## 6 Analysis of the PPR Region for the Tulip and Bowl
\begin{array}{l}\frac{1-C_{1}\cos(2\varphi_{1})}{1-C_{2}\cos(2\varphi_{1})} \approx 1-2\mu\cos(2\varphi_{1})=1-2\mu\cos(\theta_{1}),\\ \sqrt{\frac{1-C_{1}\cos(2\varphi_{1})}{1-C_{2}\cos(2\varphi_{1})}}\approx(1-2 \mu\cos(2\varphi_{1}))^{-\frac{1}{2}}\approx 1+\mu\cos(2\varphi_{1})=1+\mu\cos( \theta_{1}),\\ \mu=\frac{1}{2}(C_{2}-C_{1}),\end{array}  \tag{64}\]  
where in Equation (64), the condition given by Equation (59) was considered. With the relations in Equation (64), Equation (53) becomes  
\[ \begin{array}{l}\stackrel{{\bullet\bullet}}{{\rm w}}_{1}^{ \bullet}+\Omega_{1}^{2}{\rm w}_{1}=-2\xi\Omega_{1}(1+\mu\cos(\theta_{1})){ \rm w}_{1}^{\bullet}+2\mu\Omega_{1}^{2}\cos(\theta_{1}){\rm w}_{1}-\Gamma_{1} {\rm w}_{1}^{3}-\Gamma_{2}{\rm w}_{1}^{5}.\end{array}  \tag{65}\]  
The assumption that the damping ratio \(\xi\), the excitation coefficient \(\mu\), and the coefficients of cubic and quintic nonlinearity \(\Gamma_{1},\Gamma_{2}\) are small is incorporated in the analysis by representing these quantities in the following form:  
\[\xi=\varepsilon\xi,\mu=\varepsilon\mu,\Gamma_{1}=\varepsilon\Gamma_{1},\Gamma _{2}=\varepsilon\Gamma_{2}, \tag{66}\]  
where \(\varepsilon\) is the same small positive parameter used to obtain the slowing time. It is also assumed that the excitation frequency \(\eta_{1}\) and the excitation parameter \(\mu\) vary slowly with time, such that  
\[\frac{{\rm d}\theta_{1}}{{\rm d}t}=\eta_{1}(\tau),\mu=\mu(\tau). \tag{67}\]  
Equation (65) becomes the following after neglecting the terms in \(\varepsilon^{2}\):  
\[ \begin{array}{l}\stackrel{{\bullet\bullet}}{{\rm w}}_{1}^{ \bullet}+\Omega_{1}^{2}{\rm w}_{1}=\varepsilon\Big{[}-2\xi\Omega_{1}{\rm w}_ {1}^{\bullet}+2\mu\Omega_{1}^{2}\cos(\theta_{1}){\rm w}_{1}-\Gamma_{1}{\rm w} _{1}^{3}-\Gamma_{2}{\rm w}_{1}^{5}\Big{]}.\end{array}  \tag{68}\]  
Regarding Equation (68), the right-hand side represents a perturbation of the mathematical form \({\rm H}({\rm w}_{1},\theta_{1})\), being a periodic function in \(\theta_{1}\) with period \(2\pi\), while the left-hand side of the equation is a linear oscillator. By considering all these physical considerations and confining our attention to the investigation of the PPR region, a solution for Equation (68) is sought after in the following form to the first-order approximation in \(\varepsilon\):  
\[{\rm w}_{1}={\rm W}\cos\left(\frac{1}{2}\theta_{1}+\Psi\right), \tag{69}\]  
where \({\rm W}\) and \(\Psi\) are functions of time defined by the system of differential equations:  
\[\left\{ \begin{array}{l}\frac{{\rm d}{\rm W}}{{\rm d}t}=\varepsilon{\rm A}_{1 }(\tau,{\rm W},\psi)\\ \frac{{\rm d}{\rm W}}{{\rm d}t}=\Omega_{1}-\frac{1}{2}\eta_{1}(\tau)+\varepsilon{ \rm B}_{1}(\tau,{\rm W},\psi)\end{array} \right.. \tag{70}\]  
Differentiating the right-hand side of Equation (70) and expanding the results in powers of \(\varepsilon\) yields  
\[ \begin{array}{l}\stackrel{{\bullet\bullet}}{{\rm W}}=\frac{{ \rm d}^{2}{\rm W}}{{\rm d}t^{2}}=\varepsilon\frac{\partial{\rm A}_{1}}{ \partial\psi}\Big{(}\Omega_{1}-\frac{1}{2}\eta_{1}\Big{)}+\varepsilon^{2}....,\\ \stackrel{{\bullet\bullet}}{{\rm W}}=\frac{{\rm d}^{2}\psi}{{\rm d }t^{2}}=\varepsilon\Big{[}-\frac{1}{2}\frac{\partial\eta_{1}}{\partial\tau}+ \frac{\partial{\rm B}_{1}}{\partial\psi}\Big{(}\Omega_{1}-\frac{1}{2}\eta_{1} \Big{)}\Big{]}+\varepsilon^{2}....\end{array}  \tag{71}\]  
When differentiating the right-hand side of Equation (69) while considering the systems of Equations (70) and (71) and retaining only the first-order terms, we obtained the following:  
\[ \begin{array}{l}\stackrel{{\bullet}}{{\rm w}}_{1}^{\bullet}=-{ \rm W}\Omega_{1}\sin\biggl{(}\frac{1}{2}\theta_{1}+\psi\biggr{)}+\varepsilon \biggl{[}{\rm A}_{1}\cos\biggl{(}\frac{1}{2}\theta_{1}+\psi\biggr{)}-{\rm W}{\rm B }_{1}\sin\biggl{(}\frac{1}{2}\theta_{1}+\psi\biggr{)}\biggr{]},\end{array}