On the Stability of the Damped Mathieu Equation

The Mathieu equation $X^n$ +(a–2q cos2t)x=0 has been studied in great detail since its discovery in 1868. Classical method so analysis of equation (1) have been expounded in the text by McLachlan [1]. Periodic solutions to this equation are called Mathieu functions, known in the form of infinite series. These do not exist for all values of a and q. Given a value of q, there are only accountably in finite number of a's for which periodic solutions to the Mathieu equation exist. These a's are called characteristic numbers (each of these corresponds to a particular Mathieu function, at a given q), and the plot of a vs q for a given Mathieu function is termed its characteristic curve. The periodic solutions of eqn.(1) that reduce to cos mz or sin mz ( m being an integer) when q = 0 are called Mathieu functions of integral order. The characteristic curves for Mathieu functions of integral order separate the a-q plane into stable and unstable domains. Thus, once the characteristic curves of Mathieu functions of integral order have been plotted, the a-q plane is separated once and for all into distinct domains