Application Of Floquet Theory Research Articles

Consensus stability in multi-agent systems with periodically switched communication topology using Floquet theory

The stability of consensus in linear and nonlinear multi-agent systems with periodically switched communication topology is studied using Floquet theory. The proposed strategy is illustrated for the cases of consensus in linear single-integrator, higher-order integrator, and leader-follower consensus. In addition, the application of Floquet theory in analyzing special cases such as switched systems with joint connectivity, unstable subsystems, and nonlinear systems, including the use of feedback linearization and local linearization in the Kuramoto model, is also studied. By utilizing Floquet theory for multi-agent systems with periodically switched communication topologies, one can simultaneously evaluate the effects of each subsystem’s convergence rate and dwell time on overall behavior. Numerical simulation results are presented to demonstrate the efficacy of the proposed approach in stability analysis of all these cases.

The effective role of Dirac mass in nonlinear optical spectra of silicene

Silicene is a monolayer honeycomb system made of silicon. It is a two-dimensional topological insulator that undergoes a topological phase transition to a band insulator by applying an external electromagnetic field. All topological phase transitions in silicene come by tuning of its Dirac mass. In this paper, the tuning of the Dirac mass of silicene has been done by the application of Floquet theory. The nature of the Dirac mass in a Bloch–Seigert shift is shown in topological trivial and non-trivial phases of silicene. The collapse-revival spectra have been described by Floquet theory in silicene. The differential transmission coefficient of pump–probe spectra is shown, which have the capability of distinguishing Floquet theory oscillations in topological trivial and non-trivial phases of silicene. All these results are also valid for germanene and stanene by changing the material parameters.

Natural periodic orbit-attitude behaviors for rigid bodies in three-body periodic orbits

Trajectory design increasingly leverages multi-body dynamical structures that are based on an understanding of various types of orbits in the Circular Restricted Three-Body Problem (CR3BP). Given the more complex dynamical environment, mission applications may also benefit from deeper insight into the attitude motion. In this investigation, the attitude dynamics are coupled with the trajectories in the CR3BP. In a highly sensitive dynamical model, such as the orbit-attitude CR3BP, periodic solutions allow delineation of the fundamental dynamical structures. Periodic solutions are also a subset of motions that are bounded over an infinite time-span (assuming no perturbing factors), without the necessity to integrate over an infinite time interval. Euler equations of motion and quaternion kinematics describe the rotational behavior of the spacecraft, whereas the translation of the center of mass is modeled in the CR3BP equations. A multiple shooting and continuation procedure is employed to target orbit-attitude periodic solutions in this model. Application of Floquet theory and Poincaré mappings to identify initial guesses for the targeting algorithm are described. In the Earth–Moon system, representative scenarios are explored for axisymmetric vehicles with various inertia characteristics, assuming that the vehicles move along L1/L2 Lyapunov orbits as well as distant retrograde orbits. A rich structure of possible periodic behaviors appears to pervade the solution space in the coupled problem. The stability analysis of the attitude dynamics for the available families is included. Among the computed solutions, marginally stable and slowly diverging rotational behaviors exist and may offer interesting mission applications.

On the application of Floquet theorem in development of time-dependent Lindbladians

In this paper, the mathematical framework providing a description of completely positive and trace preserving dynamics of open quantum systems is addressed. Special case of time-dependent Lindbladian governed by periodic Hamiltonian is concerned. It is proven that appropriate trace preserving dynamical map of such periodically driven system may be constructed by application of Floquet theory. Appropriate Lindbladian and Markovian master equation in weak coupling regime and resulting dynamical map are constructed. Some examples of application of developed technique are given.

Research on the Stability of a Rotor System Based on Mechanics Properties with Application of Floquet Theory

In this paper, the single disk rotor system with a transverse open crack has been taken as an example; the vibration properties on the system have been discussed theoretically. First, the equations of the motion of this kind of system are established. Then, the conditions, positions and areas of the stable vibrations and the unstable vibrations on the system are studied by applying the Floquet Theory. Finally the influence of several parameters on stability has been discussed.

Formula omitted]-loss of derivatives for an evolution type model

This paper is devoted to the study of loss of derivatives for an evolution type model, which allows for non-integer powers of ( − Δ ) . On the one hand, we describe the influence from the orders of principle symbols. On the other hand, we explore the significant impact of ν -related oscillation from the pseudo-differential operators. In the final analysis, to demonstrate the sharpness of our estimates, counter-examples are constructed through the application of Floquet theory and instability arguments.

Quantum Interference in Time

I discuss the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory does not have the property of coherence in time, and hence cannot account for the results found. Therefore, this experiment has fundamental importance beyond the technical advances it represents. Some theoretical structures which consider the time as an observable, and thus could, in principle, have the required coherence in time, are discussed briefly, and the application of Floquet theory and the manifestly covariant quantum theory of Stueckelberg are treated in some detail. In particular, the latter is shown to account for the results in a simple and consistent way.

On the significance of a recent experiment demonstrating quantum interference in time

I discuss the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory, used by the authors in their analysis, does not have the property of coherence in time, and hence cannot account for the results found. Therefore, this experiment has fundamental importance beyond the technical advances it represents. Some theoretical structures which consider the time as an observable, and thus could, in principle, have the required coherence in time, are discussed briefly, and the application of Floquet theory and the manifestly covariant quantum theory of Stueckelberg are treated in some detail. In particular, the latter is shown to account for the results in a simple and consistent way.

