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. $$ d x / d t=x+x^{2}+y^{2}, \quad d y / d t=y-x y $$

Short answer

Answer: The nonlinear system has an unstable node at the critical point (0, 0) and a saddle point at the critical point (-1, 0).

Step-by-step solution

Find the critical points

To find the critical points of the given system of equations, set both dx/dt and dy/dt equal to 0 and solve for x and y. $$ \begin{cases} x(1 + x + y^2) = 0\\ y(1 - x) = 0 \end{cases} $$ From the second equation, we have two cases: y = 0 and 1 - x = 0. Case 1: y = 0 $$ x(1 + x) = 0, $$ So we have two critical points: (0, 0) and (-1, 0). Case 2: 1 - x = 0 $$ x = 1, $$ Thus, we have y^2 = -1, which doesn't have any real solutions. Hence, no critical points for this case. So, our critical points are (0, 0) and (-1, 0).

Linearize the system

Linearize the given system near each critical point by finding the Jacobian matrix. The Jacobian matrix for the given system is: $$ \begin{bmatrix} \frac{\partial (x + x^2 + y^2)}{\partial x} & \frac{\partial (x + x^2 + y^2)}{\partial y} \\ \frac{\partial (y - xy)}{\partial x} & \frac{\partial (y - xy)}{\partial y} \end{bmatrix} = \begin{bmatrix} 1 + 2x & 2y \\ -y & 1-x \end{bmatrix} $$ Now, we'll evaluate the Jacobian matrix at each critical point. For (0, 0): $$ J(0, 0) = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} $$ For (-1, 0): $$ J(-1, 0) = \begin{bmatrix} -1 & 0 \\ 0 & 2 \end{bmatrix} $$

Find Eigenvalues

We will now find the eigenvalues of the Jacobian matrices for each critical point. For J(0, 0): The eigenvalues are simply the diagonal elements, λ1 = 1 and λ2 = 1. For J(-1, 0): The eigenvalues are again the diagonal elements, λ1 = -1 and λ2 = 2.

Analyze the Nonlinear System

Based on the eigenvalues, we can draw conclusions about the nonlinear system around each critical point. For the critical point (0, 0), both eigenvalues are positive, suggesting that the nonlinear system has an unstable node at this point. For the critical point (-1, 0), one eigenvalue is positive and the other is negative, suggesting that the nonlinear system has a saddle point at this point.

Draw the Phase Portrait

We have shown that the nonlinear system at (0, 0) has an unstable node, and at (-1, 0) has a saddle point. A phase portrait would help confirm these conclusions. To draw the phase portrait, we will sketch the vector field of the nonlinear system and highlight the behavior around the critical points. In this case, we can already draw the conclusion that the system has an unstable node at (0, 0) and a saddle point at (-1, 0) without actually drawing the phase portrait. Nonetheless, drawing the phase portrait would help visualize this behavior and confirm our conclusions. To do so, you can use software such as MATLAB, Mathematica, or an online tool like 'PhasePortrait' on the website dfield.

Key concepts

Eigenvalues of Linear Systems

Understanding the eigenvalues of linear systems is a crucial aspect of analyzing differential equations. When dealing with a set of linear differential equations, eigenvalues help us determine the stability and type of each equilibrium point, or 'critical point', of the system. A critical point is where the system does not change; that is, the point where the derivative with respect to time is zero.

In the context of our exercise, after linearizing the system near the critical points, we calculate the eigenvalues of the resulting Jacobian matrices. The sign of the eigenvalues gives us insights into the system's behavior around those points. If both eigenvalues are positive, as with the critical point at (0, 0), it indicates an 'unstable node', meaning solutions move away from the critical point as time progresses. On the contrary, if both eigenvalues are negative, the point would be stable, attracting nearby solutions.

However, when we have eigenvalues of differing signs as with the point (-1, 0), this indicates a 'saddle point'. Here, some solutions are attracted to the point while others are repelled, depending on the direction from which they approach.
Moreover, complex eigenvalues indicate the presence of oscillatory behavior around the critical point, leading to a focus or a center, which isn't the case in our example.

Phase Portrait

A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each point in the phase plane represents the state of the system at a given time, and trajectories show how these states evolve over time. For our nonlinear system given by the coupled differential equations, the phase portrait will illustrate the behavior around the critical points identified, capturing the essence of the system's dynamics.

Phase portraits are invaluable when eigenvalues don't give a complete picture, especially in the case of higher-dimensional systems or when nonlinearities play a significant role. Even when eigenvalues indicate certain behaviors near critical points, such as nodes or saddle points, sketching a phase portrait allows us to visualize the global behavior of the system, observe possible limit cycles, and understand more complex dynamics that may occur away from the critical points.

To construct a phase portrait, one usually computes several solution trajectories with different initial conditions, focusing on how these trajectories move around critical points. The unstable node at (0, 0) of our system will be represented by trajectories moving away from this point, indicating instability. Conversely, the saddle point at (-1, 0) would show trajectories that approach the point along certain paths and move away along others.

Jacobian Matrix

The Jacobian matrix is a powerful tool in understanding the behavior of nonlinear systems near critical points. It is a matrix of first-order partial derivatives and represents the best linear approximation to a nonlinear system at a given point. For a system of two differential equations, the Jacobian matrix is a 2x2 matrix with components determined by how the equations change with respect to each variable.

In our exercise, the Jacobian matrices evaluated at the critical points provide the coefficients for the linear systems that approximate our original nonlinear system near these points. The Jacobian matrix at the critical point (0, 0) produces a simple identity matrix indicating that near (0, 0), the system behaves like a linear system with state variables growing independently at the same rate. Meanwhile, the Jacobian at (-1, 0) leads to a matrix with different diagonal elements, signaling distinct dynamics in the directions of the x and y variables.

Calculating the Jacobian matrix is a vital step in linearization, enabling us to glean the local behavior of the nonlinear system. By assessing the nature of the matrix, especially its eigenvalues, we gain insights applicable to stability analysis and how the system's trajectories behave close to equilibrium — glimpses into the larger, complex behavior of the nonlinear system.