Floquet Characteristic Exponents Research Articles

A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents

We propose a modification of a method based on Fourier analysis to obtain the Floquet characteristic exponents for periodic homogeneous linear systems, which shows a high precision. This modification uses a variational principle to find the correct Floquet exponents among the solutions of an algebraic equation. Once we have these Floquet exponents, we determine explicit approximated solutions. We test our results on systems for which exact solutions are known to verify the accuracy of our method including one‐dimensional periodic potentials of interest in quantum physics. Using the equivalent linear system, we also study approximate solutions for homogeneous linear equations with periodic coefficients.

A Note on Uniform Exponential Stability of Linear Periodic Time-Varying Systems

In this paper we derive new criterion for uniform stability assessment of the linear periodic time-varying systems $\dot x=A(t)x,$ $A(t+T)=A(t).$ As a corollary, the lower and upper bounds for the Floquet characteristic exponents are established. The approach is based on the use of logarithmic norm of the system matrix $A(t).$ Finally we analyze the robustness of the stability property under external disturbance.

Ground state of the quantum periodic Toda lattice in the large-Nlimit

In Gutzwiller's formulation of the quantum periodic Toda lattice, quantization conditions are expressed in terms of Floquet's characteristic exponents which are equal to the zeros of a Hill-type determinant. We have derived an integral equation for the density distribution of those zeros for the ground state in the large-N limit. The large-N asymptotic expansions of the conserved quantities given by this density distribution are found to be fairly good approximations to the exact values. We have also carried out the semiclassical quantization by the EBK formulation and the semiclassical eigenvalues turned out to be very close to the exact values.

Semianalytical sensitivity of Floquet characteristic exponents with application to rotary-wing aeroelasticity

The stability of a nonlinear dynamic system with periodic coefficients can be evaluated by linearizing the equations of motion of the system about a steady-state equilibrium position and then, using Floquet -Liapunov theory, by calculating the eigenvalues, or characteristi c multipliers, of the transition matrix at the end of one period, and finally by examining the real parts of the characteristic exponents corresponding to the characteristic multipliers. This article describes a methodology to calculate the sensitivity of the characteristic exponents to changes in system parameters, taken as design variables, using semianalytic derivatives rather than finite difference approximations. Linear and nonlinear contributions to such sensitivities are identified and algorithms for the calculation of each contribution are provided. The linear contribution to the sensitivity to each design variable can be calculated at a fraction of the cost of one analysis, with appropriate attention to the details of the computer implementation. The nonlinear contribution requires a fixed overhead cost of as many analyses as the number of variables that define the equilibrium position, regardless of the number of design variables with respect to which the sensitivities are sought. A potential problem in the calculation of finite difference gradients is identified and discussed. The linear portion alone is sufficiently accurate for the helicopter problem of the study.

Analysis of Dynamic Systems With Periodically Varying Parameters Via Chebyshev Polynomials

In this paper a general method for the analysis of multidimensional second-order dynamic systems with periodically varying parameters is presented. The state vector and the periodic matrices appearing in the equations are expanded in Chebyshev polynomials over the principal period and the original differential problem is reduced to a set of linear algebraic equations. The technique is suitable for constructing either numerical or approximate analytical solutions. As an illustrative example, approximate analytical expressions for the Floquet characteristic exponents of Mathieu’s equation are obtained. Stability charts are drawn to compare the results of the proposed method with those obtained by Runge-Kutta and perturbation methods. Numerical solutions for the flap-lag motion of a three-bladed helicopter rotor are constructed in the next example. The numerical accuracy and efficiency of the proposed technique is compared with standard numerical codes based on Runge-Kutta, Adams-Moulton, and Gear algorithms. The results obtained in both the examples indicate that the suggested approach is extremely accurate and is by far the most efficient one.

Non-linear resonance frequency shifts and saturation effects by oscillating and static applied fields in multi-level systems

A previously developed exact semiclassical method is used to evaluate multi-photon spectra and Floquet characteristic exponents (dressed-atom energies) for several multi-level systems interacting with both strong oscillating and static fields under varying degrees of saturation. The usefulness of characteristic exponent plots in extracting the physically significant features of the spectrum, without the necessity of evaluating portions of the complete spectrum, is demonstrated for a number of model three-level systems which are not directly amenable to study by the often used rotating-wave approximation. The effect of neighbouring non-resonant states on two-photon resonance profiles is also discussed using three to six level systems (H and Na atoms) as models.