Problem 9 - Advanced Engineering Mathematics

Question

Classify the critical point \((0,0)\) of the given linear system by computing the trace \(\tau\) and determinant \(\Delta\) and using Figure \(11.2 .12 .\) $$ \begin{aligned} &x^{\prime}=-5 x+3 y \\ &y^{\prime}=2 x+7 y \end{aligned} $$

Step-by-Step Solution

Answer

The critical point \((0,0)\) is a saddle point.

Step 1: Determine the Coefficients of the System

Identify the coefficients from the given system of equations. The system is: \(x' = -5x + 3y\) and \(y' = 2x + 7y\). Thus, the associated matrix for the system is \(A = \begin{bmatrix} -5 & 3 \ 2 & 7 \end{bmatrix}\).

Step 2: Compute the Trace of the Matrix

The trace \(\tau\) of a matrix is the sum of the elements on its main diagonal. For matrix \(A = \begin{bmatrix} -5 & 3 \ 2 & 7 \end{bmatrix}\), the trace is \(\tau = -5 + 7 = 2\).

Step 3: Compute the Determinant of the Matrix

The determinant \(\Delta\) of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is given by \(ad - bc\). Here, \(a = -5\), \(b = 3\), \(c = 2\), and \(d = 7\), thus \(\Delta = (-5 \times 7) - (3 \times 2) = -35 - 6 = -41\).

Step 4: Classify the Critical Point Using the Trace and Determinant

Based on Figure 11.2.12 and the computed values, \(\tau = 2\) and \(\Delta = -41\), since the determinant is negative, the critical point \((0,0)\) is a saddle point.

Key Concepts

Trace

The trace of a matrix is an important concept in linear algebra, especially when analyzing linear systems. It is defined as the sum of the elements along the main diagonal of a square matrix. For instance, given a matrix:\[ A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]The trace, denoted as \( \tau \), is calculated as \( \tau = a + d \). In our exercise, the matrix is\[ \begin{bmatrix} -5 & 3 \ 2 & 7 \end{bmatrix} \]Thus, the trace is \(-5 + 7 = 2\). Understanding the trace is essential, as it helps in determining the nature of equilibrium points in a linear system. The trace gives insight into the behavior of the system, particularly the tendency of trajectories in the phase plane. It indicates whether solutions are moving away from or towards an equilibrium point.

Determinant

The determinant of a matrix provides crucial information about the matrix, especially in the context of solving systems of linear equations. For a 2x2 matrix:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]The determinant, denoted \( \Delta \), is calculated as \( ad - bc \). This value can tell us a lot about the system. If the determinant is zero, it implies that the system of equations is linearly dependent or has infinitely many solutions. If it's non-zero, the system has a unique solution. In our example, we calculated the determinant as\[ \Delta = (-5 \times 7) - (3 \times 2) = -35 - 6 = -41 \]A negative determinant, as we see in this exercise, indicates specific characteristics about the critical points. These are crucial for understanding the local stability and type of each critical point, which will lead us to classify them correctly.

Saddle Point Classification

Classifying critical points involves analyzing the trace and determinant from the matrix of the linear system. The saddle point is one type of critical point characterized by its behavior in the phase space.To identify a saddle point, observe:

• The determinant is less than zero \((\Delta < 0)\)
• This indicates that the system possesses a hyperbolic structure.

Saddle points are interesting because they are inherently unstable. This means that small perturbations can lead the system away from these points rather than towards them, unlike other equilibria such as nodes or focuses that might attract trajectories. In our analysis of the linear system where \( \tau = 2 \) and \( \Delta = -41 \), the negative determinant confirms the presence of a saddle point at the origin \((0,0)\). Understanding saddle points is critical when predicting system behavior and stability.