Floquet Theory Research Articles

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Electronic states bound by repulsive potentials in graphene irradiated by a circularly polarized electromagnetic field

In the framework of the Floquet theory of periodically driven quantum systems, it is demonstrated that irradiation of graphene by a circularly polarized electromagnetic field induces an attractive area in the core of repulsive potentials. Consequently, the quasi-stationary electron states bound by the repulsive potentials appear. The difference between such field-induced states in graphene and usual systems with the parabolic dispersion of electrons is discussed and possible manifestations of these states in electronic transport and optical spectra of graphene are considered.

Microwave-optical spectroscopy of Rydberg excitons in the ultrastrong driving regime

Abstract We study the ultrastrong driving of Rydberg excitons in Cu$_2$O by a microwave field. The effect of the field is studied using optical absorption spectroscopy, and through the observation of sidebands on the transmitted laser light. A model based on Floquet theory is constructed to study the system beyond the rotating wave approximation. We obtain near quantitative agreement between theory and experiment across a 16-fold range of microwave field strengths spanning from the perturbative to the deep strong driving regime. Compared to Rydberg atoms, the large non-radiative widths of Rydberg excitons leads to new behaviour such as the emergence of an absorption continuum without ionization.

Sensing force gradients with cavity optomechanics while evading backaction

We study force-gradient sensing with a coherently driven mechanical resonator and phase-sensitive detection of motion through the two-tone backaction evading measurement of cavity optomechanics. The response of the optomechanical system, solved by numerical integration of the classical equations of motion, shows an extended region which is monotonic to changes in force gradient. We use Floquet theory to model the fluctuations, which rise only slightly above that of the usual backaction evading measurement in the presence of the mechanical drive. The monotonic response and minimal backaction are advantageous for applications such as atomic force microscopy. Published by the American Physical Society 2024

Stability of thermal convection in two superimposed miscible viscous fluids under gravitational modulation

Abstract The thermal stability of a system of two superimposed miscible viscous fluid layers, under gravitational modulation, is investigated using linear stability analysis. It is well known in the literature that in the absence of gravitational modulation, there exist two convection regimes depending on the buoyancy number: the single-cell regime, where convection is oscillating and occupies the entire volume of the two fluid layers, and the stratified regime where stationary convection occurs in each fluid layer. Using Chebyshev's spectral collocation method and Floquet's theory, the numerical results show that under gravitational modulation, single-cell convection oscillates at half the frequency of the external oscillation (subharmonic threshold), while the stratified regime oscillates at a frequency equal to that of the forcing (harmonic threshold). The critical buoyancy number corresponding to the transition between the two convection regimes depends, in addition to the viscosity ratio and the depth of the two layers, on certain dimensionless frequencies permeating this transition and also on the amplitude of oscillation via the Froude number. In the plane of the critical Rayleigh number versus dimensionless modulation frequency, the branch associated with stratified convection does not depend on the buoyancy number for equal viscosities whereas this is the case for the whole-convection regime. For different viscosities, both convection regimes depend on the buoyancy number. At last, in the stratified regime (harmonic threshold), the critical value of the viscosity contrast, corresponding to the passage from an active layer to a passive layer and vice versa, increases with the modulation frequency.

Raman-phonon-polariton condensation in a transversely pumped cavity

Phonon polaritons are hybrid states of light and matter that are typically realised when optically active phonons couple strongly to photons. We suggest a new approach to realising phonon polaritons, by employing a transverse-pumping Raman scheme, as used in experiments on cold atoms in optical cavities. This approach allows hybridisation between an optical cavity mode and any Raman-active phonon mode. Moreover, this approach enables one to tune the effective phonon–photon coupling by changing the strength of the transverse pumping light. We show that such a system may realise a phonon-polariton condensate. To do this, we find the stationary states and use Floquet theory to determine their stability. We thus identify distinct superradiant and lasing states in which the polariton modes are macroscopically populated. We map out the phase diagram of these states as a function of pump frequencies and strengths. Using parameters for transition metal dichalcogenides, we show that realisation of these phases may be practicably obtainable. The ability to manipulate phonon mode frequencies and attain steady-state populations of selected phonon modes provides a new tool for engineering correlated states of electrons.

