One of many Lagrangean line structures in turbulent flows is considered in elementary form. The example, called Shilnikov system, is constructed as simplified velocity field that contains several critical points. The solution of the autonomous Shilnikov system [74, 75] for the Lagrangean position field \(\Phi_{1}(\mathbf{X}, \tau)\), \(\Phi_{2}(\mathbf{X}, \tau), \Phi_{3}(\mathbf{X}, \tau)\)

\[ \frac{d \Phi_{1}}{d \tau}=\Phi_{1} \Phi_{3}-w \Phi_{2} \]

\[ \begin{gathered} \frac{d \Phi_{2}}{d \tau}=w \Phi_{1}+\Phi_{2} \Phi_{3} \\ \frac{d \Phi_{3}}{d \tau}=P+\Phi_{3}-\frac{1}{3} \Phi_{3}^{3}-\left(\Phi_{1}^{2}+\Phi_{2}^{2}\right)\left(1+q \Phi_{1}+e \Phi_{3}\right) \end{gathered} \]
generates such a structure, where \(\Phi_{\alpha}(\mathbf{X}, \tau)\) is the position (Cartesian coordinates) of a material point at time \(\tau\), which started at \(\mathbf{X}\) at time zero, and \(w=10, e=0.5\), \(q=0.7\) are constants and \(P>0\) is the bifurcation parameter.
(5.1) Compute the critical points \(\frac{d \Phi_{\alpha}}{d \tau}=0, \alpha=1,2,3\) for \(P>0\). Show that the critical points are on the \(\mathbf{e}_{3}\)-axis and that the location of the critical points depends only on the bifurcation parameter \(P\). Compute the value \(P_{c}\) of \(P\) separating the case of three real critical points from a single real critical point and two complex conjugate points.
(5.2) Choose two values for the bifurcation parameter \(0