Problem 25 - Advanced Engineering Mathematics

Question

A nonhomogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}+\mathbf{F}\) is given (a) In each case determine the unique critical point \(\mathbf{X}_{1}\). (b) Use a numerical solver to determine the nature of the critical point in part (a). (c) Investigate the relationship between \(\mathbf{X}_{1}\) and the critical point \((0,0)\) of the homogeneous linear system \(\mathbf{x}^{\prime}=\mathbf{A} \mathbf{x}\) $$ \begin{aligned} &x^{\prime}=0.1 x-0.2 y+0.35 \\ &y^{\prime}=0.1 x+0.1 y-0.25 \end{aligned} $$

Step-by-Step Solution

Answer

(a) Critical point: (0.5, 2). (b) Saddle point. (c) Both are saddle points, but (0.5, 2) is shifted.

Step 1: Identify the system equations

The given linear nonhomogeneous system can be written asthe following equations: 1. \( x' = 0.1x - 0.2y + 0.35 \)2. \( y' = 0.1x + 0.1y - 0.25 \).Here, the matrix \( \mathbf{A} \) is \( \begin{pmatrix} 0.1 & -0.2 \ 0.1 & 0.1 \end{pmatrix} \) and the vector \( \mathbf{F} \) is \( \begin{pmatrix} 0.35 \ -0.25 \end{pmatrix} \).

Step 2: Find the critical point \( \mathbf{X}_1 \)

The critical point \( \mathbf{X}_1 \) is found by setting the derivative equal to 0, i.e., solving the system of equations given by:- \( 0 = 0.1x - 0.2y + 0.35 \)- \( 0 = 0.1x + 0.1y - 0.25 \).Solving these equations simultaneously, we get:- From equation 2: \( 0.1x + 0.1y = 0.25 \) gives \( x + y = 2.5 \).- Substitute \( x = 2.5 - y \) into equation 1: \( 0 = 0.1(2.5 - y) - 0.2y + 0.35 \). Simplifying: \( 0 = 0.25 - 0.1y - 0.2y + 0.35 \) \( 0 = 0.6 - 0.3y \) gives \( y = 2 \).- Substitute \( y = 2 \) back into \( x = 2.5 - y \): \( x = 2.5 - 2 = 0.5 \).Thus, \( \mathbf{X}_1 = \begin{pmatrix} 0.5 \ 2 \end{pmatrix} \).

Step 3: Determine the nature of the critical point \( \mathbf{X}_1 \)

To determine the nature of the critical point \( \mathbf{X}_1 \), we examine the eigenvalues of matrix \( \mathbf{A} \).- The characteristic equation of \( \mathbf{A} \) is \( \lambda^2 - 0.2\lambda - 0.03 \).- Solving gives eigenvalues \( \lambda_1 \approx 0.25 \) and \( \lambda_2 \approx -0.05 \).- Since the eigenvalues have opposite signs, \( \mathbf{X}_1 \) is a saddle point.

Step 4: Compare with the homogeneous system

The homogeneous system \( x' = 0.1x - 0.2y \), \( y' = 0.1x + 0.1y \) has its critical point at the origin (0,0).- Both systems have the same matrix \( \mathbf{A} \), so the nature of the critical points (saddle points) is the same.- However, \( \mathbf{X}_1 = \begin{pmatrix} 0.5 \ 2 \end{pmatrix} \) is shifted due to the nonhomogeneous term \( \mathbf{F} \).

Key Concepts

Critical Points

In the context of a linear system, critical points are where the system's state doesn't change, so the derivatives are zero. To find them, we solve the system of equations with the derivatives set to zero. For example, given the equations \( x' = 0.1x - 0.2y + 0.35 \) and \( y' = 0.1x + 0.1y - 0.25 \), setting the derivatives \( x' \) and \( y' \) to zero allows us to find the critical point \( \mathbf{X}_1 \), which was calculated as \( \begin{pmatrix} 0.5 \ 2 \end{pmatrix} \). Some properties of critical points include:

• They indicate system stability or instability.
• Understanding them helps predict long-term system behavior.

Solving for critical points gives insights into how the system can be expected to evolve over time and whether it will remain steady or exhibit variability.

Numerical Solvers

Numerical solvers are computational tools that help us determine the nature of critical points and other system behaviors when analytical solutions are complex or impossible. In our system, after finding the critical point \( \mathbf{X}_1 \), a numerical solver can simulate the system to approximate how it behaves around this point.Utilizing numerical solvers comes with benefits:

• They handle complex, non-linear systems that are difficult to solve by hand.
• They provide visual insights through graphical outputs and simulations.

They act as a valuable asset in understanding and predicting system dynamics when exact solutions are not practical.

Homogeneous Linear Systems

Homogeneous linear systems are a particular type of system characterized by having all constant or non-variable terms equal to zero, as shown in the system: \[\begin{aligned}&x^{\prime}=0.1 x-0.2 y \&y^{\prime}=0.1 x+0.1 y\end{aligned}\]This system serves as a foundational topic since it helps explain behavior without external forces or shifts.Key features include:

• Critical points are usually at the origin \((0,0)\).
• The behavior of solutions is driven entirely by the matrix \(\mathbf{A}\).

Understanding homogeneous systems allows for a clearer interpretation of its nonhomogeneous counterpart, as the latter introduces additional elements, like a constant vector \(\mathbf{F}\), into the dynamics.

Saddle Points

Saddle points are special types of critical points where the system's stability is neutral – neither fully stable nor unstable. In our linear system, by calculating the eigenvalues and finding opposite signs, we determined that \( \mathbf{X}_1 \) is a saddle point. This means:

• One eigenvalue suggests a direction of stability.
• The other eigenvalue indicates instability in another direction.

This mixed stability results in trajectories moving toward the saddle point in some directions while moving away from it in others. Saddle points help predict more nuanced behaviors in dynamical systems, such as the potential for sudden transitions or bifurcations.

Eigenvalues

Eigenvalues are key to understanding system stability, as they are derived from the matrix \( \mathbf{A} \) in the linear system. By examining eigenvalues, we can identify the nature of critical points. For example, the characteristic equation of our matrix resulted in eigenvalues \( \lambda_1 \approx 0.25 \) and \( \lambda_2 \approx -0.05 \).Understanding eigenvalues allows us to:

• Predict if the system will converge or diverge over time.
• Determine if critical points are stable, unstable, or saddle points.

In essence, eigenvalues give us a mathematical lens to peer into the future behavior of the system, establishing a basis for predictions and deeper understanding of dynamic shifts in the system.