Showing papers on "Floquet theory published in 2009"
TL;DR: In this article, an analytical model is proposed to study the nonlinear dynamic behavior of rolling element bearing systems including surface defects, and the results were obtained in the form of time series, frequency responses and phase trajectories.
171 citations
TL;DR: In this article, the authors present modeling, analysis and experimental investigation for nonlinear resonances and the dynamic pull-in instability in electrostatically actuated resonators, which are induced by exciting a microstructure with nonlinear forcing composed of a dc parallel-plate electrostatic load superimposed on an ac harmonic load.
Abstract: We present modeling, analysis and experimental investigation for nonlinear resonances and the dynamic pull-in instability in electrostatically actuated resonators. These phenomena are induced by exciting a microstructure with nonlinear forcing composed of a dc parallel-plate electrostatic load superimposed on an ac harmonic load. Nonlinear phenomena are investigated experimentally and theoretically including primary resonance, superharmonic and subharmonic resonances, dynamic pull-in and the escape-from-potential-well phenomenon. As a case study, a capacitive sensor made up of two cantilever beams with a proof mass attached to their tips is studied. A nonlinear spring‐mass‐damper model is utilized accounting for squeeze-film damping and the parallel-plate electrostatic force. Long-time integration and a global dynamic analysis are conducted using a finite-difference method combined with the Floquet theory to capture periodic orbits and analyze their stability. The domains of attraction (basins of attraction) for data points on the frequency‐response curve are calculated numerically. Dover cliff integrity curves are calculated and the erosion of the safe basin of attraction is investigated as the frequency of excitation is swept passing primary resonance and dynamic pull-in. Conclusions are presented regarding the safety and integrity of MEMS resonators based on the simulated basin of attraction and the observed experimental data. (Some figures in this article are in colour only in the electronic version)
150 citations
TL;DR: In this paper, a general approach to control the coupling between a two-level system (TLS) and its surroundings, by modulating the energy-level spacing of the TLS with a high-frequency signal, is proposed.
Abstract: Using a dynamical quantum Zeno effect, we propose a general approach to control the coupling between a two-level system (TLS) and its surroundings, by modulating the energy-level spacing of the TLS with a high-frequency signal. We show that the TLS-surroundings interaction can be turned off when the ratio between the amplitude and the frequency of the modulating field is adjusted to be a zero of a Bessel function. The quantum Zeno effect of the TLS can also be observed by the vanishing of the photon reflection at these zeros. Based on these results, we propose a quantum switch to control the transport of a single photon in a one-dimensional waveguide. Our analytical results agree well with numerical results using Floquet theory.
121 citations
TL;DR: In this article, the authors apply lubrication theory to examine the possibility of using AC electric fields to exert further control over the size and shape of the pillars, and show that high frequencies may inhibit the accumulation of interfacial free charge, leading to a lowering of growth rates and wavenumbers.
Abstract: When DC electric fields are applied to a thin liquid film, the interface may become unstable and form a series of pillars. In this paper, we apply lubrication theory to examine the possibility of using AC electric fields to exert further control over the size and shape of the pillars. For perfect dielectric films, linear stability analysis shows that the influence of an AC field can be understood by considering an effective DC field. For leaky dielectric films, Floquet theory is applied to carry out the linear stability analysis, and it reveals that high frequencies may be used to inhibit the accumulation of interfacial free charge, leading to a lowering of growth rates and wavenumbers. Nonlinear simulations confirm the results of the linear stability analysis while also uncovering additional mechanisms for tuning overall pillar height and width. The results presented here may be of interest for the controlled creation of surface topographical features in applications such as patterned coatings and microelectronics.
87 citations
TL;DR: In this paper, a numerical model for predicting subway induced vibrations is presented, which fully accounts for the dynamic interaction between the train, the track, the tunnel, and the soil.
86 citations
TL;DR: In spite of the Floquet state pathologies, the explicit form of the results for the density matrix gives a consistent prescription for the statistical mechanics of periodically driven systems in the limit as N approaches infinity.
Abstract: Although the statistical mechanics of periodically driven ("Floquet") systems in contact with a heat bath has some formal analogy with the traditional statistical mechanics of undriven systems, closer examination reveals radical differences. In Floquet systems all quasienergies epsilon_{j} can be placed in a finite frequency interval 0< or =epsilon_{j}
84 citations
TL;DR: New nonequilibrium phenomena are predicted to emerge, where the optical conductivity exhibits dip and kink structures around the frequency of the pump light, a midgap absorption arising from photoinduced Floquet subbands, and a negative attenuation (gain) due to a population inversion.
