Problem 4 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

For each of the systems, determine whether the origin is stable, asymptotically stable, or unstable. \(\begin{array}{ll}\text { a. } \quad x^{\prime} & =2 x-5 y \\ & y^{\prime}=x-2 y \\ \text { c. } & x^{\prime}=-6 x-2 y \\ y^{\prime} & =2 x-1 y \\ \text { e. } & x^{\prime}=3 x-2 y \\ & y^{\prime}=2 x-1 y \\ \text { g. } & x^{\prime}=-x-5 y \\ \text { i. } & y^{\prime}=2 x-3 y \\ & x^{\prime}=-3 x+1 y \\ & y^{\prime}=2 x-2 y \\ \text { k. } & x^{\prime}=x-2 y \\ & y^{\prime}=2 x+1 y\end{array}\) \(x^{\prime}=2 x-5 y\) \(y^{\prime}=2 x-4 y\) \(x^{\prime}=-9 x+4 y\) d. \(y^{\prime}=-4 x-1 y\) \(x^{\prime}=\quad y / 2\) \(\begin{aligned} y^{\prime} &=-5 x-3 y \\ x^{\prime} &=\quad-5 y \end{aligned}\) h. \(y^{\prime}=2 x+2 y\) j. \(x^{\prime}=3 x+y\) \(\begin{aligned} y^{\prime} &=2 x+2 y \\ x^{\prime} &=6 x+4 y \end{aligned}\) l. \(y^{\prime}=2 x-y\)

Step-by-Step Solution

Answer

Stability depends on eigenvalues of matrix \( A \). Calculate for each case.

Step 1: System Representation

Each given system is represented in the form of differential equations. Rewriting them in a matrix form will help us analyze stability. Consider a system \( \begin{pmatrix} x' \ y' \end{pmatrix} = A \begin{pmatrix} x \ y \end{pmatrix} \) where \( A \) is the matrix of coefficients derived from the system of equations.

Step 2: Compute the Matrix for Each System

For each system, extract the coefficients to form matrix \( A \):- a. \( A = \begin{pmatrix} 2 & -5 \ 1 & -2 \end{pmatrix} \)- c. \( A = \begin{pmatrix} -6 & -2 \ 2 & -1 \end{pmatrix} \)- e. \( A = \begin{pmatrix} 3 & -2 \ 2 & -1 \end{pmatrix} \)- g. \( A = \begin{pmatrix} -1 & -5 \ 2 & -3 \end{pmatrix} \)- i. \( A = \begin{pmatrix} -3 & 1 \ 2 & -2 \end{pmatrix} \)- k. \( A = \begin{pmatrix} 1 & -2 \ 2 & 1 \end{pmatrix} \)

Key Concepts

Matrix Representation

In stability analysis of differential equations, matrix representation forms a cornerstone concept. When you are dealing with a system of linear differential equations, it is advantageous to express the problem using matrices. This strategy provides a more compact and manageable format, especially useful for performing further calculations such as finding eigenvalues and eigenvectors.

Consider a system:

• \(\begin{pmatrix} x' \ y' \end{pmatrix} = A \begin{pmatrix} x \ y \end{pmatrix} \)

Here, \(A\) is the matrix of coefficients that you've extracted from the given linear system of equations. Each element of the matrix corresponds directly to the coefficients from the linear equations. By rewriting the equations in this matrix form, you create a pathway to apply linear algebra techniques to study the system's properties. The matrix representation is pivotal for determining the behavior of the system and its stability.

Asymptotic Stability

Asymptotic stability is a critical concept in understanding the long-term behavior of systems described by differential equations. When analyzing stability, particularly in linear systems, we are concerned with determining whether solutions converge to an equilibrium point or diverge away over time.

The equilibrium point of interest is often the origin in the system of equations, noted as \((0,0)\). To determine asymptotic stability for linear systems, you typically check the eigenvalues of the system matrix \(A\):

• If all eigenvalues have strictly negative real parts, the system is considered asymptotically stable. Solutions will decay to zero as time progresses, indicating that perturbations diminish and the system naturally returns to equilibrium.
• If any eigenvalue has a positive real part, the system is unstable.
• If eigenvalues have zero real parts but no negative real components, the system might be stable or unstable based on further qualitative analysis.

By determining the nature of eigenvalues, one can predict how the system will behave with respect to its stability over time, which is crucial for both engineering and scientific applications.

System of Linear Equations

Understanding a system of linear equations is fundamental to stability analysis in differential equations. A system of linear equations is a collection of two or more linear equations involving the same set of variables.

When dealing with such systems, you are primarily interested in:

• Consistency - Whether there exists at least one solution fulfilling all equations, typically representing an equilibrium point or a solution curve in the context of differential equations.
• Finding the Solution - Often involves techniques like substitution, elimination, or using matrix inverses and eigenvalues, especially when applying to larger systems.

In the context of differential equations, these solutions dictate the behaviors and potential outcomes for the dynamics of the system. By translating a system of linear equations into matrix form, using techniques to solve, and then analyzing the solutions, you can uncover deeply insightful traits about system behavior, including stability, periodicity, and boundedness.