Question

Locate the unique equilibrium point of the given nonhomogeneous system, and determine the stability properties of this equilibrium point. Is it asymptotically stable, stable but not asymptotically stable, or unstable? $$ \begin{aligned} &x^{\prime}=y+2 \\ &y^{\prime}=-x+1 \end{aligned} $$

Short answer

Question: Determine the stability properties of the unique equilibrium point of the nonhomogeneous system of linear differential equations given by: $$ \begin{cases} x^{\prime} = y + 2 \\ y^{\prime} = -x + 1 \end{cases} $$ Answer: The unique equilibrium point of the given system is (1, -2). The stability properties of the equilibrium point (1, -2) are stable but not asymptotically stable.

Step-by-step solution

Finding the Equilibrium Point

First, set both derivatives equal to zero to find the equilibrium point (x₀, y₀): $$\begin{aligned} 0 &= y + 2 \\ 0 &= -x + 1 \end{aligned}$$ Solve the system of equations for x and y: $$\begin{aligned} y &= -2 \\ x &= 1 \end{aligned}$$ The unique equilibrium point of the given nonhomogeneous system is (1, -2).

Finding the Jacobian Matrix

The Jacobian matrix J is the matrix of all first-order partial derivatives of the given system. In this case, the Jacobian matrix is: $$ J = \begin{bmatrix} \frac{\partial x^{\prime}}{\partial x} & \frac{\partial x^{\prime}}{\partial y} \\ \frac{\partial y^{\prime}}{\partial x} & \frac{\partial y^{\prime}}{\partial y} \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} $$

Finding the Eigenvalues of the Jacobian Matrix

To find the eigenvalues of the Jacobian matrix, we need to solve the following equation for λ: $$\det(J - λI) = 0$$ Where I is the identity matrix. In this case, we have: $$ \det\left(\begin{bmatrix} -λ & 1\\ -1 & -λ \end{bmatrix}\right)=({-λ})(-λ) - (1)(-1) =λ^{2}+1=0 $$ The eigenvalues can be found by solving the quadratic equation: $$λ^{2} + 1 = 0$$ The eigenvalues are λ₁ = i and λ₂ = -i which are purely imaginary.

Determining Stability Properties

Now let's analyze the stability properties of the equilibrium point (1,-2): 1. If all eigenvalues have negative real parts, the equilibrium point is asymptotically stable. 2. If all eigenvalues have non-positive real parts and at least one eigenvalue has a zero real part, the equilibrium point is stable but not asymptotically stable. 3. If at least one eigenvalue has a positive real part, the equilibrium point is unstable. In this case, both eigenvalues (i and -i) have zero real parts. Thus, the equilibrium point (1, -2) is stable but not asymptotically stable.

Key concepts

Nonhomogeneous Systems

In differential equations, a nonhomogeneous system involves equations where the result is not always equal to zero, frequently involving an external force or constant term. It can be expressed as:

• Homogeneous system: \(Ax = 0\)
• Nonhomogeneous system: \(Ax = b\) where \(b eq 0\)

This differentiation is important because it affects how we analyze and solve the system. Nonhomogeneous systems often arise in real-world applications, where an external input influences the state of the systems, such as in mechanical systems with external forces or electrical circuits with source voltage.
For the given exercise, the equations \(x' = y + 2\) and \(y' = -x + 1\) are examples of a nonhomogeneous system, identifiable by the constant terms \(+2\) and \(+1\), indicating inputs that shift the system's behavior. Finding the equilibrium point involves setting the derivative terms to zero and solving the resulting equations.

Jacobian Matrix

The Jacobian matrix is a vital tool in examining non-linear differential equations, providing insights into system behavior around an equilibrium point. It consists of first-order partial derivatives, encapsulating the slope of the vector field at an equilibrium. The matrix is typically denoted by:\[J = \begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{bmatrix}\]In the exercise, the Jacobian at the equilibrium point (1, -2) is obtained by taking partial derivatives of the system equations \(x' = y + 2\) and \(y' = -x + 1\). This results in the Jacobian matrix:\[\begin{bmatrix} 0 & 1\ -1 & 0\end{bmatrix}\]This matrix helps evaluate the system's local behavior by revealing how changes in one state variable affect another, crucial for analyzing stability and determining the nature of the equilibrium.

Eigenvalues and Stability

Eigenvalues derived from the Jacobian matrix are central to determining the stability of an equilibrium point in differential equations. Here's a quick overview on what they mean:

• If all eigenvalues have negative real parts, the system is asymptotically stable — solutions that start near the equilibrium stay close and tend towards it over time.
• For eigenvalues with zero real parts, the system is stable but not asymptotically stable — it neither grows nor shrinks, maintaining a neutral equilibrium.
• If any eigenvalue has a positive real part, the system is unstable — solutions diverge from the equilibrium.

In our current exercise, the eigenvalues \(i\) and \(-i\) are purely imaginary, indicating that the system has zero real parts. This marks the equilibrium point (1, -2) as stable but not asymptotically stable, meaning that the system's state remains steady without settling into the equilibrium.

Linear Stability Analysis

Linear stability analysis offers a practical method to explore the local behavior of dynamic systems around an equilibrium point by linearizing the system. The process involves the use of the Jacobian matrix to determine the system's linear approximation and investigate small perturbations around the equilibrium.
A crucial step is examining the eigenvalues of the Jacobian matrix:

• Purely Imaginary Eigenvalues: Corresponding systems exhibit a form of oscillation, maintaining stability but not settling, as seen in our exercise with eigenvalues \(i\) and \(-i\).
• Complex with Negative Real Part: Systems return to equilibrium in a spiral, indicative of asymptotic stability.
• Complex with Positive Real Part: Lead to divergence from equilibrium, characteristic of instability.

This analysis offers deeper insights into how a system might evolve over time. For the given exercise, understanding the nature of eigenvalues informs us of the equilibrium's stability properties, leading to practical applications in engineering and physical sciences to predict system responses.