EFFECT OF EXTERNAL HARMONIC VIBRATION ON THE FORMATION OF SOLITARY WAVES IN FALLING TWO-LAYER LIQUID FILMS

Abstract We study the influence of a low-frequency harmonic vibration on the formation of the two-dimensional rolling solitary waves in vertically co-flowing two-layer liquid films. The system consists of two adjacent layers of immiscible fluids with the first layer being sandwiched between a vertical solid plate and the second fluid layer. The solid plate oscillates harmonically in the horizontal direction inducing Faraday waves at the liquid–liquid and liquid–air interfaces. We use a reduced hydrodynamic model derived from the Navier–Stokes equations in the long-wave approximation. Linear stability of the base flow in a flat two-layer film is determined semi-analytically using Floquet theory. We consider sub-millimetre-thick films and focus on the competition between the long-wavelength gravity-driven and finite wavelength Faraday instabilities. In the linear regime, the range of unstable wave vectors associated with the gravity-driven instability broadens at low and shrinks at high vibration frequencies. In nonlinear regimes, we find multiple metastable states characterized by solitary-like travelling waves and short pulsating waves. In particular, we find the range of the vibration parameters at which the system is multistable. In this regime, depending on the initial conditions, the long-time dynamics is dominated either by the fully developed solitary-like waves or by the shorter pulsating Faraday waves.

Nonlinear deterministic dynamic analysis of a GPL micropipe conveying pulsating laminar flow

In several industries, the handling of fluid-filled micropipes is common. New-brand structures are increasingly being manufactured using advanced materials. This article tackles one of the crucial dynamic analyses that an advanced fluid-filled micropipe encounters. The dynamics of a graphene nanoplatelets-reinforced micropipe (GPL micropipe) under pulsating laminar flow for the first time is examined. The Euler-Bernoulli beam model follows the modified couple stress theory (MCST) and von-Karman’s nonlinear strain to formulate the problem. The impression of the flow profile exerted by a real flow as a consequence of fluid viscosity, is taken into account. The plug flow model results are compared to those regarding real flow consideration. Using the method of multiple scales (MMS), we determine the instability area and the steady state response of the GPL micropipe subjected to the principal parametric resonance of one of its modes due to pulsating laminar flow by applying this method to the discretized couple nonlinear gyroscopic governing equations. The Floquet theory treats the linear equations and fourth-order Runge–Kutta (RK4) method attacks nonlinear ones to validate MMS findings. It is confirmed that the 0.3 % addition of the GPL to matrix phase significantly enhances its advanced attributes, making it a highly effective material. The GPL reinforcement phase in the FGO pattern increases the critical flow velocity that exhibits the micropipe to static instability by 37.98 % , and critical flow stimulation amplitude fraction by 152.19 % , while declines instability area bandwidth by 36.95 % , and steady state response by 67.17 % relative to a non-reinforced micropipe. A denser flow degrades the corresponding values by 18.40 % , 42.59 % , 41.67 % , and 227.28 % in comparison to a lighter fluid. The declared findings provide crucial insights into the key design elements affecting significantly the functioning of fluid-filled GPL micropipes system, and regulate future research and expand openings for industry applications.

Operational Modal Analysis of Self-excited Vibrations in Milling Considering Periodic Dynamics

Abstract A new method is presented to identify the dynamics of regenerative chatter from measured process vibrations in milling. This method combines the synchronous once-per-revolution sampling of process vibrations with Operational Modal Analysis to estimate the Floquet multipliers of the delayed linear time-periodic dynamics in milling, all from the natural process vibrations without external excitation. The identified multipliers quantify vibration stability, enabling chatter prediction before it occurs. In addition to this, they can be used to calibrate physics-based chatter models based on vibration measurements solely within the stable region. The method's accuracy in identifying Floquet multipliers is validated through extensive numerical simulations and two experimental case studies. The results show that chatter due to both Hopf and period-doubling bifurcations can be predicted from the process vibrations during stable cuts. Moreover, the experimental case studies demonstrate a vibration measurement system for implementing the presented method in standard milling operations and confirm its effectiveness in practice.

