Problem 16 - Advanced Engineering Mathematics

Question

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=-x\left(4-y^{2}\right) \\ &y^{\prime}=4 y\left(1-x^{2}\right) \end{aligned} $$

Step-by-Step Solution

Answer

The critical points are \((0, 0)\), \((0, 2)\), \((0, -2)\), \((1, 0)\), and \((-1, 0)\).

Step 1: Understanding Critical Points

Critical points for a system of differential equations occur where the derivatives are zero. For this system, we need to find when \(x' = 0\) and \(y' = 0\).

Step 2: Solve for \(x' = 0\)

Given \(x' = -x(4 - y^2)\), set this equation to zero. We realize that \(x' = 0\) when either \(x = 0\) or \(4 - y^2 = 0\).\[ x = 0 \quad \text{or} \quad y^2 = 4 \] This implies \(y = 2\) or \(y = -2\).

Step 3: Solve for \(y' = 0\)

Given \(y' = 4y(1 - x^2)\), set this equation to zero. We find that \(y' = 0\) when either \(y = 0\) or \(1 - x^2 = 0\).\[ y = 0 \quad \text{or} \quad x^2 = 1 \] This implies \(x = 1\) or \(x = -1\).

Step 4: Combine Conditions

Combine the conditions from steps 2 and 3 to find points where both \(x' = 0\) and \(y' = 0\). Consider each possibility from both steps:- If \(x = 0\), then \(y = 0, 2, -2\).- If \(x = 1\), then \(y = 0\).- If \(x = -1\), then \(y = 0\).

Step 5: List all Critical Points

The solutions combining these variables are:- \((0, 0)\)- \((0, 2)\)- \((0, -2)\)- \((1, 0)\)- \((-1, 0)\)

Key Concepts

Critical Points

In the context of differential equations, critical points are locations where the system essentially "pauses," meaning the values of derivatives are zero.

• For the given plane autonomous system, critical points occur when both differential equations have derivatives equating to zero: \(x' = 0\) and \(y' = 0\).
• This happens because these points represent states where the system experiences no change at that specific moment in time.

At such points, the behavior of the system is stable, unstable, or in between. Identifying and analyzing these points helps to understand the system's dynamics. For example, in the exercise, setting the equations to zero led to various combinations of \(x\) and \(y\) values, revealing the system's inherent critical points.
Finding these points is crucial in studying how the system behaves under different conditions and predicting future system behavior.

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. They are powerful tools used to model real-world behavior in physics, engineering, economics, and more.In this exercise, you are working with a set of differential equations that describe a plane autonomous system.

• The first equation is \(x' = -x(4 - y^2)\).
• The second one is \(y' = 4y(1 - x^2)\).

Both relate to how the variables \(x\) and \(y\) change over time.
This is vital in understanding how a system evolves.
Differential equations like these have balances or equilibrium points, where systems don't change. Solving them involves finding such points, which allows us to predict and analyze system dynamics.

Plane Autonomous System

A plane autonomous system is a type of dynamic system described using differential equations that define how the state variables change.

• "Plane" refers to the two-dimensional nature of the system, involving two variables, typically \(x\) and \(y\).
• "Autonomous" means that these equations do not explicitly depend on time—there is no \(t\) variable present.

The lack of time-dependence implies that the system's behavior is driven only by its current state, not when it's observed.
This forms a core feature of many systems analyzed in fields such as ecology or mechanics.
In the exercise, we examined a plane autonomous system, meaning that once the equations are solved, predictions on system behavior can be made at any moment in time, based solely on the current state, without needing additional time-related information.

Mathematical Analysis

Mathematical analysis provides the tools necessary for predicting system behavior through precise calculations and logical implications. In understanding a system of differential equations such as the one explored in the exercise, analysis encompasses:

• Identifying critical points as equilibrium states.
• Using mathematical methods to solve equations.
• Analyzing stability at these points.
• Drawing conclusions about behavior over time.

This rigorous approach allows one to draw meaningful insights about complex systems, such as whether solutions will grow, shrink, or settle over time.
The analysis in the exercise involved methodically determining where the derivatives equaled zero, thus highlighting where changes in the system cease.
Through mathematical analysis, you obtain a deeper understanding that aids in not just solving equations but truly grasping the implications of the system dynamics.