Question

Consider the system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x},\) and suppose that \(\mathbf{A}\) has one zero eigenvalue. (a) Show that \(\mathbf{x}=\mathbf{0}\) is a critical point, and that, in addition, every point on a certain straight line through the origin is also a critical point. (b) Let \(r_{1}=0\) and \(r_{2} \neq 0,\) and let \(\boldsymbol{\xi}^{(1)}\) and \(\boldsymbol{\xi}^{(2)}\) be corresponding eigenvectors. Show that the trajectories are as indicated in Figure \(9.1 .8 .\) What is the direction of motion on the trajectories?

Short answer

b) Describe the direction of motion on the trajectories of the system.

Step-by-step solution

Find the critical points

To find the critical points, we need to determine when \(\mathbf{x}^{\prime}=\mathbf{0}\). So we have \(\mathbf{A} \mathbf{x} = \mathbf{0}\). As the matrix \(\mathbf{A}\) has one zero eigenvalue, it means that there is one non-trivial solution to the equation \(\mathbf{A}\mathbf{x} = \mathbf{0}\). This implies that there is an eigenvector corresponding to the zero eigenvalue which makes the whole vector equation equal to zero. So the critical points will lie on a straight line through the origin. ##STEP 2: Find the eigenvectors and eigenvalues##

Find the eigenvectors and eigenvalues

We are given that \(r_1 = 0\) and \(r_2\neq0\), and corresponding eigenvectors \(\boldsymbol{\xi}^{(1)}\) and \(\boldsymbol{\xi}^{(2)}\). ##STEP 3: Characterize the direction of motion on the trajectories##

Characterize the direction of motion on the trajectories

Let's consider the system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}\). Since \(\textbf{A}\boldsymbol{\xi}^{(1)}=0\), we have \(\boldsymbol{\xi}^{(1)}=constant\). Also, \(\textbf{A}\boldsymbol{\xi}^{(2)}=r_{2}\boldsymbol{\xi}^{(2)}\). Divide by \(r_2\), we have \(\frac{1}{r_2}\mathbf{A}\boldsymbol{\xi}^{(2)}=\boldsymbol{\xi}^{(2)}\) which indicates that \(\frac{d\boldsymbol{\xi}}{dt}=\frac{1}{r_2}\mathbf{A}\boldsymbol{\xi}^{(2)}\). Putting the two equations together, we have \(\frac{d\mathbf{x}}{dt}=c_1\boldsymbol{\xi}^{(1)}+c_2 \frac{1}{r_2}\mathbf{A}\boldsymbol{\xi}^{(2)}\), where \(c_1\) and \(c_2\) are constants depending on the initial condition. The trajectories will be a combination of the eigenvectors \(\boldsymbol{\xi}^{(1)}\) and \(\boldsymbol{\xi}^{(2)}\). If \(c_1\neq0\), the direction of motion will be parallel to \(\boldsymbol{\xi}^{(1)}\), and the trajectories will lie in the straight line formed by \(\boldsymbol{\xi}^{(1)}\). If \(c_1=0\), the direction of motion will be parallel to \(\boldsymbol{\xi}^{(2)}\), and the trajectories will be formed by exponentiating the contribution from eigenvector \(\boldsymbol{\xi}^{(2)}\).

Key concepts

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are fundamental concepts in the study of linear algebra and are particularly important in the analysis of systems described by differential equations. When we encounter a linear system of equations like \( \mathbf{x}^{\prime} = \mathbf{A} \mathbf{x} \) where \( \mathbf{A} \) is a matrix, eigenvalues and eigenvectors give us crucial information about system dynamics.

Eigenvalues, denoted by \( r \), are scalars that result from solving the characteristic equation \( \det(\mathbf{A} - r \mathbf{I}) = 0 \), where \( \mathbf{I} \) is the identity matrix. Each eigenvalue has at least one associated eigenvector, which is a non-zero vector \( \boldsymbol{\xi} \) that satisfies the equation \( \mathbf{A}\boldsymbol{\xi} = r\boldsymbol{\xi} \).

The existence of a zero eigenvalue indicates that the matrix \( \mathbf{A} \) is not invertible and the system has infinitely many solutions along a certain line—the eigenvector corresponding to the zero eigenvalue. In the context of dynamical systems, an eigenvector provides a direction along which the dynamics are invariant. With these concepts in mind, we can begin to understand the nature of our system's critical points and their behavior.

Phase Portrait

A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each point in this plane corresponds to a state of the system, and the trajectories show how these states evolve over time.

When analyzing the phase portrait of the system \( \mathbf{x}^{\prime} = \mathbf{A} \mathbf{x} \) with one zero eigenvalue, we observe that trajectories tend to align with the eigenvectors of the matrix \( \mathbf{A} \). For instance, if we draw the phase portrait, we would see lines or curves that correspond to the eigenvectors \( \boldsymbol{\xi}^{(1)} \) and \( \boldsymbol{\xi}^{(2)} \), with the direction of the motion indicated by the sign of the corresponding non-zero eigenvalue.

The trajectory associated with the zero eigenvalue, \( \boldsymbol{\xi}^{(1)} \) in this case, will be a line passing through the origin. This line is a set of points where the system is in equilibrium, or a critical line. In contrast, the trajectory associated with the non-zero eigenvalue, \( \boldsymbol{\xi}^{(2)} \) here, will express how the system evolves over time away from equilibrium. By analyzing the phase portrait, one can predict the stability of the critical points and the long-term behavior of the system.

Dynamical Systems

Dynamical systems theory is concerned with how a particular system — described by a set of rules or equations — behaves over time. In the context of differential equations, a dynamical system can often be described by the evolution of a point in a geometric space, usually referred to as the 'state space' or 'phase space', with time.

In our exercise, the linear dynamical system is represented by \( \mathbf{x}^{\prime} = \mathbf{A} \mathbf{x} \), where \( \mathbf{x} \) is a vector that describes the state of the system and \( \mathbf{A} \) is a matrix that determines the system's rules. The critical points, or the states where the system does not change over time (i.e., \( \mathbf{x}^{\prime} = \mathbf{0} \)), play a vital role in understanding the system's behavior.

In this example, the line of critical points associated with a zero eigenvalue represents a continuous set of equilibrium states. This feature is a hallmark of certain dynamical systems, indicating that small disturbances in the direction of the zero eigenvalue will not move the system away from the critical line. However, the system's response to disturbances in other directions is determined by the other eigenvalues and eigenvectors, leading to either growth or decay — essential concepts for predicting the system's response over time.