Problem 14 - Biocalculus Calculus for the Life Sciences

Question

Sketch several solution curves in the phase plane of the system of differential equations \(d \mathbf{x} / d t=A \mathbf{x}\) using the given eigenvalues and eigenvectors of \(A .\) \(\lambda_{1}=-3, \quad \lambda_{2}=-2 ; \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{1} \\ {0}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{l}{0} \\ {1}\end{array}\right]\)

Step-by-Step Solution

Answer

This system forms a stable node at the origin; trajectories approach the origin along straight lines parallel to given eigenvectors.

Step 1: Analyze the System

The given system is \( \frac{d\mathbf{x}}{dt} = A\mathbf{x} \) with eigenvalues \( \lambda_1 = -3 \) and \( \lambda_2 = -2 \), and corresponding eigenvectors \( \mathbf{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix} \) and \( \mathbf{v}_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix} \). This indicates a stable node, meaning trajectories move towards the origin without oscillating as time goes forward since both eigenvalues are negative.

Step 2: Draw the Eigenvectors

In the phase plane, plot the eigenvectors \( \mathbf{v}_1 \) and \( \mathbf{v}_2 \). \( \mathbf{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix} \) corresponds to the x-axis, and \( \mathbf{v}_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix} \) corresponds to the y-axis. These vectors indicate the directions of the trajectories near the origin.

Step 3: Sketch Trajectories Along Eigenvectors

Since both eigenvalues are negative, trajectories along the eigenvectors will move toward the origin over time. Sketch arrows along the x-axis pointing towards the origin to represent movement along \( \mathbf{v}_1 \). Similarly, sketch arrows along the y-axis pointing towards the origin to represent movement along \( \mathbf{v}_2 \).

Step 4: Sketch General Solution Curves

The general solution is a linear combination of the eigenvectors and has the form \( \mathbf{x}(t) = c_1 e^{-3t} \mathbf{v}_1 + c_2 e^{-2t} \mathbf{v}_2 \), where \( c_1 \) and \( c_2 \) are constants. From various initial points, draw trajectories as curves that are combinations of movements along the x-axis and y-axis, all converging to the origin due to the stable nature of the node. These curves should reflect the fact they will move closer to the origin over time.

Key Concepts

Phase Plane Analysis

Phase plane analysis is a visual method used to study differential equations and their solutions. By plotting trajectories in a two-dimensional graph, you gain insight into how systems evolve over time. This kind of analysis is beneficial for examining the behavior of dynamic systems, like those described by differential equations.

Here are the key points to understand this method:

• The phase plane is constructed by using variables from the system as axes, typically denoting the state of the system.
• Together, these plots show the possible trajectories that the system can follow as time progresses.
• This approach allows one to identify critical points, stability, and the nature of the system dynamics without solving the equations analytically.

In our context, one sees how trajectories in the phase plane approach or move away from an equilibrium point, such as the origin in this exercise.

The exercise here shows a stable node scenario where the origin is a state all paths converge to over time. Understanding phase plane plots empowers you to predict long-term behavior simply by sketching—no heavy computations required!

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors of a matrix are fundamental to understanding linear transformations and systems of differential equations. They reveal the behavior of the system, particularly around equilibrium points.

In the exercise, we are given two eigenvalues, \(\lambda_1 = -3\) and \(\lambda_2 = -2\), along with their corresponding eigenvectors:

• \(\mathbf{v}_1 = \begin{bmatrix} 1 \ 0 \end{bmatrix}\) aligns with the x-axis.
• \(\mathbf{v}_2 = \begin{bmatrix} 0 \ 1 \end{bmatrix}\) aligns with the y-axis.

These elements are used to understand how solutions to the differential equation behave:

• Eigenvectors provide directions (or axes) along which the system evolves.
• Eigenvalues give the rate at which trajectories move along these eigenvectors. Negative eigenvalues, as in our case, indicate that the trajectories will decay towards the origin, showing stability.

Understanding eigenvalues and eigenvectors helps predict system behavior just by examining the matrix \(A\) without solving the entire differential equation.

Stable Node

A stable node is a concept in the qualitative analysis of differential equations where all trajectories in the phase plane converge to a single point, which is an equilibrium.

When a system has a stable node, as shown in the problem with both negative eigenvalues, it means:

• All paths in the phase plane lead towards the equilibrium point (here, the origin).
• There is no oscillation—trajectories simply smoothen out as they approach stability.

In linear systems like this exercise, the stable node behavior is easier to recognize and predict:

• The rate at which trajectories converge is determined by the magnitude of the eigenvalues. Larger magnitude indicates faster convergence.
• Because \(\lambda_1 = -3\) is more negative than \(\lambda_2 = -2\), convergence along \(\mathbf{v}_1\) will be quicker than along \(\mathbf{v}_2\).

Recognizing a stable node allows you to understand the overall system stability and predict safe states. This is powerful, as it shows no matter the starting point, the system will eventually rest steadily at the equilibrium point, embodying a sense of inherent balance.