Question

In each exercise, the eigenpairs of a \((2 \times 2)\) matrix \(A\) are given where both eigenvalues are real. Consider the phase-plane solution trajectories of the linear system \(\mathbf{y}^{\prime}=A \mathbf{y}\), where $$ \mathbf{y}(t)=\left[\begin{array}{l} x(t) \\ y(t) \end{array}\right] $$ (a) Use Table \(6.2\) to classify the type and stability characteristics of the equilibrium point at \(\mathbf{y}=\mathbf{0}\). (b) Sketch the two phase-plane lines defined by the eigenvectors. If an eigenvector is \(\left[\begin{array}{l}u_{1} \\ u_{2}\end{array}\right]\), the line of interest is \(u_{2} x-u_{1} y=0\). Solution trajectories originating on such a line stay on the line; they move toward the origin as time increases if the corresponding eigenvalue is negative or away from the origin if the eigenvalue is positive. (c) Sketch appropriate direction field arrows on both lines. Use this information to sketch a representative trajectory in each of the four phase- plane regions having these lines as boundaries. Indicate the direction of motion of the solution point on each trajectory. $$ \lambda_{1}=2, \quad \mathbf{x}_{1}=\left[\begin{array}{l} 2 \\ 0 \end{array}\right] ; \quad \lambda_{2}=1, \quad \mathbf{x}_{2}=\left[\begin{array}{l} 0 \\ 2 \end{array}\right] $$

Short answer

Answer: In the given linear system with positive eigenvalues, the equilibrium point is an unstable node. In each of the four phase-plane regions, the trajectories generally move away from the origin with the following directions: 1. Top-right region: rightward and upward. 2. Top-left region: leftward and upward. 3. Bottom-left region: leftward and downward. 4. Bottom-right region: rightward and downward.

Step-by-step solution

Classify the type and stability characteristics of the equilibrium point

Since both real eigenvalues are positive (\(\lambda_{1}=2\) and \(\lambda_{2}=1\)), the equilibrium point at \(\mathbf{y}=\mathbf{0}\) is an unstable node.

Sketch the phase-plane lines defined by the eigenvectors

For eigenvector \(\mathbf{x}_{1}=\left[\begin{array}{l}2 \\\ 0\end{array}\right]\), the line of interest is \(u_{2}x - u_{1}y = 0\), i.e., \(0x - 2y = 0\), which is \(y = 0\) (horizontal line). For eigenvector \(\mathbf{x}_{2}=\left[\begin{array}{l}0 \\\ 2\end{array}\right]\), the line of interest is \(2x - 0y = 0\), which is \(x = 0\) (vertical line). Thus, the phase-plane lines form a cross.

Sketch appropriate direction field arrows on both lines

Since the eigenvalues are positive for both eigenvectors, the trajectories on the lines should move away from the origin as time increases. On the line \(x=0\), we draw arrows pointing upward in the positive y direction. On the line \(y=0\), we draw arrows pointing to the right in the positive x direction.

Sketch representative trajectories in each of the four phase-plane regions

In the four phase-plane regions defined by the lines \(x=0\) and \(y=0\), we can sketch the representative trajectories as follows: 1. Top-right region: The trajectories move away from the origin, heading rightward and upward. 2. Top-left region: The trajectories move away from the origin, heading leftward and upward. 3. Bottom-left region: The trajectories move away from the origin, heading leftward and downward. 4. Bottom-right region: The trajectories move away from the origin, heading rightward and downward. These trajectories give us an overall view of the dynamics of the linear system \(\mathbf{y}^{\prime}=A\mathbf{y}\) with the given eigenpairs.

Key concepts

Eigenpairs of Matrices

In the study of linear algebra, eigenpairs are fundamental when analyzing different properties of matrices. An eigenpair consists of an eigenvalue and its corresponding eigenvector. To understand these concepts, imagine a non-zero vector v and a square matrix A such that when A multiplies v, the result is a scalar multiple of v. In math terms, Av = \(\lambda\)v, where \(\lambda\) is the eigenvalue.

Eigenvalues provide significant insights into the behavior of matrix transformations. They can indicate whether the system expands, contracts, or rotates. The corresponding eigenvectors show the direction in which these transformations occur. In the context of the exercise, the eigenvalues being positive means that the linear system tends to diverge from the origin, which we then visualize in the phase-plane analysis.

Equilibrium Point Stability

In differential equations, an equilibrium point is where the system rests when no external forces act upon it. Understanding equilibrium point stability is central to knowing how a system responds to small disturbances.

In the case of positive eigenvalues, as described in the exercise, the equilibrium point (\(\mathbf{y}=\mathbf{0}\)) is an unstable node. This means any small movement away from the equilibrium will result in the system moving further away, hence the term 'unstable'. By knowing that the system has an unstable node, we deduce that trajectories in the phase-plane will diverge from this point.

Linear Differential Systems

A linear differential system is a set of equations that describes the evolution of a system over time, featuring variables and their derivatives. These systems can frequently be written in matrix form as \(\mathbf{y}' = A\mathbf{y}\), where \(\mathbf{y}\) is a vector of variables and \(A\) represents the coefficients.

When analyzing such systems, it's essential to consider their linearity, which greatly simplifies their solution and interpretation. Linear systems are predictable and their solutions can be directly related to the eigenvalues and eigenvectors of matrix \(A\). The exercise provides a distinct portrayal of this relationship and by examining the eigenpairs, we can predict the behavior of \(\mathbf{y}(t)\) over time.

Direction Fields

Lastly, the concept of direction fields is fundamental in visualizing differential equations solutions without solving them explicitly. A direction field consists of little arrows drawn at various points in the plane, which represent the direction a solution curve would take if it passed through that point.

In a two-dimensional case with variables \(x\) and \(y\), such as the one appearing in the exercise, eigenvectors help us define specific lines in the phase-plane. The arrows on these lines give immediate insight into the system's dynamism: pointing towards the equilibrium for negative eigenvalues or away from it for positive eigenvalues. Sketching these arrows appropriately, as in the exercise's Step 3, allows us to predict the curves' trajectories, furnishing a graphic demonstration of the system's overall behavior without the need for complex computations.