Floquet theory is very useful for finding the response or stability of linear time-periodic equations [4] [10] [11] . Consider the linear periodic system

denotes the State Transition Matrix (STM) (fundamental solution matrix) that contains n linearly independent solutions of Equation (10) with the initial conditions as is an identity square matrix of dimension n.

As such the following conditions hold:

1) and, consequently 2)

3)

These conditions suggest that, if the solution is known for the first period, it can be designed for all time t. The matrix is called the Floquet transition matrix (FTM). The next condition considers the stability of Equation (10). Let denote the eigenvalues of. System given by Equation (10) is asymptotically stable if all lie inside the unit circle of the complex plane. If one or more of the eigenvalues of the FTM has magnitude greater than one, the system is unstable. The Floquet multipliers are the eigenvalues.

According to the Lyapunov-Floquet (L-F) theorem, STM (the fundamental matrix) of equation (10) can be written as a product of two matrices as:

where is and is a constant matrix, both, in general, are complex. There also exists factorization of the same form, where is a real constant matrix and is real. is called the L-F transformation matrix [12] . For the details on computation of the L-F transformation and its applications, we refer the reader to references [13] -[15] .

In the paper by Wooden and Sinha [15] , it is mentioned that an essential class of dynamical systems may be showed by a set of nonlinear differential equations with periodic/quasi-periodic coefficients multiplying the nonlinearity. They analyzed the system where the linear term was periodic but the nonlinear terms were quasi-periodic as given by equation

where is a matrix of constant coefficients, is matrix with time periodic coefficients and is a vector with quasi-periodic coefficients. The matrices have dimensions and the vector is dimensional. The authors used the L-F transformation and nonlinear quasi-periodic transformations to study Equation (12) and obtained resonance conditions.

In the past, Arnold [10] demonstrated normal forms of quasi-periodic nonlinear systems with time-invariant linear part. Bogoljubov et al. [11] presented the reducibility of quasi-periodic systems to approximate time-invariant forms using a small parameter strategy. Even though small parameters/perturbation type techniques have been successfully used to study stability and reducibility of quasiperiodic systems [9] , these techniques are limited by small parameters multiplying the nonlinear and/or time-varying conditions. Belhaq et al. [16] considered a homogeneous Mathieu equation with quasi-periodic linear coefficients and a constant nonlinear coefficient. The small parameter strategy of multiple scales was used twice to the system to acquire an approximate time-invariant system. Researchers have used small parameter assumption to plot stability charts for quasiperiodic systems [17] [18] .

The system studied in this research is different that the systems studied earlier. Here, we consider the system of the form given by Equation (13)

where is the time periodic matrix and matrix has the coefficients with incommensurate frequencies and is a constant matrix of appropriate dimensions. We present technique to reduce Equation (13) to a constant coefficient system without any limitation on the magnitude of i.e. small parameter assumption. This paper is organized as follows. In section two, the reducibility formulation is presented and the resonance/reducibility conditions are derived. In section three, application of this approach is presented with two examples. Section four has discussion and conclusions.

Consider the following commutative system

where

and is a system parameter The L-F transformation for matrix is

and

Applying the L-F Transformation to Equation (31) yields

It is noted that for this special case matrix is in the Jordan form. Thus

Equation (32) can be reduced to a constant coefficient system with the procedure discussed in section two. First a quasi-periodic near identity transformation can be substituted in Equation (32) where is the quasi-periodic matrix with unknown coefficients. The elements of this matrix have the form given by Equation (26) with. At this point, collecting the terms of order yields the homological equation

Expanding and in the form given by Equation (26) and collecting the terms via harmonic balance yields the elements of if the resonance condition given by Equation (28) is satisfied. Thus, we reduce the quasi-periodic system given by Equation (26) to a constant coefficient system given by Equation (31)

To compare the results we integrate Equation (29) numerically and generate the time traces and phase plane. For the reduced system given by Equation (31), it is possible to find a closed form solution as

where is the initial condition. Using the transformation it is possible to obtain the closed form solution in.

In Figure 1, the solutions of the original system and the reduced order system are compared. Time traces shown in red were obtained via numerical integration of Equation (29) and time traces in blue were obtained by solving the reduced constant coefficient system in the closed form (given by Equation (35))and mapping it back to original coordinates via quasi-periodic near identity and L-F transformation. It can be observed that the time traces match quite well. In Figure 2, phase portraits of the original and reduced system are compared. It can be seen that the phase portraits also match quite well.

Consider a Quasiperiodic Mathieu equation given by

where are incommensurate frequencies and is the small parameter. Equation (36) can be represented in the state space form as given by Equation (13) where

It is noted that the parametric excitation is strong and the L-F and modal transformation is applied to get Equation (38).

where and the detailed expression for was obtained by Mathematica™. At this stage, as before, a quasi-periodic near identity transformation can be substituted. After collecting the terms of order ε Equation (35) can be obtained

Expanding and in the form given by Equation (23) and collecting the terms via harmonic balance, elements of can be found out. It is noted that in this case the reducibility condition (given by Equation (25)) was satisfied. Thus, we could reduce the quasi-periodic system given by Equation (36) to a constant coefficient system given by Equation (40)

The solution of Equation (40) can be found in the closed form as

where and is the vector of typical initial conditions. To compare the results we integrated Equation (36) numerically and generated the time traces and phase portrait. For the reduced system, we used Equation (41) andapplied the quasi-periodic transformation, L-F and modal transformation to obtain the time traces and phase portrait of vector. Thesesolutions are compared in Figures 3 and 4. Figure 3 shows the time traces of the states and Figure 4 shows the phase portrait of the “Original” and “Reduced” states. It can be seen that these solutions match quite well.