Problem 19 - Calculus for Biology and Medicine

Question

Show that \(\begin{array}{ll}0 & \text { is an equilibrium of }\end{array}\) $$ \left[\begin{array}{l} x_{1}(t+1) \\ x_{2}(t+1) \end{array}\right]=\left[\begin{array}{ll} -1.4 & 0 \\ -0.5 & 0.1 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \end{array}\right] $$ and determine its stability.

Step-by-Step Solution

Answer

The equilibrium at 0 is unstable because the eigenvalue \(-1.4\) has magnitude greater than 1.

Step 1: Identify the System

The given system is a discrete time linear system of the form \[\begin{bmatrix} x_1(t+1) \ x_2(t+1) \end{bmatrix} = \begin{bmatrix} -1.4 & 0 \ -0.5 & 0.1 \end{bmatrix}\begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix}\] where \(A = \begin{bmatrix} -1.4 & 0 \ -0.5 & 0.1 \end{bmatrix}\) is the system matrix.

Step 2: Determine the Equilibrium Point

To find the equilibrium point, set the next state equal to the current state: \[\begin{bmatrix} x_1(t+1) \ x_2(t+1) \end{bmatrix} = \begin{bmatrix} x_1(t) \ x_2(t) \end{bmatrix}.\]At equilibrium, this reduces to \[\begin{bmatrix} -1.4 & 0 \ -0.5 & 0.1 \end{bmatrix}\begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 0 \ 0 \end{bmatrix}\]Solving this, we see \((x_1, x_2) = (0, 0)\). Thus, \(0\) is the equilibrium.

Step 3: Find Eigenvalues of A

To determine stability, find the eigenvalues of matrix \(A\). Set up the characteristic polynomial:\[\text{det}(A - \lambda I) = \begin{vmatrix} -1.4 - \lambda & 0 \ -0.5 & 0.1 - \lambda \end{vmatrix} = 0\]Solving this, we get the equation \((-1.4 - \lambda)(0.1 - \lambda) = 0\).This gives eigenvalues \(\lambda_1 = -1.4\) and \(\lambda_2 = 0.1\).

Step 4: Assess Stability of the Equilibrium

An equilibrium of a discrete system is stable if all eigenvalues of the system matrix have magnitudes less than 1.For \(\lambda_1 = -1.4\), \(|-1.4| = 1.4\) which is greater than 1.Thus, the equilibrium \(0\) is unstable because one eigenvalue's magnitude exceeds 1.

Key Concepts

Understanding Discrete Time Linear Systems

In mathematics, a **discrete time linear system** is a type of dynamic model that describes the evolution of variables over discrete time steps using linear transformations. Such systems are frequently used in engineering, economics, and physics for modeling processes that are observed at regular intervals. A discrete time linear system can often be described using a matrix equation, where the next state of the system is a linear combination of its current states.
For example, consider a system where variables are represented by a state vector, and changes from one time step to the next are governed by a system matrix. These models are simple yet powerful, being able to depict complex behavior over time while remaining computationally manageable.

• State vectors capture the current status of the system.
• System matrices define how current states transition to future states.
• Time is indexed discretely, which means computations are performed at specified intervals.

Understanding how these elements work in tandem is crucial for analyzing and predicting the behavior of dynamic systems.

Grasping the Basics of Eigenvalues

**Eigenvalues** are vital in understanding how a matrix behaves, particularly in systems of equations that model real-world processes. Eigenvalues are numbers that provide insights into the system's dynamics by revealing how transformations represented by the matrix affect vectors in its domain.
In a discrete time linear system, eigenvalues tell us how the system evolves over time. If you begin with a vector in the direction of an eigenvector, the action of the matrix is simply to scale this vector by its corresponding eigenvalue.

• They give insight into the "stretching" factors of linear transformations.
• Eigenvalues are crucial for stability analysis, determining whether solutions grow, shrink, or oscillate over time.
• For a system matrix, finding eigenvalues involves solving the characteristic polynomial derived from the matrix.

Overall, eigenvalues are essential for stability analysis, as they help categorize long-term behaviors of the equilibrium points of a system.

Exploring the System Matrix

The **system matrix** serves as a foundational element in describing discrete time linear systems. It defines how the system's state evolves from one time step to the next. This matrix is vital since it encapsulates all the parameters that govern the system's behavior.
A system matrix, often denoted as 'A,' holds the linear coefficients for a system of equations, acting on the state vector to yield a new state. In this context, the matrix essentially maps the system's dynamics:

• Each element of the matrix represents the influence of one state variable on another.
• Determining the eigenvalues and eigenvectors of the system matrix allows for analyzing the stability and behavior of the system.
• Stability conditions for the equilibrium can be directly derived by examining the system matrix.

Thus, understanding the system matrix is critical for predicting how a system behaves over time and for making design or control decisions.

Performing Stability Analysis

**Stability analysis** assesses the long-term behavior of equilibrium points within dynamic systems. In a discrete time linear system, stability concerns determine whether the system returns to an equilibrium state after a disturbance or diverges.
The key to stability lies in the eigenvalues of the system matrix. Stability analysis checks the magnitude of these eigenvalues.

• If all eigenvalues have absolute values less than one, the system's equilibrium is stable.
• If any eigenvalue has an absolute value greater than or equal to one, the system's equilibrium is unstable, suggesting potential divergence or oscillation.
• This assessment helps in understanding and controlling how a system behaves over time, particularly after any external perturbations.

In our example, the magnitude of the eigenvalue '1.4' indicates instability because it is greater than one, showing the equilibrium is not stable. Conducting stability analysis provides crucial insights for engineers and researchers in designing systems that can maintain desired states effectively.