For this special case, the general nonlinear jerk function is in the following form

$$\stackrel{\u20db}{x}+\gamma \dot{x}-\delta x\dot{x}\ddot{x}=0$$(3)

with the initial conditions

$$x\mathrm{(}0\mathrm{)}=0,\text{\hspace{0.17em}}\dot{x}\mathrm{(}0\mathrm{)}={a}_{0},\text{\hspace{0.17em}}\ddot{x}\mathrm{(}0\mathrm{)}=0$$(4)

Cubic nonlinear term is reordered

$$\delta =p\widehat{\delta}$$(5)

where *p* is the perturbation parameter.

For details of the MSLP method, see Pakdemirli et al. [8]. First, the time transformation is applied to (3)

$${\omega}^{3}{x}^{\u2034}+\gamma \omega {x}^{\prime}-p\widehat{\delta}{\omega}^{3}x{x}^{\prime}{x}^{\u2033}=0$$(6)

with the initial conditions

$$x\mathrm{(}0\mathrm{)}=0,\text{\hspace{0.17em}}{x}^{\prime}\mathrm{(}0\mathrm{)}=\frac{{a}_{0}}{\omega},\text{\hspace{0.17em}}{x}^{\u2033}\mathrm{(}0\mathrm{)}=0$$(7)

where prime is derivative with respect to the new time variable *τ*=*ωt*. Fast and slow timescales are

$${T}_{0}=\tau ,\text{\hspace{0.17em}}{T}_{1}=\text{\hspace{0.17em}}p\tau ,\text{\hspace{0.17em}}{T}_{2}={p}^{2}\tau $$(8)

Using

$$\frac{\text{d}}{\text{d}\tau}={D}_{0}+p{D}_{1}+{p}^{2}{D}_{2}+{p}^{3}{D}_{3}+\dots $$(9)

$$\frac{{\text{d}}^{2}}{\text{d}{\tau}^{2}}={D}_{0}^{2}+2p{D}_{0}{D}_{1}+{p}^{2}\mathrm{(}{D}_{1}^{2}+2{D}_{0}{D}_{2}\mathrm{)}+\dots $$(10)

$$\frac{{\text{d}}^{3}}{\text{d}{\tau}^{3}}={D}_{0}^{3}+3p{D}_{0}^{2}{D}_{1}+{p}^{2}\mathrm{(}3{D}_{0}{D}_{1}^{2}+3{D}_{0}^{2}{D}_{2}\mathrm{)}+\dots $$(11)

and substituting the expansions

$$x={x}_{0}\mathrm{(}{T}_{0},\text{\hspace{0.17em}}{T}_{1},\text{\hspace{0.17em}}{T}_{2}\mathrm{)}+p{x}_{1}\mathrm{(}{T}_{0},\text{\hspace{0.17em}}{T}_{1},\text{\hspace{0.17em}}{T}_{2}\mathrm{)}+{p}^{2}{x}_{2}\mathrm{(}{T}_{0},\text{\hspace{0.17em}}{T}_{1},\text{\hspace{0.17em}}{T}_{2}\mathrm{)}+\dots $$(12)

$$\gamma ={\omega}^{2}-p{\omega}_{1}-{p}^{2}{\omega}_{2}$$(13)

into (6) and initial conditions (7) yield after separation

$$\begin{array}{l}O\mathrm{(}1\mathrm{)}:{\omega}^{3}{D}_{0}^{3}{x}_{0}+{\omega}^{3}{D}_{0}{x}_{0}=0\text{\hspace{1em}}{x}_{0}\mathrm{(}0\mathrm{)}=0,\text{\hspace{0.17em}}{D}_{0}{x}_{0}\mathrm{(}0\mathrm{)}=\frac{{a}_{0}}{\omega},\text{\hspace{0.17em}}\\ \text{\hspace{1em}}{D}_{0}^{2}{x}_{0}\mathrm{(}0\mathrm{)}=0\end{array}$$(14)

