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Ordinary differential equations with time-periodic coefficients (so-called time-periodic systems) can be analyzed using Lyapunov–Floquet (L–F) transformations. These transformations reduce the linear part of a time-periodic equation to the time-invariant form and facilitate the application of well-established techniques tailored for time-invariant systems. In the previous work, the construction of L–F transformations relied on Chebyshev polynomials and their properties, which may often prove challenging to grasp and apply effectively. This paper endeavors to present a more intuitive and straightforward approach for computing L–F transformations. The solution of a linear time-periodic system can be expressed as a product of an exponential function and a vector-valued polynomial in time with time-periodic coefficients. Substitution of the solution reduces a time-periodic equation to an eigenvalue problem, which can be solved to obtain the general solution. Rearranging the solution yields the state transition matrix, which can be used in the Lyapunov–Floquet theorem to compute the L–F transformation. The inverse of these transformations is important for the nonlinear analysis and control and can be determined by defining the adjoint system to the time-periodic system. As examples, L–F transformations and their inverses are generated for the Mathieu equation and a double inverted pendulum subjected to a time-periodic force. In the end, the usefulness of L–F transformations is showcased by performing the bifurcation study of a nonlinear Mathieu equation using the center manifold theorem.

A track nonlinear energy sink (TNES) has been proven as an effective control strategy. However, most researches focus on numerical or experimental aspects, with relatively little analytical study of intrinsic dynamic characteristics of the TNES. This study aims to investigate the nonlinear behaviors of a harmonically excited linear structure coupled with the TNES and reveal the vibration reduction performance of the TNES from the perspective of analysis. Firstly, the motion form of the TNES system is qualitatively analyzed by the global bifurcation, the time history, the Fourier spectrum, the phase trajectory, and the Poincare map. Secondly, the amplitude-frequency response for the periodic steady-state motion is quantitatively analyzed by combining the harmonic balance with the pseudo-arc length extension method and validated through numerical solutions. Then, the stability and bifurcation feature of the periodic steady-state solution are revealed by leveraging the Floquet theorem. Finally, the damping efficiency is explored. The results demonstrate that the complex nonlinearity of the TNES system can result in the coexistence of the periodic, quasi-periodic, and chaotic motion. Saddle-node bifurcations and Hopf bifurcations are discovered in the approximate analytical solutions. Strongly modulated responses (SMR) caused by Hopf bifurcation can greatly improve the damping efficiency of the TNES. In addition, a proper increase in mass ratio can suppress the adverse effects of frequency islands or saddle-node bifurcation curves near the resonance region. This research provides the necessary theoretical basis for optimizing and designing the TNES.

We theoretically study the inverse Faraday effect, i.e., the optical induction of spin polarization with circularly polarized light, by particularly focusing on effects of band dispersions and Fermi surfaces in crystal systems with the spin-orbit interaction (SOI). By numerically solving the time-dependent Schrödinger equation of a tight-binding model with the Rashba-type SOI, we reproduce the light-induced spin polarization proportional to E02/ω3 where E0 and ω are the electric-field amplitude and the angular frequency of light, respectively. This optical spin induction is attributed to dynamical magnetoelectric coupling between the light electric field and the electron spins mediated by the SOI. We elucidate that the magnitude and sign of the induced spin polarization sensitively depend on the electron filling. To understand these results, we construct an analytical theory based on the Floquet theorem. The theory successfully explains the dependencies on E0 and ω and ascribes the electron-filling dependence to a momentum-dependent effective magnetic field governed by the Fermi-surface geometry. Several candidate materials and experimental conditions relevant to our theory and model parameters are also discussed. Our findings will enable us to engineer the magneto-optical responses of matters via tuning the material parameters.

A bstract The fact that fast oscillating homogeneous scalar fields behave as perfect fluids in average and their intrinsic isotropy have made these models very fruitful in cosmology. In this work we will analyse the perturbations dynamics in these theories assuming general power law potentials V ( ϕ ) = λ | ϕ | n /n . At leading order in the wavenumber expansion, a simple expression for the effective sound speed of perturbations is obtained c eff 2  =  ω  = ( n  − 2)/( n  + 2) with ω the effective equation of state. We also obtain the first order correction in k 2 / ω eff 2 , when the wavenumber k of the perturbations is much smaller than the background oscillation frequency, ω eff . For the standard massive case we have also analysed general anharmonic contributions to the effective sound speed. These results are reached through a perturbed version of the generalized virial theorem and also studying the exact system both in the super-Hubble limit, deriving the natural ansatz for δϕ ; and for sub-Hubble modes, exploiting Floquet’s theorem.

