1. Introduction
A matrix function 
with a square matrix of dimension 
is termed quasi-periodic with k incommensurable frequencies 
[1] [2] . A quasi-periodic function 
can be showed in the form
(1)
where a continuous function is 
of period 
in
. In addition, we can always assume that 
are independent [2] . As Moser [3] stated, the class of all almost periodic functions is not separable while 
is. The integral of a quasi-periodic function is not quasi-periodic even if the mean value of 
is zero [3] .
Let’s consider the linear equation
(2)
where 
and 
is a matrix depending quasi-periodically on time. The quasi-periodicity of 
enables it possible to raise the Equation (2) to a system of linear equations [4] [5] on 
basically writing
(3)
where the Equation (2) is acquired when the initial value for 
is zero.
Now let’s consider the matrix equation for (3), so that the lifted system [2] becomes
(4)
Equation (2) is said to be reducible whenever there is a linear time-varying change of variables
(5)
called Lyapunov-Perron (L-P) transformation [2] , which transforms the system into an equation like 
where 
is a constant matrix. As Fink [6] mentioned, if
, then all systems are reducible by the Floquet theory. This view is also applicable for common linear systems [7] like
(6)
and it means that, whenever a system like that is L-P reducible to a constant coefficients system like
(7)
This will result in many properties of the original system “such as the growth of the solutions or their boundness” being the same as those of the reduced system with constant coefficients [2] .
The primary objective of this work is to develop a practical approach for reducibility of quasi-periodic system with strong parametric periodic excitation. It is noted that in the past, the researchers have studied quasi-periodic system where the order of quasi-periodicity is less than the order of the linear term [8] [9] as given by Equation (8)
(8)
In this work, we relax this requirement. We assume the order of themost dominant “strong” periodic excitation to be of the same order as the constant term given by Equation (9) and use the L-F transformation and quasi-periodic near identity transformation to
(9)
reduce the system to a constant form. This research will present a practical approach to achievereducibility of quasi-periodic system and will lay the groundwork for future efforts in optimizing such a process.
1.1. Floquet Theory Overview
Floquet theory is very useful for finding the response or stability of linear time-periodic equations [4] [10] [11] . Consider the linear periodic system
(10)
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:
(11)
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] .
1.2. Quasi-Periodic System Reducibility
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
(12)
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)
(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.
2. Reducibility Problem Formulation
Consider the time periodic part of Equation (13),
(14)
The state transition matrix (STM) of Equation (11) can be factored as
(15)
where 
is typically
, periodic such that 
and 
is a real-valued 
constant matrix. Applying the L-F transformation 
to Equation (13) gives
(16)
Application of modal transformation 
to Equation (16) converts the constant part of the system in the Jordan form, where 
is eigenvector of R
(17)
where 
is the Jordon Form of R and 
is quasi-periodic matrix with
. We use following near-identity transformation to Equation (17)
(18)
where the unknown nonlinear function 
is quasi-periodic. Substituting Equation (18) in Equation (16), we obtain
(19)
Assuming 
and expanding Equation (19) and simplification, we obtain
(20)
Now, if
(21)
Then, Equation (20) reduces to
. Thus, quasi-periodic system given by Equation (16) will be reduced to a constant system.
Equation (21) is similar to the homological equation obtained in the normal form reduction. By collecting the coefficient of
, we get the reducibility equation
(22)
For illustration, assume all the matrices in Equation (22) are 
of the form
(23)
Equation (23) can be expanded in the scalar form as
(24)
It can be noted that Equations (21a) and (21d) can be solved as
(25)
To find the solution of Equation (24b) and (24c), elements of 
and 
will have to be expanded in multiple Fourier series as
(26)
where 
are the unknown and 
known coefficients. Term by term comparison yields
(27)
Thus, Equation (27) can be solved if
(28)
Equation (28) is called as the reducibility condition. Thus, a quasi-periodic system can be reduced to a constant system provided Equation (28) is satisfied. It is also noted that since the order of quasi-periodicity is
, we obtain solution of 
to the order
. It is possible to extend this approach to higher orders of
, if needed.
3. Applications
In this section, we present two examples—a commutative system and a quasi-periodic Mathieu equation. We apply the procedure discussed in Section two to reduce the quasi-periodic system to a constant one.
3.1. Commutative System
Consider the following commutative system
(29)
where
and 
is a system parameter The L-F transformation for matrix 
is
(30)
and
Applying the L-F Transformation 
to Equation (31) yields
(31)
It is noted that for this special case matrix 
is in the Jordan form
. Thus
(32)
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
(33)
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)
(34)
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
(35)
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.
3.2. Quasiperiodic Mathieu Equation
Consider a Quasiperiodic Mathieu equation given by
(36)
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
(37)
It is noted that the parametric excitation is strong and the L-F and modal transformation is applied to get Equation (38).
(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
(39)
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)
(40)
The solution of Equation (40) can be found in the closed form as
(41)
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.
4. Conclusions
In this work, a new approach for reducibility of quasi-periodic system analysis is presented using the L-F transformation and quasi-periodic near identity transformation. In this process, one obtains the reducibility conditions and the quasi-periodic system can be converted to a constant coefficient system provided the reducibility conditions are satisfied.
The resulting homological is expanded using multiple Fourier series which can solved for the unknown Fourier coefficients of the near-identity transformation coefficients via harmonic balance. Two examples presented a commutative system and quasi-periodic Mathieu equation. In both cases, the parametric excitation is
strong. Simulations and phase plane were plotted to compare results from numerical integration and closed form solution. It can be seen that the results matched quite well. Thus, if the reducibility conditions are satisfied then the quasi-periodic systems with strong parametric excitation can be reduced to a constant form using the L-F transformation. It is possible to analyze or control this reduced order time invariant system and map the results back using appropriate transformations to study and control original quasi-periodic system.