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A phase plane portrait is a visual representation that displays the trajectory of a dynamical system in the plane, using a two-dimensional set of axes. This graphical depiction helps you to understand how the system behaves over time, given different initial conditions. In the context of our exercise, the system of differential equations provided generates a trajectory that can be plotted in the phase plane.
A phase plane portrait helps to identify critical points, which are points where the trajectory converges or diverges. These points are essential because different types of behavior such as nodes, saddle points, spirals, and more can be observed at these positions in the system. For example, the point \(3,0\) is identified as a saddle point in this exercise.
When constructing the phase plane portrait using tools such as graphing calculators or software, you look at local portraits around each critical point. This helps to verify the behavior predicted analytically (in our case at points like \(3,0\), \(0,0\) and \(5,2\)).

Linearization is a method used to approximate non-linear systems near a critical point by a linear one. It involves calculating the Jacobian matrix at the critical point, which gives a linear system representing the behavior of the original system.
This technique is particularly useful because linear systems are easier to analyze than nonlinear systems. By studying the linearized system, you can infer the local behavior of the original nonlinear system around the critical point.
In our exercise, the system of differential equations is linearized at the critical point \(3,0\). The result is a simpler linear system \(u' = 3u - 3v\) and \(v' = -2v\), making it easier to compute and analyze the eigenvalues, and consequently the nature of \(3,0\) as a saddle point.

The Jacobian matrix is an essential tool in understanding the behavior of a dynamical system near a critical point. It contains partial derivatives of the system's equations and helps in linearization of the system.
In this exercise, the Jacobian matrix \(J\) is calculated by taking partial derivatives of the system equations with respect to both \(x\) and \(y\). For example: \(\frac{\partial}{\partial x}(-3x + x^2 - xy) = -3 + 2x - y\).
Evaluating these partial derivatives at the critical point \(3,0\), the Jacobian becomes \[J = \begin{bmatrix} 3 & -3 \ 0 & -2 \end{bmatrix},\]
which plays a crucial role in examining the nature of the critical point by allowing computation of eigenvalues.

Eigenvalues arise naturally when analyzing linearized systems via the Jacobian matrix. They are important because they determine the stability and type of behavior near critical points.
To find the eigenvalues of the Jacobian matrix, you solve the characteristic equation \(\det(J - \lambda I) = 0\). For our system, it becomes \(\lambda^2 + 2\lambda - 3 = 0\). Solving this gives eigenvalues \(\lambda_1 = 3\) and \(\lambda_2 = -2\).
These eigenvalues tell us that the critical point \(3,0\) behaves like a saddle point, since one eigenvalue is positive and the other is negative. Positive eigenvalues indicate instability and suggest that trajectories move away from the critical point, while negative ones indicate stability, guiding trajectories towards it.

Understanding saddle points is key in analyzing the local dynamics near a critical point in a differential system. A saddle point is characterized by having both attracting and repelling directions due to eigenvalues of differing signs.
In our problem, the critical point \(3,0\) is identified as a saddle point. This finding comes from the eigenvalues of the Jacobian matrix: \(\lambda_1 = 3\) (positive) and \(\lambda_2 = -2\) (negative). One direction pushes trajectories away (due to the positive eigenvalue), while the other draws them in (due to the negative eigenvalue).
This results in a dynamic where trajectories near the saddle point move away in one direction and converge in another, leading to a distinctive, 'saddle-like' behavior in the phase plane portrait.