Model reduction of high-dimensional self-excited nonlinear systems using floquet theory based parameterization method

Model reduction of high-dimensional self-excited nonlinear systems using floquet theory based parameterization method

Non-Bloch band theory for time-modulated discrete mechanical systems

This study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass–spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass–spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. This novel approach, the combination of the temporal Floquet theory for time-modulated systems and generalized Brillouin zone, enables the prediction of eigenvalue distribution under open boundary conditions and also quantitative characterization of non-Hermitian skin modes. The proposed theory is verified by some numerical experiments.

Self-resonance during preheating: The case of [formula omitted]-attractor models

In this paper, for the first time, we obtain a new class of solutions for the Hill-type differential equations, which emerge in the preheating self-resonance phase of the expanding Universe. We study, in particular, the class of symmetric and asymmetric scalar field potentials coming from the so-called α-attractor models of the early Universe cosmology. By making a series expansion of the potential and employing perturbative techniques we reformulate the Mukhanov–Sasaki equation, which captures the dynamics of the curvature perturbation in these models, into a Hill equation. This last includes higher-order terms that were never solved in the literature. Namely, those coming from the cubic and quartic contributions of the scalar field potential. Then, we derive the expressions for the Floquet exponents of the Mukhanov–Sasaki variable. Our analytical results are then compared with numerical computations, showing a good agreement and thus making this method valuable for obtaining theoretical predictions with new observational applications in the contexts of Primordial Black Holes and Scalar-Induced Gravitational Waves.

Dynamical systems theory applied to short walking trials

Human walking is an extremely complex neuromuscular activity whose simplicity disappears when an attempt is made to provide a quantitative description of the process. The dynamical systems theory provides a framework for analyzing the stability and chaotic nature of dynamical systems, employing Floquet multipliers (FM) and long and short-term Lyapunov exponents (LE), respectively. This report compares FM and LE from three methods: method A (false nearest neighbors and numerical approximation), method B (false nearest neighbors and semi-analytical technique) and method C (singular value decomposition and semi-analytical technique). Data from 33 healthy older adults with no history of falls were used to explain the dynamic system. A surrogate center of mass trajectory was calculated for the analysis of sway in the transverse plane. Results revealed methodological differences in LE and FM calculations with semi-analytical solutions providing closer approximations to observed gait behavior. The long-term LE from Methods A and B were similar, but other LE pairings differed. Method A’s short-term LE indicated chaotic gaits for all subjects, while long-term LE from Methods A and B indicated chaos for half the subjects. Method C showed non-chaotic gait for most subjects. Method B’s FM indicated over 30% of subjects had unstable gait. Method C yielded values of LE and FM that most closely matched the subjects’ gait patterns. This study offers a methodological foundation for gait analysis using short time-series data, facilitating deeper insights into both stability and chaos within gait dynamics.

Parametric oscillations of the sessile drop

Waves are formed on the surface of a sessile drop driven through substrate vibrations oriented at a slanting angle from the normal. A mathematical model is derived, which leads to an infinite system of coupled Mathieu equations governing the wave dynamics that are solved using Floquet theory. The spatial structure of the waves is described by the mode number pair $[\ell,m]$ , where $\ell$ and $m$ are the polar and azimuthal mode numbers, respectively. Limiting cases corresponding to horizontal and vertical vibrations are discussed with predictions agreeing well with prior literature. We focus our results on three drop motions – (1) harmonic $[1,1]$ rocking mode, (2) harmonic $[2,0]$ pumping mode, and (3) subharmonic rocking $[1,1]$ mode – as they depend upon the slanting angle, static contact angle, and contact-line conditions, which we assume to be either pinned or freely moving with fixed contact angle. New theoretical predictions are tested through experiments over a range of parameters, showing good agreement.

