Problem 8 - Calculus for Biology and Medicine

Question

Consider $$ \begin{array}{l} \frac{d x_{1}}{d t}=-x_{2} \\ \frac{d x_{2}}{d t}=x_{1}+x_{2} \end{array} $$ Determine the direction vectors associated with the following points in the \(x_{1}-x_{2}\) plane, and graph the direction vectors in the \(x_{1}-\) \(x_{2}\) plane: \((1,0),(0,1),(-1,0),(0,-1),(1,1),(0,0)\), and \((-2,-2)\).

Step-by-Step Solution

Answer

Direction vectors are: (0,1), (-1,1), (0,-1), (1,-1), (-1,2), (0,0), (2,-4). Plot them on an \(x_1-x_2\) plane.

Step 1: Understand the Differential System

We are given a system of differential equations:\[\begin{array}{l}\frac{d x_{1}}{d t}=-x_{2} \\frac{d x_{2}}{d t}=x_{1}+x_{2}\end{array}\]These equations describe changes in variables \(x_1\) and \(x_2\) over time. Our goal is to determine the direction of change at given points on the \(x_1 - x_2\) plane.

Step 2: Identify the Direction Vector Formula

The direction vector at any point \((x_1, x_2)\) is given by the differential equations themselves, specifically \((\frac{d x_{1}}{d t}, \frac{d x_{2}}{d t})\). This means for any point \((x_1, x_2)\), the direction vector is \((-x_2, x_1 + x_2)\).

Step 3: Calculate Direction Vectors for Each Point

We calculate the direction vectors for each specified point using the formula: - For \((1, 0)\): \((-0, 1 + 0) = (0, 1)\)- For \((0, 1)\): \((-1, 0 + 1) = (-1, 1)\)- For \((-1, 0)\): \((0, -1 + 0) = (0, -1)\)- For \((0, -1)\): \((1, 0 - 1) = (1, -1)\)- For \((1, 1)\): \((-1, 1 + 1) = (-1, 2)\)- For \((0, 0)\): \((0, 0 + 0) = (0, 0)\)- For \((-2, -2)\): \((2, -2 - 2) = (2, -4)\).

Step 4: Graphing the Direction Vectors

To graph the direction vectors on the \(x_1 - x_2\) plane, plot each point with its corresponding direction vector:- Point \((1, 0)\) with vector \((0, 1)\)- Point \((0, 1)\) with vector \((-1, 1)\)- Point \((-1, 0)\) with vector \((0, -1)\)- Point \((0, -1)\) with vector \((1, -1)\)- Point \((1, 1)\) with vector \((-1, 2)\)- Point \((0, 0)\) has zero vector, \((0, 0)\), so no direction.- Point \((-2, -2)\) with vector \((2, -4)\).

Step 5: Interpret the Zero Vector

The direction vector \((0, 0)\) at point \((0, 0)\) indicates that this point is a critical point or equilibrium, where there is no movement in either direction. This means the system doesn't change at this point.

Key Concepts

Direction Vectors

Direction vectors are crucial tools in understanding the behavior of systems described by differential equations. These vectors are simply arrows that show the direction and rate of change of a system at a particular point in the plane or space. For each point in the plane, the direction vector gives us an immediate picture of where the system is heading next.
In our exercise:

• The direction vector formula is derived directly from the differential equations, \(\left( \frac{d x_{1}}{d t}, \frac{d x_{2}}{d t} \right) = (-x_2, x_1 + x_2)\).
• This means, given any point \((x_1, x_2)\), the system's change direction at that point is indicated by the vector \((-x_2, x_1 + x_2)\).

By calculating these vectors at various points such as \((1,0)\) and \((0,1)\), we can visualize how the system behaves throughout the \(x_1-x_2\) plane. Understanding these vectors allows us to see the flow of the system and anticipate its movement under the given conditions.
This visual representation is essential for checking stability and behavior of solutions in differential equations.

Phase Plane Analysis

Phase plane analysis is a visual method used to study systems of differential equations by plotting the trajectory of solutions in a coordinate plane. It is especially useful in two-dimensional systems, where the behavior of a system can be interpreted graphically.
In our exercise, each point such as \((1,0)\) or \((-2,-2)\) represents the initial conditions of the system, and their associated direction vectors show the system's immediate behavior.

• The phase plane displays curves called trajectories, which show the path that the system's state follows over time.
• The arrows representing direction vectors give insight into the system's dynamics at each point.

By observing the phase plane, we can identify recurring patterns or cycles, stability of points, and how changes in initial conditions influence future states. This graphical analysis provides a clearer understanding of complex systems and is often combined with numerical solutions for deeper insights.

Equilibrium Points

Equilibrium points are specific locations in the phase plane where the system experiences no change, meaning the direction vector is zero at these points. In other words, an equilibrium point is where the system is at rest.

• In our differential equation system, the origin \((0,0)\) is an equilibrium point. This is evident since both equations output zero, giving a null direction vector: \((0,0)\).
• Mathematically, these are found where \( \frac{d x_{1}}{d t} = 0 \) and \( \frac{d x_{2}}{d t} = 0 \).

These points are crucial in analyzing system stability. If small disturbances around these points die out and return to equilibrium, the point is considered stable. Conversely, if disturbances grow, it's unstable. Evaluating equilibrium points in a phase plane allows us to predict long-term behavior of dynamical systems and design control strategies accordingly.