**8. Damped Cubic-Quintic Oscillator**

Let us define the following i.v.p.

$$\begin{cases} \ddot{\boldsymbol{x}} + + 2\boldsymbol{e}\dot{\boldsymbol{x}} + p\boldsymbol{\varkappa} + q\boldsymbol{\varkappa}^3 + r\boldsymbol{\varkappa}^5 = \mathbf{0}, \\\ \boldsymbol{\varkappa}(\mathbf{0}) = \boldsymbol{\varkappa}\_0 \, \boldsymbol{\&} \boldsymbol{\varkappa}'(\mathbf{0}) = \dot{\boldsymbol{\varkappa}}\_0. \end{cases} \tag{106}$$

We seek approximate analytic solution in the ansatz form

$$\mathbf{x}(t) = \exp\left(-\rho t\right)\mathbf{y}(f(t)),\tag{107}$$

with

$$f(t) = \frac{1 - \exp\left(-2(\varepsilon - \rho)t\right)}{2(\varepsilon - \rho)},\tag{108}$$

where *y* � *y t*ð Þ is the exact solution to the i.v.p.

$$\begin{cases} \ddot{\mathbf{y}} + (p - \epsilon \rho + \rho^2) \mathbf{y} + q \mathbf{y}^3 + r \mathbf{y}^5 = \mathbf{0}, \\\ y(\mathbf{0}) = \mathbf{x}\_0 \ \& \mathbf{y}'(\mathbf{0}) = \dot{\mathbf{x}}\_0 + \mathbf{x}\_0 \rho. \end{cases} \tag{109}$$

Define the residual

$$R(t) = \ddot{\varkappa} + 2e\dot{\varkappa} + p\varkappa + q\varkappa^3 + r\varkappa^5,\tag{110}$$

then, the condition *R*<sup>0</sup> ð Þ¼ 0 0 gives

$$2\rho^3 - 6\epsilon\rho^2 + \left(2p + 3q\mathbf{x}\_0^2 + 4r\mathbf{x}\_0 + 4\epsilon^2\right)\rho - 2p\epsilon - 2q\mathbf{x}\_0^2\epsilon - 2r\mathbf{x}\_0^4\varepsilon = 0.\tag{111}$$

Some real roots of Eq. (111) give the value of *ρ*. For *x*<sup>0</sup> ¼ 0, the default value of *ρ* could be chosen as *ρ* ¼ ð Þ 2*=*3 *ε*.

**Example 9.**

Let

$$\begin{cases} \ddot{\mathbf{x}} + \mathbf{0}.05\dot{\mathbf{x}} + \mathbf{9}\mathbf{6}.6289\mathbf{x} - 3.5\mathbf{x}^3 - \mathbf{0}.8\mathbf{x}^5 = \mathbf{0}, \\ \qquad \mathbf{x}(\mathbf{0}) = \mathbf{1} \,\&\mathbf{x}'(\mathbf{0}) = \mathbf{0}. \end{cases} \tag{112}$$

The approximate analytical solution of the i.v.p. is given by

*x*appð Þ*t*

$$=\frac{1.00785e^{-0.0259\text{M}t}(\text{cm}(f(t)|m)-0.00261773\text{dm}(f(t)|m)\text{sn}(f(t)|m))}{\left[1+\frac{0.0107805(\text{cm}(f(t)|m)-0.00261773\text{dm}(f(t)|m)\text{sn}(f(t)|m)}{\left(1+5.14\times10^{-9}\text{sn}(f(t)|m)^2\right)^{2}}+\frac{0.00785e^{-0.0259\text{m}(f(t)|m)\text{sn}(f(t)|m)^2}}{\left(1+5.14\times10^{-9}\text{sn}(f(t)|m)^2\right)^{2}}\right]}+\frac{1.00785e^{-0.0259\text{m}(f(t)|m)\text{sn}(f(t)|m)\text{sn}(f(t)|m)^2}}{\left(1+5.14\times10^{-9}\text{sn}(f(t)|m)^2\right)^{4}}\tag{112}$$

where

$$f(t) = 6807.39 - 6807.39e^{0.00142017t} \text{ \& } m = -0.000750607.\tag{114}$$

**Figure 9.** *A comparison between RK4 solution and (a) Solution (113) and (b) Zuñiga solution [17].*

The distance error as compared to the RK4 numerical solution reads

$$\max\_{0 \le t \le 5} \left| \boldsymbol{\omega}\_{\text{app}}(t) - \boldsymbol{\omega}\_{\text{RK}4}(t) \right| = 0.0561216. \tag{115}$$

In **Figure 9**, we make a comparison between our solution, RK4 solution, and Zuñiga solution given in Ref. [17]. It is clear that the accuracy of our solution is better than the solution of Zuñiga [17].

### **9. Realistic physical applications**

The above solutions could be applied to various fields of physics and engineering such as they could be used for describing the behavior of oscillations in RLC electronic circuits, plasma physics etc. In the below section, the above solution will be devoted for studying oscillations in various plasma models.

#### **9.1 Nonlinear oscillations in RLC series circuits with external source**

In the RLC series circuits consisting of a linear resistor with resistance *R* in Ohm unit, a linear inductor with inductance *L* in Henry unit, and nonlinear capacitor with capacitance *C* in Farady unit as well as external applied voltage *E* in voltage unit, the Kirchhoff's voltage law (KVL) could be written as

$$L\partial\_t i'(t) + i(t)R + \mathfrak{s}q + aq^2 = E,\tag{116}$$

where the relation between the current the charge is given by *<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup>tq* � *<sup>q</sup>*\_, *<sup>i</sup>* <sup>0</sup> � *<sup>∂</sup>ti*, the coefficients ð Þ *a*, *s* are related to the nonlinear capacitor, and *E* represents the voltage of the battery which is constant. By reorganizing Eq. (116), the following constant forced and damped Helmholtz equation could be obtained as

$$
\ddot{q} + 2\chi \dot{q} + aq + \beta q^2 = F,\tag{117}
$$

with *<sup>γ</sup>* <sup>¼</sup> *<sup>R</sup>=*ð Þ <sup>2</sup>*<sup>L</sup>* , *<sup>α</sup>* <sup>¼</sup> <sup>1</sup>*=*ð Þ *LC* , *<sup>β</sup>* <sup>¼</sup> <sup>1</sup>*<sup>=</sup> Cq*0*<sup>L</sup>* , and *<sup>F</sup>* <sup>¼</sup> *<sup>E</sup>=<sup>L</sup>* where *<sup>q</sup>*<sup>0</sup> <sup>¼</sup> *q t*ð Þ <sup>¼</sup> <sup>0</sup> is the initial charge value at *<sup>t</sup>* <sup>¼</sup> 0, €*<sup>q</sup>* � *<sup>∂</sup>*<sup>2</sup> *<sup>t</sup> <sup>q</sup>*, and *<sup>q</sup>*\_ � *<sup>∂</sup>tq*.

The solution of Eq. (117) can be devoted for interpreting and analyzing the oscillations that can generated in the RLC circuit.
