91Ó°ÊÓ

To find the fixed points, we need to set both \(\dot{x}\) and \(\dot{y}\) to 0 and solve the resulting system of algebraic equations: \[ \begin{cases} 1+y-e^{-x}=0\\ x^3-y=0 \end{cases} \] First, express \(y\) from the second equation: \(y=x^3\). Now substitute this expression of \(y\) into the first equation: \[ 1+x^3-e^{-x} = 0 \] At this point, we need to solve this equation for \(x\). Unfortunately, the equation is not solvable analytically, so we need to find the fixed points numerically. Using numerical methods (e.g., Newton-Raphson or a graphical approach), we can find that the fixed point is approximately \(x \approx 1.2785\). Then, using the expression \(y = x^3\), we can find the corresponding \(y \approx 2.0893\). So the fixed point is approximately \((1.2785, 2.0893)\).

Now, let's linearize the system near the fixed point \((x^*, y^*)\) by computing the Jacobian matrix: \[ J(x,y) = \begin{bmatrix} \frac{\partial}{\partial x}(1+y-e^{-x}) & \frac{\partial}{\partial y}(1+y-e^{-x}) \\ \frac{\partial}{\partial x}(x^3-y) & \frac{\partial}{\partial y}(x^3-y) \end{bmatrix} = \begin{bmatrix} e^{-x} & 1 \\ 3x^2 & -1 \end{bmatrix} \] Evaluate the Jacobian at the fixed point \((1.2785, 2.0893)\): \[ J(1.2785,2.0893) \approx \begin{bmatrix} 0.2783 & 1\\ 4.9254 & -1 \end{bmatrix} \] Next, find the eigenvalues of the Jacobian matrix by solving the characteristic equation \(|J - \lambda I| = 0\). This yields: \[ (0.2783-\lambda)(-1-\lambda) - (1)(4.9254) = \lambda^2 + \lambda - 3.6536 = 0 \] By solving this quadratic equation using the quadratic formula, we find two eigenvalues: \(\lambda_1 \approx -2.0273\) and \(\lambda_2 \approx 1.7489\). Since there is one positive eigenvalue, the fixed point \((1.2785, 2.0893)\) is a saddle.

Now, using our linear analysis, we can sketch the local behavior around the fixed point. Given that the fixed point is a saddle, this means that the trajectories in its vicinity will be attracted along one eigendirection and repelled along another eigendirection, which can be found by the eigenvectors associated with the eigenvalues. In the neighborhood of the saddle point, the phase portrait will resemble the classic "saddle" shape.

Now that we have the local behavior around the fixed point, we can use our knowledge of the system's behavior to complete the phase portrait. To do this, we can sketch curves that represent the trajectories of the system. By simulating the system numerically with different initial conditions, we can visualize how the trajectories evolve over time and fill in the rest of the phase portrait. The saddle behavior around the fixed point should be clearly visible, while other trajectories should resemble the general behavior of the system.