Quasi-periodic Oscillators Research Articles

Reduced-order dynamics of coupled self-induced stochastic resonance quasiperiodic oscillators driven by varying noise intensities

Reduced-order dynamics of coupled self-induced stochastic resonance quasiperiodic oscillators driven by varying noise intensities

Thermo-magnetic convection and convective magneto-gravitational parametric resonance in a horizontal ferrofluid layer under time quasi-periodic vibrations: A balance harmonic analysis

Convective instability in a horizontal ferromagnetic liquid layer with rigid boundaries is investigated in the presence of both an uniform magnetic field and a vertical quasi-periodic forcing having two incommensurate frequencies ω1 and ω2. A linearized convective instability analysis predicting the onset of convection is performed and by means of a Galerkin projection truncated to the first order, the linear dynamic of the system is reduced to a damped quasi-periodic Mathieu oscillator. Using a balance harmonic analysis, it turns out that at sufficiently low vibration frequencies, we recover the critical instability parameters related to the unmodulated case where the convection is mainly due to the magnetic mechanism. This feature means that the buoyancy mechanism does not contribute into the convective cells formation and consequently heating from below or from above does not alter the onset of convection. However, at sufficiently high vibration frequencies, both the magnetic and buoyancy mechanisms are operative and here we are dealing with a magneto-gravitational parametric resonance convection. The transition between these convective modes is occurred via bicritical states giving rise to cusp points in the magnetic Rayleigh number besides discontinuities in its corresponding critical wave number. In addition, it is shown that a proper tuning of the irrational frequency ratio ω=ω2/ω1 provides a control of the convection onset as well as the nature of its mechanism. Moreover, special attention is devoted to highlight the effect of the Prandtl number on the dynamic of the system at different values of the frequency ratio ω.

Dynamic perturbation analysis of fractional order differential quasiperiodic Mathieu equation

The paper investigates the influence of parameters on the stability of fractional order differential quasiperiodic Mathieu equations. First, we use the perturbation method to obtain approximate expressions (i.e., transition curves) for the stability and unstable region boundaries of the equation. After obtaining the approximate expression of the transition curve, we use Lyapunov's first method to analyze the stability of the fractional order differential quasiperiodic Mathieu system, thereby obtaining the conditions for the stability of the fractional order differential quasiperiodic Mathieu equation system. Second, by comparing the approximate expressions of the transition curve of the steady-state periodic solution of the quasiperiodic Mathieu oscillator under different parameter conditions, we obtained the conclusion that the fractional order differential term exists in the form of equivalent stiffness and equivalent damping in the fractional order differential quasiperiodic Mathieu system. By comparison, we have summarized the general forms of equivalent linear damping and equivalent stiffness of the system. Through this general form, we can define an approximate expression for the thickness of unstable regions to better study the characteristics of fractional order differential quasiperiodic Mathieu systems. Finally, the influence of the parameters of the fractional order differential quasiperiodic Mathieu equation on the transition curve of the equation was intuitively analyzed through numerical simulation, to analyze the stability changes in the equation.

Effect of Horizontal Quasi-Periodic Oscillation on the Interfacial Instability of Two Superimposed Viscous Fluid Layers in a Vertical Hele-Shaw Cell

We investigate the effect of horizontal quasi-periodic oscillation on the stability of two superimposed immiscible fluid layers confined in a horizontal Hele-Shaw cell. To approximate real oscillations, a quasi-periodic oscillation with two incommensurate frequencies is considered. Thus, the linear stability analysis leads to a quasi-periodic oscillator, with damping, which describes the evolution of the amplitude of the interface. Two types of quasi-periodic instabilities occur: the low-wavenumber Kelvin-Helmholtz instability and the large-wavenumber resonances. We mainly show that, for equal amplitudes of the superimposed accelerations, and for a low irrational frequency ratio, there is competition between several resonance modes allowing a very large selection of the wavenumber from lower to higher values. This is a way to control the sizes of the waves. Furthermore, increasing the frequency ratio has a stabilizing effect for both types of instability whose thresholds are found to correspond to quasi-periodic solutions using the frequency spectrum. For a ratio of the two superimposed displacement amplitudes equal to unity and less than unity, the number of resonances and competition between their modes also become significant for the intermediate values of the ratio of frequencies. The effects of other physical and geometrical parameters, such as the damping coefficient, density ratio, and heights of the two fluid layers, are also examined.

Emergence of extreme events in a quasiperiodic oscillator.

