Poincaré maps, we observed a quasiperiodic transition from low dimensional chaos to order, as the amplitude of an applied mechanical stimulus was increased. This transition was accompanied by a reduction in Kolmogorov entropy, a quantity used to determine the degree of disorder in a dynamical system.

Chaotic sedimentation of particle pairs in a vertical channel at low Reynolds number: Multiple states and routes to chaos

The sedimentation of a pair of rigid circular particles in a two-dimensional vertical channel containing a Newtonian fluid is investigated numerically, for terminal particle Reynolds numbers (ReT) ranging from 1 to 10, and for a confinement ratio equal to 4. While it is widely admitted that sufficiently inertial pairs should sediment by performing a regular DKT oscillation (Drafting-Kissing-Tumbling), the present analysis shows in contrast that a chaotic regime can also exist for such particles, leading to a much slower sedimentation velocity. It consists of a nearly horizontal pair, corresponding to a maximum effective blockage ratio, and performing a quasiperiodic transition to chaos while increasing the particle weight. For less inertial regimes, the classical oblique doublet structure and its complex behavior (multiple stable states and hysteresis, period-doubling cascade and chaotic attractor) are recovered, in agreement with previous work [Aidun, C. K. and Ding, E.-J., “Dynamics of particle sedimentation in a vertical channel: Period-doubling bifurcation and chaotic state,” Phys. Fluids 15, 1612 (2003)]. As a consequence of these various behaviors, the link between the terminal Reynolds number and the non-dimensional driving force is complex: it contains several branches displaying hysteresis as well as various bifurcations. For the range of Reynolds number considered here, a global bifurcation diagram is given.