Abstract: We present a framework to determine nonequilibrium steady states in strongly correlated electron systems in the presence of dissipation. This is demonstrated for a correlated electron (Falicov-Kimball) model attached to a heat bath and irradiated by an intense pump light, for which an exact solution is obtained with the Floquet method combined with the nonequilibrium dynamical mean-field theory. On top of a Drude-like peak indicative of photometallization as observed in recent pump-probe experiments, new nonequilibrium phenomena are predicted to emerge, where the optical conductivity exhibits dip and kink structures around the frequency of the pump light, a midgap absorption arising from photoinduced Floquet subbands, and a negative attenuation (gain) due to a population inversion.
83 citations
TL;DR: In this paper, a generalized Van Vleck (GVV) nearly degenerate high-order perturbation theory for the treatment of the Floquet Hamiltonian was proposed, allowing the reduction of the infinite-dimensional Floquet matrix to an $N\ifmmode\times\else\texttimes\fi{}N$ effective Hamiltonian, where $N$ is the number of eigenstates under consideration.
Abstract: We present a Floquet treatment of multiphoton quantum interference in a strongly driven superconducting flux qubit. The periodically time-dependent Schr\"odinger equation can be reduced to an equivalent time-independent infinite-dimensional Floquet matrix eigenvalue problem. For resonant or nearly resonant multiphoton transitions, we extend the generalized Van Vleck (GVV) nearly degenerate high-order perturbation theory for the treatment of the Floquet Hamiltonian, allowing the reduction of the infinite-dimensional Floquet matrix to an $N\ifmmode\times\else\texttimes\fi{}N$ effective Hamiltonian, where $N$ is the number of eigenstates under consideration. The GVV approach allows accurate treatment of ac Stark shift, power broadening, time-dependent and time-averaged transition probability, etc., well beyond the rotating wave approximation. We extend the Floquet and GVV approaches for numerical and analytical studies of the multiphoton resonance processes and quantum interference phenomena for the superconducting flux qubit system $(N=2)$ driven by intense ac fields.
78 citations
TL;DR: In this paper, an ab initio nonperturbative time-dependent $R$-matrix theory for ultrafast atomic processes is described, which enables investigations of the interaction of few-femtosecond and -attosecond pulse lasers with complex multielectron atoms and atomic ions.
Abstract: We describe an ab initio nonperturbative time-dependent $R$-matrix theory for ultrafast atomic processes. This theory enables investigations of the interaction of few-femtosecond and -attosecond pulse lasers with complex multielectron atoms and atomic ions. A derivation and analysis of the basic equations are given, which propagate the atomic wave function in the presence of the laser field forward in time in the internal and external $R$-matrix regions. To verify the accuracy of the approach, we investigate two-photon ionization of Ne irradiated by an intense laser pulse and compare current results with those obtained using the $R$-matrix Floquet method and an alternative time-dependent method. We also verify the capability of the current approach by applying it to the study of two-dimensional momentum distributions of electrons ejected from Ne due to irradiation by a sequence of 2 as light pulses in the presence of a 780 nm laser field.
70 citations
TL;DR: The effect is quite general, and occurs whenever the Floquet multipliers governing the stability of the limit cycle are complex, with the amplitude of the oscillations increasing as the instability boundary is approached.
Abstract: We study the effects of intrinsic noise on chemical reaction systems, which in the deterministic limit approach a limit cycle in an oscillatory manner. Previous studies of systems with an oscillatory approach to a fixed point have shown that the noise can transform the oscillatory decay into sustained coherent oscillations with a large amplitude. We show that a similar effect occurs when the stable attractors are limit cycles. We compute the correlation functions and spectral properties of the fluctuations in suitably comoving Frenet frames for several model systems including driven and coupled Brusselators, and the Willamowski-Rossler system. Analytical results are confirmed convincingly in numerical simulations. The effect is quite general, and occurs whenever the Floquet multipliers governing the stability of the limit cycle are complex, with the amplitude of the oscillations increasing as the instability boundary is approached.
TL;DR: In this paper, a special case of the Lyapunov-Floquet (L-F) transformation is used to transform an anisotropic rotor into a time-invariant system using the transition matrix and Floquet eigenvectors as a basis.
05 Jan 2009
TL;DR: In this article, the authors explored the validity of the time-invariant assumption by comparison to direct periodic techniques, which retain all periodic system information, and found that the direct periodic methods produce almost identical results to time invariant methods after MBC under all conditions studied.