$$\begin{array}{l}O\mathrm{(}p\mathrm{)}:{\omega}^{3}{D}_{0}^{3}{x}_{1}+{\omega}^{3}{D}_{0}{x}_{1}=-3{\omega}^{3}{D}_{0}^{2}{D}_{1}{x}_{0}-{\omega}^{3}{D}_{1}{x}_{0}+\omega {\omega}_{1}{D}_{0}{x}_{0}\\ \text{\hspace{1em}}+\widehat{\delta}{\omega}^{3}{x}_{0}\mathrm{(}{D}_{0}{x}_{0}\mathrm{)}\mathrm{(}{D}_{0}^{2}{x}_{0}\mathrm{)}\text{\hspace{1em}}{x}_{1}\mathrm{(}0\mathrm{)}=0,\text{\hspace{0.17em}}{D}_{0}{x}_{1}\mathrm{(}0\mathrm{)}+{D}_{1}{x}_{0}\mathrm{(}0\mathrm{)}=0,\text{\hspace{0.17em}}\\ \text{\hspace{1em}}{D}_{0}^{2}{x}_{1}\mathrm{(}0\mathrm{)}+2{D}_{0}{D}_{1}{x}_{0}\mathrm{(}0\mathrm{)}=0\end{array}$$(15)

$$\begin{array}{l}O\mathrm{(}{p}^{2}\mathrm{)}:{\omega}^{3}{D}_{0}^{3}{x}_{2}+{\omega}^{3}{D}_{0}{x}_{2}=-3{\omega}^{3}{D}_{0}^{2}{D}_{1}{x}_{1}-3{\omega}^{3}{D}_{0}{D}_{1}^{2}{x}_{0}\\ \text{\hspace{1em}}-3{\omega}^{3}{D}_{0}^{2}{D}_{2}{x}_{0}-{\omega}^{3}{D}_{1}{x}_{1}-{\omega}^{3}{D}_{2}{x}_{0}+\omega {\omega}_{1}\mathrm{(}{D}_{0}{x}_{1}+{D}_{1}{x}_{0}\mathrm{)}\\ \text{\hspace{1em}}+\omega {\omega}_{2}{D}_{0}{x}_{0}+\widehat{\delta}{\omega}^{3}[{x}_{0}\mathrm{(}{D}_{0}{x}_{0}\mathrm{)}\mathrm{(}{D}_{0}^{2}{x}_{1}+2{D}_{0}{D}_{1}{x}_{0}\mathrm{)}\\ \text{\hspace{1em}}+{x}_{0}\mathrm{(}{D}_{0}{x}_{1}+{D}_{1}{x}_{0}\mathrm{)}\mathrm{(}{D}_{0}^{2}{x}_{0}\mathrm{)}+{x}_{1}\mathrm{(}{D}_{0}{x}_{0}\mathrm{)}\mathrm{(}{D}_{0}^{2}{x}_{0}\mathrm{)}]\end{array}$$(16)

The solution at the first order is

$${x}_{0}=A{e}^{i{T}_{0}}+cc$$(17)

where

$$A=\frac{1}{2}a{e}^{i\lambda}$$(18)

is the polar representation of complex amplitudes. In terms of the real amplitudes and phases, the first-order solution is

$${x}_{0}=a\mathrm{cos}\mathrm{(}{T}_{0}+\lambda \mathrm{)}$$(19)

Applying the initial conditions yield

$$a\mathrm{(}0\mathrm{)}=\frac{-{a}_{0}}{\omega},\text{\hspace{0.17em}}\lambda \mathrm{(}0\mathrm{)}=\frac{\pi}{2}$$(20)

Equation (17) is substituted into (15) and secular terms are eliminated

$$2{D}_{1}A+\frac{i{\omega}_{1}A}{{\omega}^{2}}-i\widehat{\delta}{A}^{2}\overline{A}=0$$(21)

