Problem 8 - Advanced Engineering Mathematics

Question

In Problems, find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=y^{2}-x \\ &y^{\prime}=x^{2}-y \end{aligned} $$

Step-by-Step Solution

Answer

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

Step 1: Define Critical Points

Critical points of a system are where both derivatives equal zero. For our system, we need to solve the equations \( x' = y^2 - x = 0 \) and \( y' = x^2 - y = 0 \).

Step 2: Solve for y in Terms of x

Start with the first equation, \( y^2 - x = 0 \), to express \( y \) in terms of \( x \). This yields \( y^2 = x \) or \( y = \pm \sqrt{x} \).

Step 3: Substitute y into the Second Equation

Use the expressions for \( y \) from Step 2 in the second equation, \( x^2 - y = 0 \). Substitute \( y = \sqrt{x} \) and \( y = -\sqrt{x} \).

Step 4: Solve the Equation for Each Possibility

For \( y = \sqrt{x} \), substitute into the second equation: \( x^2 - \sqrt{x} = 0 \). Re-arrange to get \( x^2 = \sqrt{x} \). Substitute \( y = -\sqrt{x} \) into \( x^2 + \sqrt{x} = 0 \), which simplifies to the same result since \(-\sqrt{x}\) only applies to negative roots.

Step 5: Simplify the Equation

In both cases, solve \( x^2 = x^{1/2} \). Re-write as \( x^{4/2} = x^{1/2} \), leading to \( x^{3/2} = 1 \). Thus, \( x = 0 \) or \( x = 1 \).

Step 6: Find Corresponding y Values and Critical Points

For \( x = 0 \), then \( y = \pm \sqrt{0} = 0 \), giving critical point \((0,0)\). For \( x = 1 \), substitute back to find \( y = \pm \sqrt{1} = \pm 1 \), giving critical points \((1, 1)\) and \((1, -1)\).

Key Concepts

Understanding Critical Points

In the realm of autonomous systems, critical points, also known as equilibrium points, are vital for understanding how solutions to differential equations behave. Critical points are the values where the rate of change of each variable in the system is zero. In simpler terms, it's the point at which the system is at rest, and no changes occur over time.

To find the critical points, we set the derivatives equal to zero. Take, for example, our system where

• \( x' = y^2 - x = 0 \)
• \( y' = x^2 - y = 0 \)

By solving these simultaneous equations, we find the solutions that give us the coordinates of the critical points. This process involves algebraic manipulation to express variables and find intersections where both conditions are satisfied.

Identifying critical points helps us to predict long-term behavior in the system, such as where a system naturally rests or how it might oscillate about an equilibrium. They are foundational for performing further analysis like stability and behavior prediction of the system dynamics.

System of Differential Equations Essentials

A system of differential equations involves multiple interrelated equations that describe dynamic processes involving rates of change. In this context, every equation captures how a variable changes over time relative to other variables. Such systems are called autonomous if they do not explicitly depend on the independent variable, typically time.

For the given problem, the system is expressed as

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

Here, each equation influences the outcome of the other. By examining this system, students learn to understand the dynamic interplay between different variables and how they can together describe a complex behavior.

In real-world scenarios, these systems can model populations, chemical reactions, physical systems, and more. Understanding their behavior requires deep analysis, allowing for insights into stability, oscillations, and trends.

Phase Plane Analysis Concepts

Phase plane analysis is an invaluable tool for visualizing the behavior of a system of differential equations. The phase plane itself is a graphical representation, where each axis represents one of the system's variables, and trajectories on this plane depict how those variables evolve.

Using phase plane diagrams, students can visually identify equilibrium or critical points and analyze the stability and type of those points, such as whether they attract or repel solutions. This graphical approach provides intuitive insights into system dynamics which can sometimes be missed in purely algebraic solutions.

A phase plane analysis can show various features:

• Critical points where trajectories meet or converge.
• Limit cycles, which indicate repeating or oscillating behavior.
• The overall systemic flow and direction fields, showing state changes.

By assessing these characteristics, we gain a deeper understanding of the system's inherent properties and long-term behavior. Thus, phase plane analysis extends the understanding garnered from equations, offering a more comprehensive view of how systems behave.