Question

(a) Determine all critical points of the given system of equations. (b) Find the corresponding linear system near each critical point. (c) Find the eigenalues of each linear system. What conclusions can you then draw about the nonlinear system? (d) Draw a phase portrait of the nonlinear system to confirm your conclusions, or to extend them in those cases where the linear system does not provide definite information about the nonlinear system. $$ d x / d t=x-y^{2}, \quad d y / d t=y-x^{2} $$

Short answer

2. What is the Jacobian matrix? 3. What are the eigenvalues for each critical point? 4. What type of critical points are (0,0) and (1,1)? 5. How can we draw a phase portrait of the nonlinear system? Answers: 1. The critical points of the system are (0,0) and (1,1). 2. The Jacobian matrix is given by: $$ J(x,y) = \begin{pmatrix} 1 & -2y\\ -2x & 1 \end{pmatrix} $$ 3. The eigenvalues are λ₁ = 1 (double) for the critical point (0,0), and λ₂ = -1, λ₃ = 3 for the critical point (1,1). 4. The critical point (0,0) is an unstable improper node, and the critical point (1,1) is a saddle point, which is also unstable. 5. To draw a phase portrait of the nonlinear system, we can use computer software like MATLAB or Mathematica, or sketch the directions of the vector field while considering the behavior of trajectories around the critical points.

Step-by-step solution

Step 1. Determine all critical points:

To find the critical points, we need to set both dx/dt and dy/dt equal to zero and solve the system of equations. $$ \begin{cases} x - y^2 = 0\\ y - x^2 = 0 \end{cases} $$ Since both equations are equal to zero, we can set them equal to each another: $$ x^2 = y^2 $$ It implies that \(x = y\) or \(x = -y\). Substituting these conditions back into original equations, we obtain the critical points: \((0,0)\) and \((1,1)\).

Step 2. Find the corresponding linear system near each critical point:

To find the linear system near each critical point, we need to compute the Jacobian matrix of the original system: $$ J(x,y) = \begin{pmatrix} \frac{\partial (x-y^2)}{\partial x} & \frac{\partial (x-y^2)}{\partial y}\\ \frac{\partial (y-x^2)}{\partial x} & \frac{\partial (y-x^2)}{\partial y} \end{pmatrix} = \begin{pmatrix} 1 & -2y\\ -2x & 1 \end{pmatrix} $$ Evaluate the Jacobian matrix at the critical points \((0,0)\) and \((1,1)\): $$ J(0,0) = \begin{pmatrix} 1 & 0\\ 0 & 1 \end{pmatrix}, \quad J(1,1) = \begin{pmatrix} 1 & -2\\ -2 & 1 \end{pmatrix} $$

Step 3. Find the eigenvalues of each linear system:

To find the eigenvalues of each linear system, we need to compute the determinant: $$ \det(J - \lambda I) = (1-\lambda)^2 - (-2y)(-2x) = \lambda^2 - 2\lambda + 1 - 4xy $$ For the critical points \((0,0)\) and \((1,1)\), we have: $$ \det(J(0,0) - \lambda I) = \lambda^2 - 2\lambda + 1\\ \det(J(1,1) - \lambda I) = \lambda^2 - 2\lambda - 3 $$ The eigenvalues are the roots of these characteristic equations: $$ \lambda^2 - 2\lambda + 1 = (\lambda - 1)^2\\ \lambda^2 - 2\lambda - 3 = (\lambda+1)(\lambda-3) $$ Thus, eigenvalues are \(\lambda_1 = 1\) (double) for the critical point \((0,0)\) and \(\lambda_2 = -1, \lambda_3 = 3\) for the critical point \((1,1)\).

Step 4. Analyze the nonlinear system based on the eigenvalues:

Since eigenvalues for the critical point \((0,0)\) are positive and double, the point \((0,0)\) is an unstable improper node. For the critical point \((1,1)\), since one eigenvalue is positive and the other is negative, the point \((1,1)\) is a saddle point which is also unstable.

Step 5. Draw a phase portrait of the nonlinear system:

Drawing the phase portrait would confirm that \((0,0)\) is an unstable improper node and \((1,1)\) is a saddle point. In this case, the phase portrait can be drawn using computer software like MATLAB or Mathematica, or by sketching the directions of the vector field and considering the behavior of trajectories around the critical points. It would show that trajectories near \((0,0)\) move away from the point, and the system's state moves along its eigendirections at the saddle point \((1,1)\).