**2. Duffing equation**

Let us consider the standard (undamping) Duffing equation in the absence both friction (2*ε x*\_ ) and excitation (Fð Þ*t* ) forces [34, 35]

$$
\ddot{\varkappa} + p\varkappa + q\varkappa^3 = 0,\\
\varkappa = \varkappa(t),\tag{4}
$$

which is subjected to the following initial conditions

$$
\boldsymbol{\mathfrak{x}}(\mathbf{0}) = \boldsymbol{\mathfrak{x}}\_0 \otimes \boldsymbol{\mathfrak{x}}'(\mathbf{0}) = \dot{\boldsymbol{\mathfrak{x}}}\_0. \tag{5}
$$

The general solution of Eq. (4) maybe written in terms of any of the twelve Jacobian elliptic functions.

For example, let us assume

$$\mathbf{x}(t) = c\_1 \mathbf{cn}\left(\sqrt{a}t + c\_2, m\right). \tag{6}$$

By inserting solution (6) in Eq. (4), we get

$$2\ddot{\mathbf{x}} + p\mathbf{x} + q\mathbf{x}^3 = \left(c\_1^3 q - 2c\_1 m \alpha \right) \mathbf{cn}^3 + \left(2c\_1 m \alpha + c\_1 p - c\_1 \alpha \right) \mathbf{cn},\tag{7}$$

where cn <sup>¼</sup> cn ffiffiffi *ω* p ð Þ *t* þ *c*2, *m :*

Equating to zero the coefficients of cn *<sup>j</sup>* gives an algebraic system whose solution gives

$$
\omega = \sqrt{p + qc\_1^2} \text{ and } m = \frac{qc\_1^2}{2(p + qc\_1^2)}.\tag{8}
$$

Thus, the general solution of Eq. (4) reads

$$\mathbf{x}(t) = \text{cn}\left(\sqrt{p + qc\_1^2}t + c\_2, \frac{qc\_1^2}{2(p + qc\_1^2)}\right). \tag{9}$$

The values of the constants *c*<sup>1</sup> and *c*<sup>2</sup> could be determined from the initial conditions given in Eq. (5).

**Definition 1**. The number <sup>Δ</sup> <sup>¼</sup> *<sup>p</sup>* <sup>þ</sup> *qx*<sup>2</sup> 0 � �<sup>2</sup> <sup>þ</sup> <sup>2</sup>*qx*\_ 2 <sup>0</sup> is called the discriminant of the i.v.p. (4) -(5). Below three cases will be discussed depending on the sign of the discriminant Δ.
