Problem 16 - Advanced Engineering Mathematics

Question

In Problems, 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

Critical points: (0, 0), (1, 2), (-1, 2), (1, -2), (-1, -2).

Step 1: Identify the system of equations

The given autonomous system consists of two equations: 1. \( x' = -x(4 - y^2) \) 2. \( y' = 4y(1 - x^2) \). We need to find the critical points where both derivatives are zero.

Step 2: Set derivatives to zero

For critical points, set \( x' = 0 \) and \( y' = 0 \). - From \( x' = -x(4 - y^2) = 0 \), either \( x = 0 \) or \( 4 - y^2 = 0 \).- From \( y' = 4y(1 - x^2) = 0 \), either \( y = 0 \) or \( 1 - x^2 = 0 \).

Step 3: Solve for critical points from x' equation

- If \( x = 0 \), substitute into the \( y' = 0 \) equation: \( 4y(1 - 0) = 0 \) gives \( y = 0 \). This results in the critical point \((0, 0)\).- If \( 4 - y^2 = 0 \), then \( y^2 = 4 \), so \( y = 2 \) or \( y = -2 \). We need further cases for \( y = 2 \) and \( y = -2 \).

Step 4: Analyze equations for specific y-values

- For \( y = 2 \), substituting into \( y' = 4y(1 - x^2) = 0 \) gives non-zero results unless solved simultaneously with \(x = 0\) already checked, or \( 1 - x^2 = 0 \) which gives \( x^2 = 1 \) so \( x = 1 \) or \( x = -1 \). Thus, critical points are \((1, 2)\) and \((-1, 2)\).- For \( y = -2 \), following the same substitution leads to \((1, -2)\) and \((-1, -2)\).

Step 5: List critical points

Combine all results:- \((0, 0)\)- \((1, 2)\)- \((-1, 2)\)- \((1, -2)\)- \((-1, -2)\)

Key Concepts

Understanding Critical Points

Critical points in a system of autonomous differential equations are where the system does not change. These are solutions or nodes where the rate of change for all involved variables becomes zero. In simpler terms, it's where the system "pauses," thoroughly analyzing interactions between variables if one exists at these points.

To find critical points, you calculate when each derivative in the system equals zero. For the given equations \( x' = -x(4 - y^2) \) and \( y' = 4y(1 - x^2) \), set both derivatives to zero:

• From \( x' = -x(4 - y^2) = 0 \), solutions arise when either \( x = 0 \) or \( 4 - y^2 = 0 \).
• From \( y' = 4y(1 - x^2) = 0 \), solutions arise when either \( y = 0 \) or \( 1 - x^2 = 0 \).

By setting derivatives to zero, you effectively pause the system's progress and find equilibrium where all changes halt, aiding in determining potential steady-state solutions for the system.

Exploring Differential Equations

Differential equations describe how quantities change in relation to each other. Specifically, autonomous differential equations are time-independent, meaning the behavior of variables depends only on their current state, not time.

Our system includes:

• \( x' = -x(4 - y^2) \)
• \( y' = 4y(1 - x^2) \)

These show how each variable evolves related to one another, making them essential in modeling systems where time-free dynamic interactions matter, like population models in ecology or certain mechanical systems. Understanding this interaction allows us to predict system behavior and identify points of sudden change or stability.

Differential equations can become complicated swiftly. However, focusing on the zero-derivatives between entities, like in our task, effectively finds critical points, illuminating stability zones in dynamic processes.

Plane Analytical Geometry Basics

In plane analytical geometry, we examine geometric shapes via algebraic equations. Critical points calculated in context become recognizable positions on a plane, cultivated from variable systems like the one being analyzed.

The geometry of our autonomous system can be considered in the coordinate plane using the derived critical points. By solving \( x = 0 \), \( y^2 = 4 \), and \( x^2 = 1 \), our solutions: \((0,0), \ (1, 2), \ (-1, 2), \ (1, -2), \ (-1, -2)\) become points of intersection where indicated behavior holds.

• \((0,0)\): Trivial point often representing neutral equilibrium.
• \(\{(1,2), (-1,2), (1,-2), (-1,-2)\}\): Points where the surface interaction hits maximizing constraints, suggesting further dynamics.

Understanding these relationships between points facilitates our broader comprehension of spatial behavior in systems, crucial for fields like physics or engineering, where prediction of behavior within given constraints in the coordinate space enhances problem-solving frameworks.