Dynamic stability and response of morphing wing structure with time-varying stiffness

Abstract Morphing, adaptable or smart structures are being used in mechanical and aerospace applications in recent years. These structures often have the property of time-varying stiffness or inertial properties, which can cause parametric instability issues that are not well understood. This paper examines the dynamic stability and response of a morphing aircraft wing with periodically time-varying structural stiffness. The wing is modeled as a beam with coupled bending-torsion motion, and parametrically excited stiffness. Aerodynamic loads introduce aerodynamic damping and aerodynamic stiffness to the wing structure. The dynamic and aeroelastic equation of motion resembles a coupled, damped Mathieu-type equation but differs with asymmetric damping and stiffness matrices, and symmetric inertial matrix. Further, these equations are functions of airspeed, magnitude and frequency of parametric excitation. Initially, dynamic stability of the wing is analyzed using Floquet theory, and the instability regions are numerically quantified by stability charts. Subsequently, dynamic responses in the stable and unstable regions are investigated with a Floquet-based Harmonic balance method. The findings reveal that, at zero airspeed, the combination instabilities are eliminated by varying the bending and torsional stiffness with equal magnitudes and frequency. However, as airspeed increases, instability regions shift unevenly, leading to the emergence of new instabilities. Furthermore, the response analysis within stable regions uncovers several unfavorable zones for operating the variable stiffness, where response decay is slower. The results clearly show that parametric excitation can cause unusual phenomena that significantly impact the operation of morphing wings with variable stiffness, which needs to be thoroughly investigated for successful implementation.

Cascades of heterodimensional cycles via period doubling

A heterodimensional cycle is formed by the intersection of stable and unstable manifolds of two saddle periodic orbits that have unstable manifolds of different dimensions: connecting orbits exist from one periodic orbit to the other, and vice versa. The difference in dimensions of the invariant manifolds can only be achieved in vector fields of dimension at least four. At least one of the connecting orbits of the heterodimensional cycle will necessarily be structurally unstable, meaning that is does not persist under small perturbations. Nevertheless, the theory states that the existence of a heterodimensional cycle is generally a robust phenomenon: any sufficiently close vector field (in the C1-topology) also has a heterodimensional cycle.We investigate a particular four-dimensional vector field that is known to have a heterodimensional cycle. We continue this cycle as a codimension-one invariant set in a two-parameter plane. Our investigations make extensive use of advanced numerical methods that prove to be an important tool for uncovering the dynamics and providing insight into the underlying geometric structure. We study changes in the family of connecting orbits as two parameters vary and Floquet multipliers of the periodic orbits in the heterodimensional cycle change. In particular the Floquet multipliers of one of the periodic orbits change from real positive to real negative prior to a period-doubling bifurcation. We then focus on the transitions that occur near this period-doubling bifurcation and find that it generates new families of heterodimensional cycles with different geometric properties. Our careful numerical study suggests that further two-parameter continuation of the ‘period-doubled heterodimensional cycles’ gives rise to an abundance of heterodimensional cycles of different types in the limit of a period-doubling cascade.Our results for this particular example vector field make a contribution to the emerging bifurcation theory of heterodimensional cycles. In particular, the bifurcation scenario we present can be viewed as a specific mechanism behind so-called stabilisation of a heterodimensional cycle via the embedding of one of its constituent periodic orbits into a more complex invariant set.

Floquet-Engineering Topological Phase Transition in Graphene Nanoribbons by Light

Abstract Quasi-one-dimensional (1D) graphene nanoribbons (GNRs) play a crucial role in advancement of next-generation devices. Recent studies have suggested their potential to exhibit unique symmetry-protected topological phases defined by a Z 2 invariant. By employing both the tight-binding model and the Floquet theory, our investigation demonstrates the effective control of the topological phase within quasi-1D armchair GNRs (AGNRs) using elliptically polarized light, unveiling rich topological phase diagrams. Specifically, we observe that varying the amplitude of the light can induce transitions in the band gap (E g) of AGNRs, leading to multiple changes in the system’s Z 2 invariant. Furthermore, for heterojunctions composed of different AGNR segments, the junction state can be either created or eliminated by the application of elliptically polarized light.