Extreme events are unusual and rare large-amplitude fluctuations can occur unexpectedly in nonlinear dynamical systems. Events above the extreme event threshold of the probability distribution of a nonlinear process characterize extreme events. Different mechanisms for the generation of extreme events and their prediction measures have been reported in the literature. Based on the properties of extreme events, such as those that are rare in the frequency of occurrence and extreme in amplitude, various studies have shown that extreme events are both linear and nonlinear in nature. Interestingly, in this Letter, we report on a special class of extreme events which are nonchaotic and nonperiodic. These nonchaotic extreme events appear in between the quasiperiodic and chaotic dynamics of the system. We report the existence of such extreme events with various statistical measures and characterization techniques.

Embedding classical dynamics in a quantum computer

We develop a framework for simulating measure-preserving, ergodic dynamical systems on a quantum computer. Our approach provides an operator-theoretic representation of classical dynamics by combining ergodic theory with quantum information science. The resulting quantum embedding of classical dynamics (QECD) enables efficient simulation of spaces of classical observables with exponentially large dimension using a quadratic number of quantum gates. The QECD framework is based on a quantum feature map that we introduce for representing classical states by density operators on a reproducing kernel Hilbert space, $\mathcal{H}$. Furthermore, an embedding of classical observables into self-adjoint operators on $\mathcal{H}$ is established, such that quantum mechanical expectation values are consistent with pointwise function evaluation. In this scheme, quantum states and observables evolve unitarily under the lifted action of Koopman evolution operators of the classical system. Moreover, by virtue of the reproducing property of $\mathcal{H}$, the quantum system is pointwise-consistent with the underlying classical dynamics. To achieve a quantum computational advantage, we project the state of the quantum system onto a finite-rank density operator on a ${2}^{n}$-dimensional tensor product Hilbert space associated with $n$ qubits. By employing discrete Fourier-Walsh transforms of spectral functions, the evolution operator of the finite-dimensional quantum system is factorized into tensor product form, enabling implementation through an $n$-channel quantum circuit of size $O(n)$ and no interchannel communication. Furthermore, the circuit features a state preparation stage, also of size $O(n)$, and a quantum Fourier transform stage of size $O({n}^{2})$, which makes predictions of observables possible by measurement in the standard computational basis. We prove theoretical convergence results for these predictions in the large-qubit limit, $n\ensuremath{\rightarrow}\ensuremath{\infty}$. In light of these properties, QECD provides a consistent simulator of the evolution of classical observables, realized through projective quantum measurement, which is able to simulate spaces of classical observables of dimension ${2}^{n}$ using circuits of size $O({n}^{2})$. We demonstrate the consistency of the scheme in prototypical dynamical systems involving periodic and quasiperiodic oscillators on tori. These examples include simulated quantum circuit experiments in Qiskit Aer, as well as actual experiments on the IBM Quantum System One.

Auto-correlation functions of astrophysical processes, and their relation to Gaussian processes

Context. Accounting for the effects of stellar magnetic phenomena is indispensable to fully exploit radial velocities (RVs) obtained using modern exoplanet-hunting spectrometers. Correlated time variations are often mitigated by non-trivial noise models in the framework of Gaussian processes. These models rely on fitting kernel functions that are motivated on mathematical grounds, and whose physical interpretation is often elusive. Aims. We aim to establish a clear connection between stellar magnetic activity affecting RVs and their corresponding correlations with physical parameters, and compare this connection with kernels used in the literature. Methods. We use simple activity models to investigate the relationship between the physical processes generating the signals and the covariances typically found in data, and to demonstrate the qualitative behaviour of this relationship. We use the StarSim code to calculate RVs of an M dwarf with different realistic evolving spot configurations. The auto-correlation function (ACF) of a synthetic data set shows a very specific behaviour and is explicitly related to the kernel. Gaussian process regression is performed using the quasi-periodic (QP) and simple harmonic oscillator kernels of the george and celerite codes, respectively. Comparison of the resulting kernels with the exact ACFs allows us to cross-match the kernel hyper-parameters with the introduced physical values, study the overall capabilities of the kernels, and improve their definition. Results. We find that the QP kernel provides a more straightforward interpretation of the physics. It is able to consistently recover both the introduced rotation period Prot and the spot lifetime. Our study indicates that the performance can be enhanced by fixing the form factor w and adding a physically motivated cosine term with period Prot∕2, where the contribution to the ACF for the different spot configurations differs significantly. The newly proposed quasi-periodic with cosine (QPC) kernel leads to significantly better model likelihoods, can potentially distinguish between different spot configurations, and can thereby improve the sensitivity of RV exoplanet searches.

