Linear Quasi-periodic Systems Research Articles

A Direct Approach to Compute the Lyapunov–Perron Transformation for Linear Quasi-periodic Systems

A Direct Approach to Compute the Lyapunov–Perron Transformation for Linear Quasi-periodic Systems

Approximate Lyapunov–Perron Transformations: Computation and Applications to Quasi-Periodic Systems

Abstract A new technique for the analysis of dynamical equations with quasi-periodic coefficients (so-called quasi-periodic systems) is presented. The technique utilizes Lyapunov–Perron (L–P) transformation to reduce the linear part of a quasi-periodic system into the time-invariant form. A general approach for the construction of L–P transformations in the approximate form is suggested. First, the linear part of a quasi-periodic system is replaced by a periodic system with a “suitable” large principal period. Then, the state transition matrix (STM) of the periodic system is computed in the symbolic form using Floquet theory. Finally, Lyapunov–Floquet theorem is used to compute approximate L–P transformations. A two-frequency quasi-periodic system is studied and transformations are generated for stable, unstable, and critical cases. The effectiveness of these transformations is demonstrated by investigating three distinct quasi-periodic systems. They are applied to a forced linear quasi-periodic system to generate analytical solutions. It is found that the closeness of the analytical solutions to the exact solutions depends on the principal period of the periodic system. A general approach to obtain the stability bounds on linear quasi-periodic systems with stochastic perturbations is also discussed. Finally, the usefulness of approximate L–P transformations is presented by analyzing a nonlinear quasi-periodic system with cubic nonlinearity using time-dependent normal form (TDNF) theory. The closed-form solution generated is found to be in good agreement with the exact solution.

Lyapunov–Perron Transformation for Quasi-Periodic Systems and Its Applications

Abstract This paper depicts the application of symbolically computed Lyapunov–Perron (L–P) transformation to solve linear and nonlinear quasi-periodic systems. The L–P transformation converts a linear quasi-periodic system into a time-invariant one. State augmentation and the method of normal forms are used to compute the L–P transformation analytically. The state augmentation approach converts a linear quasi-periodic system into a nonlinear time-invariant system as the quasi-periodic parametric excitation terms are replaced by “fictitious” states. This nonlinear system can be reduced to a linear system via normal forms in the absence of resonances. In this process, one obtains near identity transformation that contains fictitious states. Once the quasi-periodic terms replace the fictitious states they represent, the near identity transformation is converted to the L–P transformation. The L–P transformation can be used to solve linear quasi-periodic systems with external excitation and nonlinear quasi-periodic systems. Two examples are included in this work, a commutative quasi-periodic system and a non-commutative Mathieu–Hill type quasi-periodic system. The results obtained via the L–P transformation approach match very well with the numerical integration and analytical results.

On the reducibility of linear quasi-periodic systems with Liouvillean basic frequencies and multiple eigenvalues

In this paper we consider the linear quasi-periodic systemx˙=(A+ϵP(t))x,x∈Rd, where A is a d×d constant matrix of elliptic type and has multiple eigenvalues, P(t) is analytic quasi-periodic with respect to t with basic frequencies ω=(1,α), where α is irrational, and ϵ is a small perturbation parameter. Under suitable non-resonant condition, non-degeneracy condition and 0≤β(α)<r⁎, where β(α)=limsupn→∞ln⁡qn+1qn, qn is the sequence of denominations of the best rational approximations for α∈R∖Q, 0<r⁎<r, r is the initial radius of analytic strip, it is proved that for most sufficiently small ϵ, this system can be reduced to a constant system x˙=A⁎x, where A⁎ is a constant matrix close to A. As some applications, we apply our results to quasi-periodic Hill's equations, three dimensional skew symmetric systems and n coupled Schrödinger equations to study the Lyapunov stability of the equilibrium and the existence of quasi-periodic solutions.

Reducibility and Analysis of Linear Quasi-Periodic Systems Via Normal Forms

Abstract This article introduces a technique to accomplish reducibility of linear quasi-periodic systems into constant-coefficient linear systems. This is consistent with congruous proofs common in literature. Our methodology is based on Lyapunov–Floquet transformation, normal forms, and enabled by an intuitive state augmentation technique that annihilates the periodicity in a system. Unlike common approaches, the presented approach does not employ perturbation or averaging techniques and does not require a periodic system to be approximated from the quasi-periodic system. By considering the undamped and damped linear quasi-periodic Hill-Mathieu equation, we validate the accuracy of our approach by comparing the time-history behavior of the reduced linear constant-coefficient system with the numerically integrated results of the initial quasi-periodic system. The two outcomes are shown to be in exact agreement. Consequently, the approach presented here is demonstrated to be accurate and reliable. Moreover, we employ Floquet theory as part of our analysis to scrutinize the stability and bifurcation properties of the undamped and damped linear quasi-periodic system.

On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters

In this paper we consider a real analytic linear quasi-periodic system of 2-dimension, whose coefficient matrix depends on a small parameter C m C^m -smoothly and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption with respect to the small parameter, we prove that the system is reducible for many of the sufficiently small parameters.

Stability of periodic motions of quasilinear systems

The stability of linear and quasilinear systems with periodic coefficients is analyzed. The properties of stability are established in terms of matrix-valued Lyapunov functions. An algorithm is developed to set up matrix-valued Lyapunov functions for linear quasiperiodic systems. A numerical example is given to illustrate the application of the algorithm

Analytic families of reducible linear quasi-periodic differential equations

In this paper we study the existence of analytic families of reducible linear quasi-periodic differential equations in matrix Lie algebras. Under suitable conditions we show, by means of a Kolmogorov–Arnold–Moser (KAM) scheme, that a real analytic quasi-periodic system close to a constant matrix can be modified by the addition of a time-free matrix that makes it reducible to constant coefficients. If the system depends analytically on external parameters, then this modifying term is also analytic.As a major application, we prove the analyticity of resonance tongue boundaries in Hill's equation with a small quasi-periodic forcing. Several consequences for the spectrum of Schrödinger operators with quasi-periodic forcing are derived. In particular, we prove that, generically, the spectrum of Schrödinger operators with a small real analytic and quasi-periodic potential has all spectral gaps open and, therefore, it is a Cantor set. Some other applications are included for linear quasi-periodic systems on .

Reducible Linear Quasi-periodic Systems with Positive Lyapunov Exponent and Varying Rotation Number

A linear system in two dimensions is studied. The coefficients are 2π-periodic in three angles, θj, j=1, 2, 3, and these angles are linear with respect to time, with incommensurable frequencies. The system has positive Lyapunov coefficients and the rotation number changes in a continuous way when some parameter moves. A lift to T3×R2, however, is only of class Lp, for any p<2.

On uniformly stable linear quasiperiodic systems

In a finite-dimensional complex space, we consider a system of linear differential equations with quasiperiodic skew-Hermitian matrix. The space of solutions of this system is a sum of one-dimensional invariant subspaces. Over a torus defined by a quasiperiodic matrix of the system, we investigate the corresponding one-dimensional invariant bundles (nontrivial in the general case). We find conditions under which these bundles are trivial and the system can be reduced to diagonal form by means of the Lyapunov quasiperiodic transformation with a frequency module coinciding with the frequency module of the matrix of the system.

Behavior of solutions of linear quasiperiodic systems near resonances

Behavior of solutions of linear quasiperiodic systems near resonances