ANALYSIS OF PERIODICALLY VARYING GEAR MESH SYSTEMS WITH COULOMB FRICTION USING FLOQUET THEORY

This article presents a new analytical model of a gear pair with time varying mesh stiffness, viscous damping and sliding friction parameters. Unlike previous models, the excitation consists of three separate terms, namely the unloaded transmission error, time-invariant external torque and the periodically varying sliding friction force. A Coulomb friction model is considered using first a quasi-static mean transmitted load that is represented by the Meissner equation. Then, a truly dynamic force between gear teeth is described that leads to a triangular function, and after appropriate substitutions, this assumes the form of the Bessel equation of the one-third order. For the damped Meissner equation, the forced vibration response is found with the application of Floquet theory. Exact integrals are calculated for the state transition matrix in a piecewise manner, instead of using the Fourier series expansion, thus eliminating the mode truncation errors. From the state transition matrix, unstable zones are identified and the actual forced response of the system is found in terms of dynamic transmission error for these zones. With the aid of an example, the significance of sliding friction on system response and stability is examined. Finally, key advantages and the need for analytical methods are demonstrated for such systems.

The harmonic Floquet oscillator

The harmonic oscillator interacting with a monochromatic electromagnetic field is one of the few soluble models in the application of Floquet theory. After recalling the form of the Floquet wave functions and eigenenergies we show how they can be derived as a limiting case of the solution for two oscillators with a bilinear coupling. We also derive a solution in the acceleration frame and examine the high-frequency regime. Finally, the model offers an opportunity to test the performances of a coupled-channel formulation of Floquet theory.

Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems

In this paper, a new analysis technique in the study of dynamical systems with periodically varying parameters is presented. The method is based on the fact that all linear periodic systems can be replaced by similar linear time-invariant systems through a suitable periodic transformation known as the Liapunov-Floquet (L-F) transformation. A general technique for the computation of the L-F transformation matrices is suggested. In this procedure, the state vector and the periodic matrix of the linear system equations are expanded in terms of the shifted Chebyshev polynomials over the principal period. Such an expansion reduces the original differential problem to a set of linear algebraic equations from which the state transition matrix (STM) can be constructed over the period in closed form. Application of Floquet theory and eigenanalysis to the resulting STM yields the L-F transformation matrix in a form suitable for algebraic manipulations. The utility of the L-F transformation in obtaining solutions of both linear and nonlinear dynamical systems with periodic coefficients is demonstrated. It is shown that the application of L-F transformation to free and harmonically forced linear periodic systems directly provides the conditions for internal and combination resonances and external resonances, respectively. The application of L-F transformation to quasilinear periodic systems provides a dynamically similar quasilinear systems whose linear parts are time-invariant and the solutions of such systems can be obtained through an application of the time dependent normal form theory. These solutions can be transformed back to the original coordinates using the inverse L-F transformation. Two dynamical systems, namely, a commutative system and a Mathieu type equation are considered to demonstrate the effectiveness of the method. It is shown that the present technique is virtually free from the small parameter restriction unlike averaging and perturbation procedures and can be used even for those systems for which the generating solutions do not exist in the classical sense. The results obtained from the proposed technique are compared with those obtained via the perturbation method and numerical solutions computed using a Runge-Kutta type algorithm.

Analysis of quasilinear dynamical systems with periodic coefficients via Liapunov—Floquet transformation

In this paper, a new analysis technique in the study of general quasilinear systems with periodically varying parameters is presented. The method is based on the fact that all quasilinear periodic systems can be replaced by similar systems whose linear parts are time-invariant, via the well-known Liapunov-Floquet (L-F) transformation. A general technique for the computation of the L-F transformation matrices is outlined. In this technique, the state vector and the periodic matrix of the linear system equations are expanded in terms of the shifted Chebyshev polynomials over the principal period. Such an expansion reduces the original problem to a set of linear algebraic equations from which the state transition matrix can be constructed over the period as an explicit function of time. Application of Floquet theory and use of symbolic software yields the L-F transformation matrix in a form suitable for algebraic manipulations. Once the transformation has been applied, the solution of the resulting system is obtained through an application of the time-dependent normal form theory. The method is suitable for both numerical and symbolic computations and in some cases approximate closed form solutions can be obtained. Two simple examples of quasilinear periodic systems—namely, a commutative system with quadratic nonlinearity and a Mathieu equation with cubic non-linearity—are used to demonstrate the effectiveness of the method. For verification, results obtained from the proposed technique are compared with the numerical solutions computed using a standard Runge-Kutta type algorithm. It is shown that the present technique is applicable to systems where the periodic matrix does not contain a small parameter, which is not the case with averaging and perturbation procedures. It can also be used even for those systems for which the generating solutions do not exist in the classical sense.

On the Stability of Perfect Viscoelastic Columns

The stability problem associated with an Euler-Bernoulli beam made of an arbitrary linear viscoelastic material is formulated. The three parameter and the Kelvin-Voigt models are analyzed in the presence of constant as well as periodic loads. The application of a finite time stability concept is shown for the constant loading case when the traditional stability criterion fails to make sense. For the case of a periodic loading, the stability diagrams are obtained through an application of Floquet theory. It is found that the addition of periodic loads may significantly alter the stability behavior of a column which is originally subjected to a constant load only.

Dynamic Stability of the Flexible Connecting Rod of a Slider Crank Mechanism

The partial differential equation of motion of the flexible connecting rod of a slider crank is derived, under the assumption of small deflections. Application of the Galerkin procedure, leads to a system of linear ordinary differential equations, with respect to the modal coordinates of vibration of the rod. For periodic solutions, the foregoing system reduces to a system of coupled Hill equations. Application of Floquet theory, determines those values of the parameters: speed, input torque, geometry, and material properties that constitute the boundaries between the regions of stability and instability.