Question

In Exercises \(21-30\), perform steps i) and ii) for the linear system $$ \mathbf{x}^{\prime}=\mathbf{A x}, $$ where \(\mathbf{A}\) is the given matrix. i) Find the type of the equilibrium point at the origin. Do this without the aid of any technology or pplane6. You may, of course, check your answer using pplane. ii) Use pplane6 to plot several solution curves - enough to fully illustrate the behavior of the system. You will find it convenient to use the "linear system" choice from the Gallery menu. If the eigenvalues are real, use the "Keyboard input" option to include straight line solutions starting at \(\pm 10\) times the eigenvectors. If the equilibrium point is a saddle point, compute and plot the separatrices. $$ \left[\begin{array}{rr} -1 & 0 \\ 0 & -1 \end{array}\right] $$

Short answer

The equilibrium point at the origin is a stable node.

Step-by-step solution

Identify the Matrix A

The given matrix \(\mathbf{A}\) is \(\begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix}\).

Find the Eigenvalues

To find the eigenvalues, solve the characteristic equation \(\text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0\). For the given matrix, this becomes: \[\begin{bmatrix} -1-\lambda & 0 \ 0 & -1-\lambda \end{bmatrix}\] Setting the determinant of this matrix to zero gives:\((-1 - \lambda)^2 = 0\)Thus, the eigenvalue \(\lambda = -1\) with multiplicity 2.

Determine the Equilibrium Type

Since both eigenvalues are real and negative \(\lambda = -1\), the origin is a stable node (also known as an attractor). This suggests that trajectories near the equilibrium point will converge towards the origin over time.

Plotting Solution Curves

Using pplane6, plot the solution trajectories of the system. Since the equilibrium is a stable node, all solution trajectories will converge towards the origin. Use various initial points to observe how different starting positions affect the rate and path of convergence.

Key concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, especially useful in analyzing systems of linear differential equations. Given a square matrix \( \mathbf{A} \), the eigenvalues are scalars \( \lambda \) such that there exists a non-zero vector \( \mathbf{v} \), called the eigenvector, that satisfies the equation: \[\mathbf{A} \mathbf{v} = \lambda \mathbf{v}\] To compute the eigenvalues of matrix \( \mathbf{A} \), you solve the characteristic equation: \[\text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0\] In our given example, we solve this for a 2x2 matrix with \( \lambda = -1 \) appearing twice, indicating it has a degenerate eigenvalue with a multiplicity of 2. Eigenvectors provide direction along which transformations involving the matrix \( \mathbf{A} \) stretch or shrink the space. Essentially, they offer insight into the system's behavior in the phase plane.

Equilibrium Point Classification

Classifying equilibrium points in a linear system involves analyzing the eigenvalues of the associated matrix. These eigenvalues help determine the local stability of the equilibrium point. For the matrix provided, both eigenvalues are \( \lambda = -1 \), which are real and identical, leading us to classify the equilibrium as a stable node. This means that trajectories in the system will converge towards the equilibrium point (the origin), regardless of their starting point. Here’s a brief rundown of the classifications:

• Real and negative eigenvalues: Stable node (attractor)
• Real and positive eigenvalues: Unstable node (repeller)
• Real but with opposite signs: Saddle point
• Complex with negative real parts: Spiral sink
• Complex with positive real parts: Spiral source

Phase Plane Analysis

Phase plane analysis is a visual method used to understand the behavior of systems of differential equations. It involves plotting solution curves on a plane—typically the \(x-y\) plane—where each point represents a state of the system. For the system given by matrix \( \mathbf{A} \), the solution trajectories reveal how the system evolves over time.In the current example, since the equilibrium point is a stable node, phase plane analysis shows all trajectories heading towards the origin. This convergence means that small perturbations in initial conditions result in paths that closely follow the eigenvector directions before eventually arriving at the equilibrium.Using tools like pplane6 enables us to visualize these dynamics efficiently, allowing for a clearer understanding of the behavior of the system.

Stable Node Behavior

Stable nodes are significant in system dynamics, representing points where solutions converge over time. Here, the matrix \( \begin{bmatrix} -1 & 0 \ 0 & -1 \end{bmatrix} \) has eigenvalues of \( \lambda = -1 \). This indicates that the origin functions as a stable node, characterized by:

• All trajectories move towards the node, no matter where they start in the phase plane.
• The speed of convergence depends on the distance from the origin and the orientation of the eigenvectors.

In practical terms, a stable node implies that the system naturally stabilizes at the equilibrium point, providing predictability and robust stability despite disturbances. Understanding behavior like this is crucial for applications requiring reliability and safety, including engineering and physics.