Abstract: The inherent periodic behavior of an operating wind turbine is not well accommodated by common time-invariant analysis and control techniques. A multi-blade coordinate transformation (MBC) helps to overcome this issue for rotors with three or more blades by mapping the dynamic state variables into a non-rotating reference frame. A number of researchers have applied MBC for modal analyses and individual blade pitch controller designs. They do so by assuming the transformed system model from MBC is time-invariant, which is not often the case. The paper explores the validity of the time-invariant assumption by comparison to direct periodic techniques, which retain all periodic system information. In a modal analysis study, eigenvalues of a system after MBC are compared to direct Floquet modes. In an individual blade pitch control design study, a linear quadratic regulation (LQR) design after MBC is compared to direct periodic LQR. A 5-MW three-bladed wind turbine model is used to quantify performance differences. Normal operating conditions are considered as well as conditions selected to increase the harmonics that are unfiltered by MBC. It is found that the direct periodic methods produce almost identical results to timeinvariant methods after MBC under all conditions studied. MBC is recommended for threebladed turbines, which can be followed by Floquet analysis or periodic control design methods if necessary.
TL;DR: It is shown that for typical spin system and experimental parameters a (13)C linewidth of approximately 12 Hz can be obtained for TPPM decoupling in an organic solid or a protein, which indicates that decoupled techniques are still one of the limiting factors in the achievable linewsidths.
Abstract: A theoretical description of the two-pulse phase-modulated (TPPM) decoupling sequence in magic-angle spinning NMR is presented using a triple-mode Floquet approach. The description is formulated in the radio-frequency interaction-frame representation and is valid over the entire range of possible parameters leading to the well-known results of continuous-wave (cw) decoupling and XiX decoupling in the limit of a phase change of 0 degrees and 180 degrees , respectively. The treatment results in analytical expressions for the heteronuclear residual coupling terms and the homonuclear spin-diffusion terms. It also allows the characterization of all resonance conditions that can contribute in a constructive or a destructive way to the residual linewidth. Some of the important resonance conditions are described for the first time since they are not accessible in previous treatments. The combination of the contributions from the residual couplings and the resonance conditions to the effective Hamiltonian, as obtained in a Floquet description, is shown to be required to describe the decoupling behavior over the full range of parameters. It is shown that for typical spin system and experimental parameters a (13)C linewidth of approximately 12 Hz can be obtained for TPPM decoupling in an organic solid or a protein. This is a major contribution to the experimentally observed linewidths of around 20 Hz and indicates that decoupling techniques are still one of the limiting factors in the achievable linewidths.
TL;DR: In this paper, a method of incorporating the discrete dipole approximation (DDA) method with the point matching method to formulate the T-matrix for modelling arbitrarily shaped microsized objects is presented.
Abstract: We present a method of incorporating the discrete dipole approximation (DDA) method with the point matching method to formulate the T-matrix for modelling arbitrarily shaped microsized objects. The T-matrix elements are calculated using point matching between fields calculated using vector spherical wave functions and DDA. When applied to microrotors, their discrete rotational and mirror symmetries can be exploited to reduce memory usage and calculation time by orders of magnitude; a number of optimization methods can be employed based on the knowledge of the relationship between the azimuthal mode and phase at each discrete rotational point, and mode redundancy from Floquet's theorem. A ‘reduced-mode’ T-matrix can also be calculated if the illumination conditions are known.
Posted Content•
TL;DR: In this paper, a unified Floquet theory including Lyapunov transformations and their various stability preserving properties was developed, and a canonical Floquet decomposition on time scales in terms of the generalized exponential function was established.
Abstract: In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system's poles in the complex plane. We include several examples to show the utility of this theory.
TL;DR: In this article, a three-dimensional linear BiGlobal instability of two-dimensional states over a periodic array of T-106/300 low-pressure turbine (LPT) blades is investigated for Reynolds numbers below 5000.
Abstract: Three-dimensional linear BiGlobal instability of two-dimensional states over a periodic array of T-106/300 low-pressure turbine (LPT) blades is investigated for Reynolds numbers below 5000. The analyses are based on a high-order spectral/hp element discretization using a hybrid mesh. Steady basic states are investigated by solution of the partial-derivative eigenvalue problem, while Floquet theory is used to analyse time-periodic flow set-up past the first bifurcation. The leading mode is associated with the wake and long-wavelength perturbations, while a second shortwavelength mode can be associated with the separation bubble at the trailing edge. The leading eigenvalues and Floquet multipliers of the LPT flow have been obtained in a range of spanwise wavenumbers. For the most general configuration all secondary modes were observed to be stable in the Reynolds number regime considered. When a single LPT blade with top to bottom periodicity is considered as a base flow, the imposed periodicity forces the wakes of adjacent blades to be synchronized. This enforced synchronization can produce a linear instability due to long-wavelength disturbances. However, relaxing the periodic restrictions is shown to remove this instability. A pseudo-spectrum analysis shows that the eigenvalues can become unstable due to the non-orthogonal properties of the eigenmodes. Three-dimensional direct numerical simulations confirm all perturbations identified herein. An optimum growth analysis based on singular-value decomposition identifies perturbations with energy growths O(10 5 ).
