Problem 12 - Advanced Engineering Mathematics

Question

Critical points are (1,0) and \((-1,0),\) and $$\mathbf{g}^{\prime}(\mathbf{X})=\left(\begin{array}{rr} 2 x & -2 y \\ 0 & 2 \end{array}\right)$$ At \(\mathbf{X}=(1,0), \tau=4, \Delta=4,\) and so \(\tau^{2}-4 \Delta=0 .\) We can conclude that (1,0) is unstable but we are unable to classify this critical point any further. At \(\mathbf{X}=(-1,0), \Delta=-4<0\) and so (-1,0) is a saddle point.

Step-by-Step Solution

Answer

The point (1,0) is unstable but indeterminate, and (-1,0) is a saddle point.

Step 1: Understanding the Critical Points

The critical points given are (1,0) and (-1,0). These are the points where we will evaluate the matrix \( \mathbf{g}^{\prime}(\mathbf{X}) \).

Step 2: Evaluating the Jacobian at Critical Point (1,0)

For \( \mathbf{X} = (1,0) \), substitute the values into the Jacobian matrix \( \mathbf{g}^{\prime}(\mathbf{X}) = \begin{pmatrix} 2x & -2y \ 0 & 2 \end{pmatrix} \). This results in \( \begin{pmatrix} 2 & 0 \ 0 & 2 \end{pmatrix} \).

Step 3: Finding the Eigenvalues for (1,0)

The eigenvalues of a 2x2 matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \) are found using the characteristic equation \( \lambda^2 - \tau\lambda + \Delta = 0 \). Here, \( \tau = a + d = 4 \) and \( \Delta = ad - bc = 4 \). Calculating \( \tau^2 - 4\Delta = 16 - 16 = 0 \), which indicates the eigenvalues are the same, and thus further classification needs other information.

Step 4: Evaluating the Jacobian at Critical Point (-1,0)

For \( \mathbf{X} = (-1,0) \), substitute into the Jacobian matrix to get \( \begin{pmatrix} -2 & 0 \ 0 & 2 \end{pmatrix} \).

Step 5: Finding the Eigenvalues for (-1,0)

Using \( \tau = -2 + 2 = 0 \) and \( \Delta = (-2)(2) - (0)(0) = -4 \). Since \( \Delta < 0 \), (-1,0) is classified as a saddle point because there is one positive and one negative eigenvalue.

Key Concepts

Critical Points

Critical points in a differential equation are the places where the derivative equals zero. These are pivotal because they often indicate the behavior of the system at those points. In our exercise, the critical points given are

• (1,0)
• (-1,0)

At these locations, we study the system's stability by analyzing the Jacobian matrix and its eigenvalues. Exploring critical points allows us to understand various solutions of the equation, such as whether they are stable, unstable, or lead to some form of periodic behavior. For this problem, they serve as locations to test the nature of the solutions.

Jacobian Matrix

The Jacobian matrix is a crucial tool for studying differential equations' behavior near critical points. It represents the gradient of a vector function and helps us understand how small changes near the critical point influence the system. Typically, the Jacobian matrix is defined for a function of multiple variables as:\[\begin{pmatrix} \\frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1}{\partial x_n} \\vdots & \ddots & \vdots \\frac{\partial f_m}{\partial x_1} & \cdots & \frac{\partial f_m}{\partial x_n} \\end{pmatrix}\]In our exercise, it was given by\[\begin{pmatrix} 2x & -2y \ 0 & 2 \end{pmatrix}\]Evaluating the Jacobian at each critical point allows us to derive insight into the stability and behavior of the system dynamic around those points. By substituting (1,0) and (-1,0) into this matrix, we can assess how the system changes when perturbed.

Eigenvalues

Eigenvalues are mathematical constructs that help us determine the stability of critical points. These values are derived from the Jacobian and give us clues about the behavior of the system's solutions. For a 2x2 matrix, eigenvalues \( \lambda \) are found using the characteristic equation:\[\lambda^2 - \tau\lambda + \Delta = 0\]where:

• \( \tau \) is the trace of the matrix (sum of the diagonal entries)
• \( \Delta \) is the determinant

At point (1,0), both eigenvalues were equal, making classification ambiguous. However, at (-1,0), the determinant was negative, indicating one positive and one negative eigenvalue. These results show that each point can have significantly different behaviors.

Saddle Point

A saddle point in the context of differential equations indicates a point where the system behaves unstably in one direction but could be stable in another. It occurs when a critical point has both positive and negative eigenvalues, leading to trajectories that move away and towards the point simultaneously. In our exercise, the critical point (-1,0) was identified as a saddle point due to the negative determinant of the Jacobian matrix, confirming one positive and one negative eigenvalue. Understanding saddle points is essential in physics and engineering as they can imply instabilities or oscillations in systems such as bridges, circuits, and ecosystems.