Question

The coefficient matrix contains a parameter \(\alpha\). In each of these problems: (a) Determine the eigervalues in terms of \(\alpha\). (b) Find the critical value or values of \(\alpha\) where the qualitative nature of the phase portrait for the system changes. (c) Draw a phase portrait for a value of \(\alpha\) slightly below, and for another value slightly above, each crititical value. $$ \mathbf{x}^{\prime}=\left(\begin{array}{ll}{-1} & {\alpha} \\ {-1} & {-1}\end{array}\right) \mathbf{x} $$

Short answer

Answer: The critical value of α is 0. When α is slightly above the critical value, the phase portrait exhibits a spiral sink with orbits spiraling inward. When α is slightly below the critical value, the phase portrait exhibits a stable node with orbits moving towards the stable node.

Step-by-step solution

Find eigenvalues

First, find the eigenvalues of the given matrix by setting the determinant of the matrix minus λ times the identity matrix equal to zero, and solve for λ. $$ \text{det} \left( \begin{pmatrix} -1 - \lambda & \alpha \\ -1 & -1 - \lambda \end{pmatrix} \right) = 0 $$ This determinant can be written as: $$ (-1-\lambda)(-1-\lambda) - (-1)\alpha = 0 $$ Simplify the equation to get a quadratic equation in λ: $$ \lambda^2 + 2\lambda + 1 - \alpha = 0 $$ Now, let's solve for λ using the quadratic formula: $$ \lambda_{1,2} = \frac{-2 \pm \sqrt{(-2)^2-4(1)(1-\alpha)}}{2} = -1 \pm \sqrt{\alpha} $$

Find critical values of α

The critical values of α are the ones where the qualitative nature of the phase portrait changes. We evaluate the eigenvalues for different values of α to determine their behavior. 1. If α > 0, then we have complex conjugate eigenvalues because the discriminant is positive. This means that the system will have spiraling behavior (either spiraling inward or outward, depending on the sign of the real part of the eigenvalues). 2. If α = 0, then we have a single repeated real eigenvalue, which leads to a degenerate node. 3. If α < 0, then we have two distinct negative real eigenvalues. This will result in a stable node. Thus, our critical value of α is 0, as it is the point where the qualitative behavior of the system changes.

Draw phase portraits for α slightly below, and slightly above the critical value

Now that we know α = 0 is the critical value, let's examine the phase portrait for α slightly below, and slightly above this critical value. 1. When α > 0 (i.e. `alpha = 0.1`), the phase portrait will have complex conjugate eigenvalues, leading to a spiral sink (since the real part of the eigenvalues is negative). The phase portrait orbits will be spiraling inward. 2. When α < 0 (i.e. `alpha = -0.1`), the phase portrait will have two distinct negative real eigenvalues, leading to a stable node. The phase portrait orbits will be moving towards the stable node. Note that you can use software like Matlab or Python to visualize these phase portraits for different values of α, but the analysis above should provide a good understanding of the behavior for α values slightly above or below the critical value.

Key concepts

Differential Equations

When dealing with systems like the one presented in the exercise, we are working with differential equations. These equations involve derivatives, which describe how a quantity changes over time.
For example, if we have a system expressed as \( \mathbf{x}^{\prime} = A \mathbf{x} \), this is a linear ordinary differential equation. The vector \( \mathbf{x} \) represents the state of the system, while the matrix \( A \) contains the coefficients that define how changes in \( \mathbf{x} \) depend on its current state.
In simple terms, differential equations let us model dynamic systems in mathematics, physics, engineering, and many other fields. They help us understand how a system evolves and predict future states. By finding the eigenvalues of the coefficient matrix, which will be a part of the exercise, we can gain insights into the long-term behavior of these systems.

Phase Portrait

A phase portrait is a graphical representation of the trajectories of a dynamical system in its phase space as time progresses.
Specifically, for a 2D system like the one described in the exercise, the phase portrait shows how the system evolves for different initial conditions. Each point in the phase space represents a possible state of the system. When you plot the trajectories or paths traced by the system states over time, you get the phase portrait.
Understanding phase portraits can be crucial because they visually depict the behavior of the system. For example, they can show whether the system has equilibrium points, stable or unstable behaviors, or whether it spirals inwards or outwards. These insights can help in identifying what happens to the system over time without solving the differential equation analytically.

Critical Values

Critical values are the specific points at which the nature of the system's behavior changes. In the exercise, the critical value of \( \alpha \) is found where the nature of the phase portrait transitions.
Identifying critical values involves finding points where the mathematical properties of the system change significantly. In this case, it is where the discriminant of the eigenvalue equation changes, affecting whether the eigenvalues are real or complex.
For example, when \( \alpha = 0 \), the system transitions from having complex eigenvalues, showing oscillatory behavior, to real repeated eigenvalues, symbolizing a different dynamic. Understanding these critical points helps in predicting and analyzing the stability or instability of a system.

Stability Analysis

Stability analysis involves examining whether solutions to a differential equation remain close to a point, diverge away, or oscillate around it over time.
In the context of the given system, stability analysis is conducted by assessing the eigenvalues derived from the coefficient matrix. The real parts of these eigenvalues are crucial. If the real part is negative, it implies stability, meaning solutions tend to remain close to equilibrium or a stable point over time. If positive, the system is unstable and solutions will tend to move away from the equilibrium.
By studying the eigenvalues as a function of \( \alpha \), we can determine how system behavior changes with different \( \alpha \) values. Doing so allows us to precisely know when the system switches from stable to unstable behavior, providing deeper insights into the dynamic characteristics of the system.