This paper proposes a local resonance-type pentagonal phononic crystal beam structure for practical engineering applications to achieve better vibration and noise reduction. The energy band, transmission curve, and displacement field corresponding to the vibration modes of the structure are calculated based on the finite element method and Bloch-Floquet theorem. Furthermore, an analysis is conducted to understand the mechanism behind the generation of bandgaps. The numerical analysis indicates that the pentagonal unit oscillator creates a low-frequency bandgap between 60–70 Hz and 107–130 Hz. Additionally, the pentagonal phononic crystal double-layer beam structure exhibits excellent vibration damping, whereas the single-layer beam has poor vibration damping. The article comparatively analyzes the effects of different parameters on the bandgap range and transmission loss of a pentagonal phononic crystal beam. For instance, increasing the thickness of the lead layer leads to an increase in the width of the bandgap. Similarly, increasing the thickness of the rubber layer, intermediate plate, and total thickness of the phononic crystals results in a bandgap at lower frequencies. By adjusting the parameters, the beam can be optimized for practical engineering purposes.

We consider the Schrödinger equation , . The operator includes a smooth potential , which is assumed to be time -periodic. Let be the fundamental solution of this linear ODE system on . Then, according to the terminology from Lyapunov–Floquet theory, is the monodromy operator. We prove that is unitarily conjugated to , where is diagonal in the standard Fourier basis, while is a compact operator with an arbitrarily small norm.

A spatial modes filtering (SMF) finite-element time-domain (FETD) method with periodic boundary condition (PBC) is proposed for efficiently analyzing the electromagnetic characteristics of 3-D periodic structures with partial fine structures. The system matrices of FETD become asymmetrical because of the introduction of PBC, which prevents the system eigenvalue analysis. By decomposing the system matrix into PBC-independent and PBC-related parts, the unstable spatial modes under the given large time step can be found and removed from the symmetrical PBC-independent system matrices. Then the system matrix equation and time marching of the SMF-FETD and SMF-FETD method based on local eigenvalue solution (LES-SMF-FETD) with PBC are obtained. Numerical results illustrate the efficiency and effectiveness of the SMF-FETD method with PBC based on non-uniform mesh for analyzing the transport properties of 3-D periodic structures.

This paper uses the scaled boundary finite element method (SBFEM) to study the wave propagation in a phononic crystal (PC). The SBFEM is a general semi-analytical method where a problem domain is divided into subdomains satisfying the scaling requirement. It offers the advantages of the finite element method (FEM) and the boundary element method (BEM), avoiding some drawbacks and making it very attractive for PC applications. In this paper, the SBFEM is formulated using the Bloch–Floquet theory to model periodic PC unit cells. This is an unprecedented modeling, since it is the first paper to use this methodology. The combined use of SBFEM with the Bloch–Floquet theorem provides a robust and efficient framework to accurately design and study PCs, enabling applications such as vibration control and acoustic insulation. The interest in elastic metamaterials (EMs) and PCs started in many engineering applications as vibration and noise control devices around a decade ago. PCs consist of two or more different materials periodically distributed, producing stop band or band gaps characteristic, where no elastic/acoustic waves propagate. The effect of Bragg scattering is analyzed through the dynamic responses obtained for different cases. The results are computed in the form of elastic band structure, forced response, and wave mode shapes. The SBFEM results are compared with those obtained by the FEM and plane wave expansion (PWE) method. Analyses were performed for various frequency ranges, such as f = 0 - 250 Hz, f = 0 - 300 Hz, f = 0 - 10 kHz, and f = 0 - 15 kHz. The analysis range depends on the geometric properties and the frequency range in which vibration attenuation through band gaps is desired. The relative errors of the natural frequencies calculated using SBFEM and FEM were computed for two cases: SBFEM I and SBFEM II. It was observed that for the SBFEM II case, the errors remained within 0.42 % . A systematic error analysis was conducted, along with a systematic mesh convergence study, and a simple sensitivity analysis of the filling fraction with respect to the width of the generated band gaps was also performed. For the analyzed cases, comparing computational times shows that SBFEM is considerably more efficient than FEM. In all analyses performed for PCs, it was demonstrated that the SBFEM exhibits higher efficiency and better performance compared to the FEM, establishing itself as a highly effective method for the design and analysis of PCs and EMs.

Dielectric metasurfaces have emerged as a promising alternative to their plasmonic counterparts due to lower ohmic losses, which hinder sensing applications and nonlinear frequency conversion, and their larger flexibility to shape the emission pattern in the visible regime. To date, the computational cost of full-wave numerical simulations has forced the exploitation of the Floquet theorem, which implies infinitely periodic structures, in designing such devices. In this work, we show the potential pitfalls of this approach when considering finite-size metasurfaces and beam-like illumination conditions, in contrast to the typical infinite plane-wave illumination compatible with the Floquet theorem.

With the development of wireless sensor networks (WSNs), energy constraints and network security have become the main problems. This paper discusses the dynamic of the Susceptible, Infected, Low-energy, Susceptible model under pulse charging (SILS-P) in wireless rechargeable sensor networks. After the construction of the model, the local stability and global stability of the malware-free T-period solution of the model are analyzed, and the threshold R0 is obtained. Then, using the comparison theorem and Floquet theorem, we obtain the relationship between R0 and the stability. In order to make the conclusion more intuitive, we use simulation to reveal the impact of parameters on R0. In addition, the paper discusses the continuous charging model, and reveals its dynamic by simulation. Finally, the paper compares three charging strategies: pulse charging, continuous charging and non-charging and obtains the relationship between their threshold values and system parameters.