Multiplicative signal induces double and quasi-periodic vibrational resonance in the overdamped oscillator

The vibrational resonance in the overdamped oscillator with the triple-well potential driven by a multiplicative low-frequency force and an additive high-frequency force is investigated by numerical simulation. The double VR and quasi-periodic oscillator are observed. There exists a resonant range, which is increased with the amplitude and frequency of the high-frequency force, but the response amplitude decreases as the high-frequency force becomes strong.

Effect of the localization on the response of a quasi-periodic electromagnetic oscillator array for vibration energy harvesting

Vibration energy harvesting by exploiting the multimodal approach in a quasi-periodic system is proposed. The quasi-periodic system, based on electromagnetic transduction, consists of two weakly coupled magnets mechanically guided by two elastic beams. Mistuning is achieved by varying the mechanical stiffness of one of the beams. These imperfections will lead to the vibration energy localization in regions close to the imperfections which will be exploited to maximize the harvested energy.

Peculiar Emission Line Generation from Ultra-Rapid Quasi-Periodic Oscillations of Exotic Astronomical Objects

The purpose of this article is to alert Astronomers, particularly those using spectroscopic surveys, to the fact that exotic astronomical objects (e.g. quasars or active galactic nuclei) that send ultra-rapid quasi periodic pulses of optical light would generate spectroscopic features that look like emission lines. This gives a simple technique to find quasi periodic pulses separated by times smaller than a nanosecond. One should look for emission lines that cannot be identified with known spectral lines in spectra. Such signals, generated by slower pulses, could also be found in the far infra-red, millimeter and radio regions, where they could be detected as objects unusually bright in a single narrow-band filter or channel. The outstanding interest of the technique comes from its simplicity so that it can be used to find ultra-rapid quasi-periodic oscillators in large astronomical surveys. A very small fraction of objects presently identified as Lyman alpha emitters that do not have other spectral features to confirm the Lyman alpha redshift, may possibly be quasi-periodic oscillators. However this is only a hypothesis that needs more observations for confirmation.

Effect of vertical quasi-periodic vibrations on the stability of the free surface of a fluid layer

In this paper we examine the effect of vertical quasi-periodic oscillations on the stability of the free surface of a horizontal liquid layer. The quasi-periodic motion considered here is characterized by two incommensurate frequencies, $ \Omega_{1}$ and $ \Omega_{2}$ , i.e. the ratio $ \omega=\Omega_{2}/\Omega_{1}$ is irrational. The linear quasi-periodic oscillator, corresponding to the governing equations of the Faraday instability, is treated using the harmonic balance method developed by Rand et al. and Zounes and Rand, and by numerical methods using the Floquet analysis and knowing that an irrational number can be approximated by a rational number. We determine the marginal stability curves in terms of reduced amplitude forcing and wave number for inviscid and viscous liquids. For both cases, we show that the neutral stability curves depend strongly on the frequency ratio of oscillations, $ \omega$ , when this parameter is below $ \sqrt{2}$ . Beyond this value, there is no effect of $ \omega$ and the first resonance occurs always at the wave number corresponding to the value of the natural frequency squared, $ \delta=0.25$ . Below the value $ \omega=\sqrt{2}$ , variations of the wave number as a function of $ \omega$ and $ \Omega_{1}^{}$ are presented for the inviscid case. However, for the viscous case, we show the existence of the bicritical points and we present the instability threshold versus $ \Omega_{1}^{}$ for different values of $ \omega$ .

Synchronization of multi-frequency noise-induced oscillations

Using a model system of FitzHugh-Nagumo type in the excitable regime, the similarity between synchronization of self-sustained and noise-induced oscillations is studied for the case of more than one main frequency in the spectrum. It is shown that this excitable system undergoes the same frequency lockings as a self-sustained quasiperiodic oscillator. The presence of noise-induced both stable and unstable limit cycles and tori, as well as their tangential bifurcations, are discussed. As the FitzHugh-Nagumo oscillator represents one of the basic neural models, the obtained results are of high importance for neuroscience.

Neutron star equation of state

New observations of neutron stars from sources such as radio pulsars, X-ray binaries, quasi-periodic oscillators, X-ray bursters and thermally-emitting isolated neutron stars, can ultimately constrain the dense matter equation of state. Observable quantities include neutron star masses, radii, rotation rates, radiation radii, redshifts, moments of inertia, temperatures and ages. Several ways in which these quantities can lead to theoretical limits are discussed.

