Another form is the algebraic form of the Mathieu equation is
In this case, for a scalar particle in this background metric, one gets the Mathieu equation which is a special case of the Heun equation.
If the right side of Equation (19) is equal to zero, it becomes the classical Mathieu equation. The stability chart of the Mathieu equation is well known [13-15], and the region near [delta] = 0 in the [epsilon] - [delta] parametric plane is shown in Figure 5.
Equation (24) is a Mathieu equation without external force.
Note that the stability condition will be still valid for the case of [delta] < 0 according to the research conclusion for the Mathieu equation. We denote two functions: