Problem 13 - 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 \(A .\) \(\lambda_{1}=2, \quad \lambda_{2}=-2 ; \quad \mathbf{v}_{1}=\left[ \begin{array}{l}{3} \\ {1}\end{array}\right] \quad \mathbf{v}_{2}=\left[ \begin{array}{l}{1} \\ {1}\end{array}\right]\)

Step-by-Step Solution

Answer

Sketch trajectories diverging along \( \mathbf{v}_1 \) and converging along \( \mathbf{v}_2 \); origin is a saddle point.

Step 1: Understand the System Dynamics

The system is given by \( \frac{d \mathbf{x}}{dt} = A \mathbf{x} \) where \( A \) is a matrix. The eigenvalues \( \lambda_1 = 2 \) and \( \lambda_2 = -2 \) tell us that we have solutions that behave exponentially. A positive eigenvalue \( \lambda_1 = 2 \) means solutions grow exponentially in the direction of \( \mathbf{v}_1 \), and a negative eigenvalue \( \lambda_2 = -2 \) means solutions decay exponentially in the direction of \( \mathbf{v}_2 \).

Step 2: Identify Solution Behavior

With \( \lambda_1 = 2 \) and the eigenvector \( \mathbf{v}_1 = \begin{bmatrix} 3 \ 1 \end{bmatrix} \), solutions will move away from the origin along this vector. With \( \lambda_2 = -2 \) and eigenvector \( \mathbf{v}_2 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \), solutions will move towards the origin along this vector.

Step 3: Sketch Eigenvectors in the Phase Plane

Draw the vector \( \mathbf{v}_1 = \begin{bmatrix} 3 \ 1 \end{bmatrix} \) in the phase plane. This vector points in the direction where solutions will grow, forming trajectories that move outward. Draw the vector \( \mathbf{v}_2 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \) where solutions will decay, forming trajectories that move inward toward the origin.

Step 4: Plot Trajectories in the Phase Plane

Use the direction of eigenvectors to draw several solution curves. Start some trajectories near \( \mathbf{v}_1 \), showing they diverge exponentially. Start some trajectories near \( \mathbf{v}_2 \), showing convergence to the origin. Paths will tend to align with \( \mathbf{v}_2 \) as they approach the origin and with \( \mathbf{v}_1 \) as they move away.

Step 5: Analyze the Stability of the Origin

With one positive and one negative eigenvalue, the origin is a saddle point. Solutions approach the origin along \( \mathbf{v}_2 \) and depart along \( \mathbf{v}_1 \), confirming the system's unstable nature.

Key Concepts

Eigenvalues

Eigenvalues play a crucial role in understanding the system of differential equations. They are values that indicate how the system will behave over time, particularly whether the system is stable, unstable, or oscillatory. In our case, the matrix \( A \) has two eigenvalues: \( \lambda_1 = 2 \) and \( \lambda_2 = -2 \).

Eigenvalues can be thought of as stretching or shrinking factors along certain directions.

• A positive eigenvalue, such as \( \lambda_1 = 2 \), means the system experiences exponential growth in the direction of its associated eigenvector.


• A negative eigenvalue, like \( \lambda_2 = -2 \), implies exponential decay.

This behavior allows us to predict the trajectories of the system, as we can foresee where solutions tend to expand or contract within the phase plane.

Eigenvectors

Eigenvectors are fundamental for understanding the direction of these exponential behaviors dictated by eigenvalues. They determine the line along which solutions either grow or decay.

For the given problem:

• The eigenvector associated with \( \lambda_1 = 2 \) is \( \mathbf{v}_1 = \begin{bmatrix} 3 \ 1 \end{bmatrix} \). In the phase plane, this means solutions start moving away from the origin along this vector.


• On the other hand, the eigenvector for \( \lambda_2 = -2 \) is \( \mathbf{v}_2 = \begin{bmatrix} 1 \ 1 \end{bmatrix} \), causing solutions to decay or move towards the origin.

This highlights the role of vectors as directors of pathways for solutions, emphasizing key paths of movement within the system dynamics.

Phase Plane Analysis

Phase plane analysis offers a visual depiction of the trajectories of differential equations, helping us understand the system's behavior at a glance. It's like a map of possible paths that solutions can take over time.

In our analysis:

• The eigenvector \( \mathbf{v}_1 \) represents a path from which solutions rapidly escape the origin, leading to outward trajectories.


• Conversely, \( \mathbf{v}_2 \) directs solutions into the origin, forming inward trajectories.

Drawing these vectors in the phase plane, we sketch trajectories that start around these vectors. The interplay of these paths shows the dynamic balance between the forces of growth and decay, dictated by positive and negative eigenvalues.

Stability Analysis

Stability analysis lets us evaluate the equilibrium points of the system and determine if they are stable or unstable. It provides insight into how systems behave as time progresses. In our context, the origin is the equilibrium point.

With eigenvalues \( \lambda_1 = 2 \) and \( \lambda_2 = -2 \), we can easily assess the stability:

• The presence of a positive eigenvalue indicates divergence from the origin, implying instability in certain directions.


• The negative eigenvalue signals convergence towards the origin in other directions.

As a result, the origin is classified as a saddle point, where solutions approach along one direction and diverge along another. This mixed behavior classifies the system as inherently unstable.