*2.1. Monodromy Matrix*

Stability of periodic solutions in dynamic systems can be represented by using a mathematical tool. Suppose the solution *x*<sup>∗</sup> is a periodic feature of constant frequency *f* and its period *T* as time evolves. The problem of instability of periodic solutions have been studied within the framework of the methods developed by [7]. The trajectories of Equation (1) can be defined as *x* := *φ*(*<sup>t</sup>*, *<sup>z</sup>*). Equation (1) has periodic solution with *z* as an initial value, i.e., *φ*(*t* + *T*, *z*) = *φ*(*<sup>t</sup>*, *<sup>z</sup>*). The trajectory progresses to the regular orbit *x*<sup>∗</sup> = *φ*(*<sup>t</sup>*, *z*<sup>∗</sup>) as Equation (1) is perturbed with *z*<sup>∗</sup> + *d*0. The distance between trajectory progression and the periodic orbit is

$$d(t) = \phi(t, z^\* + d\_0) - \phi(t, z^\*) \tag{2}$$

where *z*<sup>∗</sup> is the specific initial value for a specific solution. The distance with each period *T* is calculated by *d*(*T*) and the linear representation with Taylor expansion becomes

$$d(T) = \frac{\partial \phi(T, z^\*)}{\partial z} d\_0 \tag{3}$$

This linear approximation is not applicable to large disturbances such as *d*1 in Figure 1. In Equation (3), the matrix

$$M = \frac{\partial \phi(T, z^\*)}{\partial z} \tag{4}$$

governs the growth or decay as the initial disturbance *d*0 is applied to the periodic solution. The matrix form of Equation (4) is the monodromy matrix. The characteristics of monodromy matrix are directly related to the behavior of the periodic solution and determined by its eigenvalues [7].

**Figure 1.** A periodic response of system and stable and unstable solution for initial disturbance *d*0 and *d*1.