TL;DR: In this paper, it has been shown that the Bolotin's method will result with the enlargement of the instability region for the gyroscopic system, which may contradict the results based upon the Floquet's method.
Abstract: Some published papers used Bolotin's method for stability boundaries of spinning rotor with parametrically excited gyroscopic system. However, the original work of the method does not indicate that the method can be used to determine the stability of gyroscopic system. This paper intends to highlight the differences in the results using Bolotin's method. As counter examples, dynamic stability of a special case of the parametrically excited gyroscopic system, a simple gyroscopic rotor under parametric excitation and a rotating Timoshenko shaft subjected to periodic axial forces varying with time are analyzed and discussed by using Bolotin's method and Floquet's method respectively to indicate the differences. The causation of these differences is attributed to the differences in the assumptions of Bolotin's and Floquet's methods as that the assumption of Floquet multipliers in Bolotin's method cannot be satisfied for the gyroscopic system. In this paper it has been shown that the Bolotin's method will result with the enlargement of the instability region for the gyroscopic system, which may contradict the results based upon the Floquet's method.
TL;DR: It is shown that such closed-form expression accurately predicts frequency shift under weak pulling, quasi-lock, as well as locked conditions, and an estimation of the main spectrum components of the pulled response is also derived.
Abstract: This paper presents a hybrid numerical-analytical approach to evaluate and quantify injection pulling effects in RF oscillators. The method employs the Floquet nu1(t) eigenvector to project the perturbation signal into the phase domain. An original closed-form expression for the frequency shift induced by small-signal harmonic perturbations is derived. It is shown that such closed-form expression accurately predicts frequency shift under weak pulling, quasi-lock, as well as locked conditions. An estimation of the main spectrum components of the pulled response is also derived. The proposed macromodeling approach has the peculiarity to be applicable to any oscillator topology.
TL;DR: In this article, the spectrum of the Hill operator is characterized by knowing one of its eigenfunctions, and applications are given to nonlinear stability of a class of periodic problems.
Abstract: This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in $${L^{2}_{\rm per}[0, \pi]}$$
. We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.
TL;DR: A coordinate-independent description of oscillator linear response through a decomposition scheme derived independently of any Floquet theoretic results is constructed.
Abstract: We construct a coordinate-independent description of oscillator linear response through a decomposition scheme derived independently of any Floquet theoretic results. Trading matrix algebra for a simpler graphical methodology, the text will present the reader with an opportunity to gain an intuitive understanding of the well-known phase noise macromodel. The topics discussed in this paper include the following: orthogonal decompositions, AM-PM conversion, and nonhyperbolic oscillator noise response.
TL;DR: In this paper, the authors investigated the long time behavior of the L 2 -energy of solutions to wave equations with variable speed of propagation and proposed an approach that combines estimates for higher order derivatives of the coefficient with a stabilisation property.
01 Jan 2009
TL;DR: In this paper, the structural sensitivity of the nonlinear periodic oscillation arising in the wake of a circular cylinder for Re47 was analyzed by performing a structural stability analysis of the problem.
Abstract: In this paper we study the structural sensitivity of the nonlinear periodic oscillation arising in the wake of a circular cylinder for Re47. The sensibility of the periodic state to a spatially localised feedback from velocity to force is analysed by performing a structural stability analysis of the problem. The sensitivity of the vortex shedding frequency is analysed by evaluating the adjoint eigenvectors of the Floquet transition operator. The product of the resulting neutral mode with the nonlinear periodic state is then used to localise the instability core. The results obtained with this new approach are then compared with those derived by Giannetti & Luchini [8]. An excellent agreement is found comparing the present results with the experimental data of Strykowski & Sreenivasan [7].
TL;DR: In this article, the authors focus on the delay induced Hopf bifurcation in a dual model of Internet congestion control algorithms which can be modeled as a time-delay system described by a one-order delay differential equation (DDE).