Exponential averaging under rapid quasiperiodic forcing

We derive estimates on the magnitude of the interaction between a wide class of analytic partial differential equations and a high-frequency quasiperiodic oscillator. Assuming high regularity of initial conditions, the equations are transformed to an uncoupled system of an infinite dimensional dynamical system and a linear quasiperiodic flow on a torus; up to coupling terms which are exponentially small in the smallest frequency of the oscillator. The main technique is based on a careful balance of similar results for ordinary differential equations by Simó, Galerkin approximations and high regularity of the initial conditions. Similar finite order estimates assuming less regularity are also provided. Examples include reaction-diffusion and non-linear Schrödinger equations.

The quasi-periodic Hamiltonian Hopf bifurcation

We consider the quasi-periodic dynamics of non-integrable perturbations of a family of integrable Hamiltonian systems with normally 1 : −1 resonant invariant tori. In particular, we focus on the supercritical quasi-periodic Hamiltonian Hopf bifurcation and address the persistence problem of the singular foliation into invariant quasi-periodic tori near the bifurcation point. With the help of KAM theory, we show that the singular torus foliation survives a small perturbation and that the persisting tori in this foliation form Cantor families. A leading example is the Lagrange top near gyroscopic stabilization weakly coupled with a quasi-periodic oscillator.

Unwrapping trajectories of a quasi-periodic forced oscillator can give maps of the real line with SNA

Abstract We start from a simple model of quasi-periodic forced oscillator and show how it is possible to find corresponding maps of the two dimension torus. One of these constructions is frequently used to analyse the apparition of strange non-chaotic attractors (SNA in brief). SNA are complex attractors, their trajectories seem to be chaotic but without the sensitivity to initial conditions. Some trajectories can be unwrapped from the torus giving a simple iteration of the real line. It leads us to suppose that iterations from a simple continuous increasing map with quasi-periodic displacement can exhibit SNA.

Bursters and quasi-periodic solutions of a self-excited quasi-periodic Mathieu oscillator

Abstract In this paper the conditions of occurrence of quasi-periodic (QP) solutions and bursting dynamics in a self-excited quasi-periodic Mathieu Oscillator are discussed. The quasi-periodic excitation consists of two periodic excitations; one with a very slow frequency and the other with a frequency resonant with the proper frequency of the oscillator. The fast dynamics are initially averaged. The complimentary quasi-static solutions of the modulation equations of amplitude and phase are determined and their stability is analyzed. Numerical simulations and power spectra are shown to complete the theoretical analysis.

Dynamics in coupled oscillators with recurrent/random forcing, a PDE approach

The current paper is devoted to the study of coupled oscillators with recurrent/random forcing. Special attention is given to the solutions having the same recurrence/randomness as that of the forcing (recurrent/random solutions for short). By embedding coupled oscillators into coupled parabolic equations, it establishes a general theorem on the existence of recurrent/random solutions. It also finds conditions under which such solutions are unique. When the recurrent forcing is actually quasi-periodic or almost periodic, recurrent solutions are refereed to as quasi-periodic or almost periodic solutions in a weak sense and they are quasi-periodic or almost periodic in the classical sense under the uniqueness conditions. In addition, applications of the general theory to coupled Duffing type oscillators and Josephson junctions are considered and the results obtained extend several existing ones for quasi-periodic Duffing oscillators.

Analog computation with rings of quasiperiodic oscillators: the microdynamics of cognition in living machines

We describe experimental results to demonstrate the wide-ranging computational ability of quasiperiodic oscillators built from rings of differentiating Schmitt triggers. We describe a theoretical model based on necklace functions to compute the number of states supportable by a ring circuit of a given size. Experimental results are presented to demonstrate that probabilistic state machines can be built from these ring circuits. Other experimental results are given to demonstrate that the rings can model spiking neural network circuits.

Direction of coupling from phases of interacting oscillators: an information-theoretic approach.

A directionality index based on conditional mutual information is proposed for application to the instantaneous phases of weakly coupled oscillators. Its abilities to distinguish unidirectional from bidirectional coupling, as well as to reveal and quantify asymmetry in bidirectional coupling, are demonstrated using numerical examples of quasiperiodic, chaotic, and noisy oscillators, as well as real human cardiorespiratory data.