Problem 15 - Advanced Engineering Mathematics

Question

In Problems, 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

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

Step 1: Identify Critical Points

A critical point occurs where both derivatives are zero: \(x' = 0\) and \(y' = 0\). So, we start by setting \(x' = 0\) and \(y' = 0\).

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

Solve the equation \(x(1-x^2-3y^2)=0\). This can be true if either \(x=0\) or \(1-x^2-3y^2=0\).

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

Solve the equation \(y(3-x^2-3y^2)=0\). This can be true if either \(y=0\) or \(3-x^2-3y^2=0\).

Step 4: Identify Intersection of Conditions

Combine the conditions from \(x' = 0\) and \(y' = 0\), consider cases where \(x = 0\), \(y = 0\), and when both equations satisfy the conditions simultaneously.

Step 5: Case 1: \(x = 0\)

Substitute \(x=0\) into \(y(3-x^2-3y^2)=0\) which becomes \(y(3-3y^2) = 0\). Thus, \(y = 0\) or \(3-3y^2 = 0\), leading to \(y = \pm1\). So the critical points are \((0, 0)\), \((0, 1)\), \((0, -1)\).

Step 6: Case 2: \(y = 0\)

Substitute \(y=0\) into \(x(1-x^2-3y^2)=0\) which becomes \(x(1-x^2)=0\). Thus, \(x = 0\) or \(1-x^2 = 0\), leading to \(x = \pm1\). So the critical points are \((0, 0)\), \((1, 0)\), \((-1, 0)\).

Step 7: Case 3: Solve Simultaneous Equations

If both \(1-x^2-3y^2=0\) and \(3-x^2-3y^2=0\) simultaneously, subtract the second from the first to get \(-2 = 0\), which is false. Hence, no additional solutions.

Step 8: List all Critical Points

Combine all unique solutions from above cases: \((0, 0), (0, 1), (0, -1), (1, 0), (-1, 0)\).

Key Concepts

Autonomous Systems

Autonomous systems are a special class of systems in the field of differential equations where time is not explicitly involved in the evolution of the system. This means the rates of change for the variables do not depend directly on the time variable. Instead, they depend solely on the current values of the variables themselves.
The function for each variable is defined by an equation that involves only the variables present, such as:

• In our exercise, we have functions for \(x'\) and \(y'\), representing the rates of change of \(x\) and \(y\).
• Each of these functions depends only on the variables \(x\) and \(y\).

By studying autonomous systems, we can understand how systems evolve over time based solely on their current state. This makes it easier to analyze the general behavior of the systems without having to track individual trajectories over time.

Differential Equations

Differential equations are mathematical equations that involve derivatives, which represent how a quantity changes over time. These are fundamental tools in modeling natural phenomena and engineering systems because they can describe how dynamic processes occur.
Differential equations can be classified into different types, such as ordinary differential equations (ODEs), where the functions involved depend only on one variable, typically time. In our autonomous system exercise:

• The differential equations \(x' = x(1-x^2-3y^2)\) and \(y' = y(3-x^2-3y^2)\) describe how the values of \(x\) and \(y\) change.
• They are used to find critical points by setting the rate of change to zero.

This allows us to identify conditions where the system does not change, providing insights into the system's overall behavior.

Phase Plane Analysis

Phase plane analysis is a graphical method used to study the behavior of autonomous systems by plotting the trajectories of the system in a two-dimensional plane. Each axis represents one of the variables.

• The phase plane helps visualize how different initial conditions lead to different trajectories over time.
• Critical points, where the system changes nature, often become apparent in this plane.

In the given exercise, the critical points, such as \((0, 0)\), are plotted on the phase plane, allowing us to see where the system equilibrates or where it switches directions.
Identifying critical points and observing their behavior in the phase plane helps us understand stability and ongoing system behavior better.

System of Equations

A system of equations involves multiple equations that are solved together to find values satisfying all equations simultaneously. This is common in cases where multiple variables interact, each governed by different rules or constraints.
In the problem, we solve the system formed by \(x' = 0\) and \(y' = 0\) to find the critical points:

• These equations can be solved independently or simultaneously, depending on how they are structured.
• From the solution, we obtain a set of conditions or values where both \(x\) and \(y\) do not change, known as critical points.

By solving systems of equations, particularly in autonomous systems, we can determine states where the system reaches a steady state or stability, crucial for further analysis like stability investigation.

Stability Analysis

Stability analysis is vital for understanding the behavior of critical points in a system, particularly whether they can maintain their state when subjected to small disturbances.
In essence, stability analysis answers the question: if a system is slightly disturbed, will it return to its previous state, move away, or reach a new state?

• Critical points are evaluated to see if they are stable, unstable, or semi-stable.
• Techniques such as eigenvalue analysis or linearization around the critical points are used.

For the exercise’s critical points, stability analysis helps determine which points might be attractors (pulling trajectories towards them) or repellors (pushing trajectories away). Understanding stability is crucial for predicting long-term behavior of dynamical systems.