Problem 15 - Advanced Engineering Mathematics

Question

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x\left(1-x^{2}-3 y^{2}\right) \\ &y^{\prime}=y\left(3-x^{2}-3 y^{2}\right) \end{aligned} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the System

We are given a system of two differential equations, \( x' = x(1 - x^2 - 3y^2) \) and \( y' = y(3 - x^2 - 3y^2) \). The goal is to find critical points, i.e., points where both \( x' = 0 \) and \( y' = 0 \).

Step 2: Set Equations to Zero

Set \( x' = 0 \) which yields: \( x(1 - x^2 - 3y^2) = 0 \). This gives two possibilities: either \( x = 0 \) or \( 1 - x^2 - 3y^2 = 0 \). Similarly, set \( y' = 0 \): \( y(3 - x^2 - 3y^2) = 0 \). This gives \( y = 0 \) or \( 3 - x^2 - 3y^2 = 0 \).

Step 3: Solve Equations Where x = 0 and y = 0

For \( x = 0 \), the second equation \( y(3 - 3y^2) = 0 \) implies \( y = 0 \) or \( y^2 = 1 \). Thus, possible points are \((0, 0)\), \((0, 1)\), and \((0, -1)\). Similarly, for \( y = 0 \), the first equation \( x(1 - x^2) = 0 \) implies \( x = 0 \) or \( x^2 = 1 \). Thus, the possible points are \((0, 0)\), \((1, 0)\), and \((-1, 0)\).

Step 4: Solve Equation When Neither x nor y is Zero

For both \( x \) and \( y \) not equal to zero, set \( 1 - x^2 - 3y^2 = 0 \) and \( 3 - x^2 - 3y^2 = 0 \). These two equations are incompatible because they imply \( 1 = 3 \), which is not possible. Thus, no additional critical points exist from this case.

Step 5: Compile All Critical Points

Collect all points found: \((0, 0)\), \((0, 1)\), \((0, -1)\), \((1, 0)\), and \((-1, 0)\). These are all the critical points of the system.

Key Concepts

Autonomous Systems

Autonomous systems refer to a type of differential equation system where the system's behavior is time-independent. In simpler terms, this means that the change in the system is dependent only on the current state, not on when the change occurs. This feature allows autonomous systems to be easier to analyze, particularly in terms of stability and behavior over time.

Autonomous systems often take the form of \[ \begin{aligned} \frac{dx}{dt} &= f(x, y), \\frac{dy}{dt} &= g(x, y) \end{aligned} \]Where the functions \(f\) and \(g\) specify the rules for change in each variable. By setting these equations equal to zero, we attempt to find equilibrium points, which don't change over time. These points are crucial for understanding the long-term behavior of a system.

Plane Autonomous System

A plane autonomous system is a specific type of autonomous system where the state of the system is described using two variables, often interpreted as a point on a plane. The system can be analyzed by plotting vector fields or trajectories to visually understand how those points or states evolve.

In the case of our exercise, the system is defined by two differential equations:\[ \begin{aligned} &x^{\prime}=x\left(1-x^{2}-3y^{2}\right) \ &y^{\prime}=y\left(3-x^{2}-3y^{2}\right) \end{aligned} \]The beauty of plane autonomous systems lies in their geometric interpretations. By finding critical points and analyzing their nature (whether they are stable, unstable, or semi-stable), we can draw conclusions about the system's behavior without needing explicit solutions to the differential equations.

Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They play a key role in modeling continuous phenomena such as growth, decay, or oscillation, which are commonplace in physics, biology, and engineering.

For a plane autonomous system, differential equations describe how the state variables change relative to each other, rather than over time explicitly.

In our example, the differential equations \[ x' = x(1 - x^2 - 3y^2) \] and \[ y' = y(3 - x^2 - 3y^2) \]serve to model the interactions between variables \(x\) and \(y\). Understanding the role of each component in these equations is crucial to solving for critical points and ultimately, understanding the system's behavior.

Critical Points Analysis

Critical points are solutions to a system of differential equations where all derivatives are zero; essentially, they represent the state where there is no change. Identifying these critical points involves setting the equations to zero and solving for the system state.

In the given exercise, critical points are found at:

• \((0, 0)\)
• \((0, 1)\)
• \((0, -1)\)
• \((1, 0)\)
• \((-1, 0)\)

These points offer deep insights into the system's dynamic behavior.

Once determined, further analysis is often conducted to classify these points as stable, unstable, or saddle points.
Such classifications help predict how the system will evolve from nearby states. Critical points analysis is thus an essential tool in understanding complex dynamical systems.