Abstract: This paper focuses on the delay induced Hopf bifurcation in a dual model of Internet congestion control algorithms which can be modeled as a time-delay system described by a one-order delay differential equation (DDE). By choosing communication delay as the bifurcation parameter, we demonstrate that the system loses its stability and a Hopf bifurcation occurs when communication delay passes through a critical value. Moreover, the bifurcating periodic solution of the system is calculated by means of the perturbation method. Discussion of stability of the periodic solutions involves the computation of Floquet exponents by considering the corresponding Poincare–Lindstedt series expansion. Finally, numerical simulations for verifying the theoretical analysis are provided.
TL;DR: It is shown that Bloch oscillations are persistent if the interaction is modulated harmonically with suitable frequency and phase, and collective coordinates whose Hamiltonian dynamics permits one to predict a whole family of stable solutions.
Abstract: We investigate Bloch oscillations of interacting cold atoms in a mean-field framework. In general, atom-atom interaction causes dephasing and destroys Bloch oscillations. Here we show that Bloch oscillations are persistent if the interaction is modulated harmonically with suitable frequency and phase. For other modulations, Bloch oscillations are rapidly damped. We explain this behavior in terms of collective coordinates whose Hamiltonian dynamics permits one to predict a whole family of stable solutions. In order to describe also the unstable cases, we carry out a stability analysis for Bogoliubov excitations. Using Floquet theory, we are able to predict the unstable modes as well as their growth rate, found to be in excellent agreement with numerical simulations.
TL;DR: Hybrid spectral-spatial representations are introduced in this article to rapidly calculate periodic scalar and dyadic Green's functions of the Helmholtz equation for 2D and 3D configurations with a 1D (linear) periodicity.
Abstract: Hybrid spectral-spatial representations are introduced to rapidly calculate periodic scalar and dyadic Green's functions of the Helmholtz equation for 2D and 3D configurations with a 1D (linear) periodicity. The presented schemes work seamlessly for any observation location near the array and for any practical array periodicities, including electrically small and large periodicities. The representations are based on the expansion of the periodic Green's functions in terms of the continuous spectral integrals over the transverse (to the array) spectral parameters. To achieve high convergence and numerical efficiency, the introduced integral representations are cast in a hybrid form in terms of: (i) a small number of contributions due to sources located around the unit cell of interest; (ii) a small number of symmetric combinations of the Floquet modes; and (iii) an integral evaluated along the steepest descent path (SDP). The SDP integral is regularized by extracting the singular behavior near the saddle point of the integrand and integrating the extracted components in closed form. Efficient quadrature rules are established to evaluate this integral using a small number of quadrature nodes with arbitrary small error for a wide range of structure parameters. Strengths of the introduced approach are demonstrated via extensive numerical examples.
Journal Article•
TL;DR: In this article, the authors compared the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients, and showed that the model controlled by a time periodic control on its coefficients can achieve the same growth rate as a stationary model with averaged coefficients.
Abstract: We study the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients. Our motivation comes from the comparison between experimental tumor growth curves in mice endowed with intact or disrupted circadian clocks, known to exert their influence on the cell division cycle. We compare the growth rate of the model controlled by a time-periodic control on its coefficients with the growth rate of stationary models of the same nature, but with averaged coefficients. We firstly derive a delay differential equation which allows us to prove several inequalities and equalities on the growth rates. We also discuss about the necessity to take into account the structure of the cell division cycle for chronotherapy modeling. Numerical simulations illustrate the results.
TL;DR: In this paper, quasi-periodic tori under a normal internal resonance, possibly with multiple eigenvalues, were studied and two non-degeneracy conditions played a role, one generalizes invertibility of the Floquet matrix and prevents drift of the lower dimensional torus.
TL;DR: The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution, for these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies.
Abstract: We present an algorithm for the computation of reducible invariant tori of discrete dynamical systems that is suitable for tori of dimensions larger than 1. It is based on a quadratically convergent scheme that approximates, at the same time, the Fourier series of the torus, its Floquet transformation, and its Floquet matrix. The Floquet matrix describes the linearization of the dynamics around the torus and, hence, its linear stability. The algorithm presents a high degree of parallelism, and the computational effort grows linearly with the number of Fourier modes needed to represent the solution. For these reasons it is a very good option to compute quasi-periodic solutions with several basic frequencies. The paper includes some examples (flows) to show the efficiency of the method in a parallel computer. In these flows we compute invariant tori of dimensions up to 5, by taking suitable sections.
TL;DR: It is shown that the predator-prey system with an Ivlev-type functional response and impulsive control strategies containing a biological control and a chemical control with the same period, but not simultaneously is permanent under some conditions.