In the MSLP as outlined in [8], first *D*_{1}*A*=0 is selected and if the frequency correction is real, this choice is admissible. If *ω*_{1} turns out to be complex, then *D*_{1}*A*≠0 which implies *ω*_{1}=0 and secularities are eliminated by choosing *D*_{1}*A*. For (21), selection of

$${D}_{\text{1}}A=\text{0}\Rightarrow A=A\text{(}{T}_{\text{2}}\text{)}$$(22)

produces

$${\omega}_{1}=\widehat{\delta}A\overline{A}{\omega}^{2}=\frac{1}{4}\widehat{\delta}{a}^{2}{\omega}^{2}$$(23)

which is suitable because *ω*_{1} is real. The solution at order (*p*) is

$${x}_{1}=B{e}^{i{T}_{0}}+\frac{\widehat{\delta}}{24}{A}^{3}{e}^{3i{T}_{0}}+cc=b\mathrm{cos}\mathrm{(}{T}_{0}+\theta \mathrm{)}+\frac{\widehat{\delta}}{96}{a}^{3}\mathrm{cos}\mathrm{(}3{T}_{0}+3\lambda \mathrm{)}$$(24)

where

$$B=\frac{1}{2}b{e}^{i\theta}$$(25)

The initial conditions at this order imply

$$b\mathrm{(}0\mathrm{)}=-\frac{\widehat{\delta}}{32{\omega}^{3}}{a}_{0}^{3},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\theta \mathrm{(}0\mathrm{)}=\lambda \mathrm{(}0\mathrm{)}=\frac{\pi}{2}$$(26)

At the last order, (17) and (24) are inserted into (16) and secular terms are eliminated

$$2{D}_{2}A+i\mathrm{(}\frac{{\omega}_{2}A}{{\omega}^{2}}-\widehat{\delta}{A}^{2}\overline{B}-\widehat{\delta}A\overline{A}B+\frac{7{\widehat{\delta}}^{2}}{24}{A}^{3}{\overline{A}}^{2}\mathrm{)}=0$$(27)

According to the method, first *D*_{2}*A*=0 should be tried. This choice leads to a real *ω*_{2} and therefore admissible. After algebraic manipulations, (27) yields

$$a=-\frac{{a}_{0}}{\omega},\text{\hspace{0.17em}}b=-\frac{\widehat{\delta}}{32{\omega}^{3}}{a}_{0}^{3},\text{\hspace{0.17em}}\theta =\lambda =\frac{\pi}{2},\text{\hspace{0.17em}}{\omega}_{2}={\omega}^{2}\mathrm{(}-\frac{7{\widehat{\delta}}^{2}{a}^{4}}{384}+\frac{\widehat{\delta}ab}{2}\mathrm{)}$$(28)

The frequency is

$${\omega}^{2}=\gamma +p\frac{\widehat{\delta}}{4}{a}_{0}^{2}-{p}^{2}\frac{{\widehat{\delta}}^{2}}{384{\omega}^{2}}{a}_{0}^{4}$$(29)

Solving for the frequency yields

$$\omega =\sqrt{\frac{1}{2}\mathrm{(}\gamma +\frac{\delta}{4}{a}_{0}^{2}\mathrm{)}+\frac{1}{2}\sqrt{{\mathrm{(}\gamma +\frac{\delta}{4}{a}_{0}^{2}\mathrm{)}}^{2}-\frac{{\delta}^{2}}{96}{a}_{0}^{4}}}$$(30)

The final solution in terms of this frequency is

$$\begin{array}{l}x\mathrm{(}t\mathrm{)}=-\frac{{a}_{0}}{\omega}\mathrm{cos}\mathrm{(}\omega t+\frac{\pi}{2}\mathrm{)}-\frac{\delta {a}_{0}^{3}}{96{\omega}^{3}}[3\mathrm{cos}\mathrm{(}\omega t+\frac{\pi}{2}\mathrm{)}\\ \text{}+\mathrm{cos}\mathrm{(}3\omega t+\frac{3\pi}{2}\mathrm{)}]+O\mathrm{(}{p}^{2}\mathrm{)}\end{array}$$(31)

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