Faraday instability of viscous liquid films on a heated substrate with Maxwell–Cattaneo heat flux

Faraday instability of viscous liquid films with Maxwell–Cattaneo (MC) heat flux on an infinite, heated horizontal substrate subject to vertical time-varying periodic vibration is investigated theoretically. The MC effect means that the response of the heat flux to a temperature gradient obeys a relaxation time law rather than a classical Fourier time law. Applying the classic Floquet theory to linear analysis, a recursive relation is obtained. When considering the MC effect, a new phenomenon appears at a large wave number k. The neutral stability curves branch new tongues that turn left rather than right as before, but the tongues still move up and right as the wave number increases. Furthermore, typical harmonic (H) and subharmonic (SH) alternation behavior continues to exist. Interestingly, the first tongue of a branch is H or SH, implying that there is a transition following the branches. However, near the critical wave number kc of a branch, the SH and H almost overlap. As Cattaneo number C increases, the tongue-like unstable zones of branches become wider, and the critical wave number kc of the appeared branch becomes small. As the driving frequency ω decreases, the branch tongues become elongated and the critical wave number kc of the appeared branch becomes small.

Dynamic characteristics analysis and vibration control of coupled bending-torsional system for axial flow hydraulic generating set

Aiming at the vibration problems of shaft system for axial flow hydraulic generating sets caused by complex excitations, the coupled bending-torsional dynamic equations of rotor-runner system are established considering multi-effect including hydraulic, mechanical, electrical and other external vibration sources. Whether taking the torsional degree of freedom into account, the stability and disparate destabilization laws of system periodic solutions are analyzed using the shooting method combined with Floquet theory. On this basis, the influence of coupling bending-torsional effect on system dynamic behaviors is discussed through bifurcation diagrams, Poincaré maps, trajectories, time domains and frequency spectrums. Furthermore, in order to alleviate the unsteady vibration of rotor as well as runner, the magnetorheological fluid damper is introduced into the shaft system, and numerical simulations are performed on models of coupled bending-torsional systems with or without the magnetorheological fluid damper. The analyses indicate that the interaction between bending and torsional vibration of the shaft system will expand the chaotic sustained range for rotor. Meanwhile, the amplitude of the rotor-runner system is magnified owing to the torsional vibration, which leads to local rubbing faults of the system under specific operating conditions, while the intervention of the magnetorheological fluid damper can significantly suppress the nonlinear motion of rotor and runner, reduce the amplitude of system vibration, so as to avoid the occurrence of friction defects effectively. The above research results can provide conducive references for fault diagnosis, optimized design, and vibration control of axial flow hydraulic generating sets.

Individually tunable tunnelling coefficients in optical lattices using local periodic driving

Abstract Ultracold atoms in optical lattices have emerged as powerful quantum simulators of translationally invariant systems with many applications in e.g. strongly-correlated and topological systems. However, the ability to locally tune all Hamiltonian parameters remains an outstanding goal that would enable the simulation of a wider range of quantum phenomena. Motivated by recent advances in quantum gas microscopes and optical tweezers, we here show theoretically how local control over individual tunnelling links in an optical lattice can be achieved by incorporating local time-periodic potentials. We propose to periodically modulate the on-site energy of individual lattice sites and employ Floquet theory to demonstrate how this provides full individual control over the tunnelling amplitudes in one dimension. We provide various example configurations realising interesting topological models such as extended Su–Schrieffer–Heeger models that would be challenging to realise by other means. Extending to two dimensions, we demonstrate that local periodic driving in a Lieb lattice engineers a two-dimensional (2D) network with fully controllable tunnelling magnitudes. In a three-site plaquette, we show full simultaneous control over the relative tunnelling amplitudes and the gauge-invariant flux piercing the plaquette, providing a clear stepping stone to building a fully programmable 2D tight-binding model. We also explicitly demonstrate how utilise our technique to generate a magnetic field gradient in 2D. This local modulation scheme is applicable to many different lattice geometries.