Along any solution curve that asymptotically approaches the saddle, either in forward or backward time, the value of \( H \) must be the same as at the saddle. Find a point on each of the 3 solution curves, colored black in the plot above, that asymptotically approach the saddle in either backwards or forwards time or both. Answers are \( x, y \) coordinates as a vector. **Phase Portrait and Differential Equations** In the nonlinear system of differential equations: \[ \begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} y \\ x^2 - x \end{bmatrix} \] The quantity \[ H(x, y) = \frac{1}{2}y^2 + \frac{1}{2}x^2 - \frac{1}{3}x^3 \] is constant along solution curves. **Graph Explanation** The graph illustrates the phase portrait of the given differential system. Key features include: - **Arrows (Vector Field):** Red arrows depict the direction of the vector field, indicating the flow direction within the phase space. - **Solution Curves:** The blue curves represent the trajectories or solution curves of the system. These curves show how the system evolves over time. - **Critical Points:** The graph highlights a center (blue point) and a saddle point (red point). The center is surrounded by closed orbits, while the saddle point has trajectories that approach and veer away. - **Conservation of H(x, y):** The function \( H(x, y) \) remains constant on each blue solution curve, indicating conserved quantities in the system. This visualization provides a geometric interpretation of the system's behavior, showcasing stability and the nature of the critical points.