Maximum Interval of Stability and Convergence of Solution of a Forced Mathieu’s Equation

This paper investigates the maximum interval of stability and convergence of solution of a forced Mathieu’s equation, using a combination of Frobenius method and Eigenvalue approach. The results indicated that the equilibrium point was found to be unstable and maximum bounds were found on the derivative of the restoring force showing sharp condition for the existence of periodic solution. Furthermore, the solution to Mathieu’s equation converges which extends and improves some results in literature.


Introduction
Consider a harmonically forced Mathieu's equation defined by ( ) For small ε , this equation describes a simple harmonic oscillator whose frequency is a periodic function of time with the boundary condition as; is the second derivative with respect to time, f is the amplitude of a periodic driving force, w and ε are the Mathieu's parameters and λ is the angular frequency of the periodic driving force.
Mathieu's equation is a special case of a linear second order homogenous equation [1]. In [2], Equation (1) was discussed in connection with problem of vibrations in elliptical membrane and developed the leading terms of the series known as Mathieu's function. Mathieu's function was further investigated by a number of researchers who found a considerable amount of results. [3] [4] [5] [6] wrote that Mathieu's differential equation occurs in two main categories involving elliptical geometrics, such as analysis of vibrating modes, elliptical membrane, the propagation modes of elliptic pipes and the oscillation of water in a lake of elliptic shape. Mathieu's equation arises after separating the wave equation using elliptic coordinates. Secondly, problems involving periodic motion are the trajectory of an electron in a periodic array of atoms [7].
Stability is an important concept in linear and nonlinear analysis. For instance, roughly speaking, a physical system is stable if small changes at sometimes cause only a small change in the behavior of the system in future [8]. Analytically, stability is determined by the interval placed on the total derivative of the system form by the given differential equation.  , ,

Preliminaries
, , x t t x for later times.
x is a regular singular point and Then Frobenius method is effective at regular singular point of the form.
and that are analytic at 0 x = , then they will have Maclaurin series expansion with radius of convergence 1 0 r > and 2 0 r > respectively. That is Then the point 0 0 x = is called a regular singular point of (5).
with radius of convergence R, then term by term differential and integration of the power series is permitted and does not change the radius of convergence that is; Theorem 2.6. Let A be an n n * matrix and let the eigenvalue of A be de- Re A λ ≤ and all the eigenvalues of A with real part zero are simple, then zero is a stable fixed point of (10) Re A λ < , then zero is a globally asymptotically stable solution of (10) (iii) If there is an eigenvalue of A with positive real part, then zero is unstable

Stability Analysis of Mathieu Equation
We consider the equation where p is a positive constant. Equation (11) can be written as (12) can be written as The equivalent system is given by (14) can be reduced in matrix form as (15) can further be written as For the eigenvalue of A we compute ip λ = ± (20) Since the eigenvalue is in complex form with real part equals to zero, then the equilibrium point is unstable.

Convergence of Mathieu Equation
We consider the Mathieu equation of the form; n n p a a n k n k Given the value of 0 a , we can evaluate 2 a , 4 a , etc. The odd n a are completely independent and as far as getting a solution is concerned, we can put them all to zero. This independent of the odd and even n a is a consequence of the fact that odd and even solution of the differential equation are possible.
In order to generate these odd/even solution, it is easiest to put 1 0 a = in order not to create extra solution by merely using some of the key solution into that of 0 k = .
Define an additional argument for solving the ODE

Conclusions
From our result, we observed that the solution existed using the Frobenius method and also periodic. The solution converged at the equilibrium point but unfortunately this convergence did not imply asymptotic stability, and the converse is true. The solution was observed to be unbounded for the given parameters w-ε. Since the solution is unbounded, we concluded that the corresponding equilibrium point in w-ε plane is unstable. This can be seen in Figure 3 where     In Figure 2, the solution was also periodic using the second solution function values and independent variable values. The starting point of the trajectory was far from the origin hence showing instability of the system.
In Figure 3, phase portrait of Mathieu's equation was obtained by relating X 2 and X 1 . The phase portrait was seen to be far away from the equilibrium point.
This shows that the solution of Mathieu's equation is unstable with the maximum displacement coinciding with the maximum displacement of X 1 in Figure   1. These values represent the maximum interval of stability of the system.