Problem 7 - Advanced Engineering Mathematics

Question

Find all critical points of the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=x+x y \\ &y^{\prime}=-y-x y \end{aligned} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Critical Points

To find the critical points of the system, set the derivatives equal to zero, which corresponds to setting the right-hand sides of the given equations to zero. This gives us the system of equations:\[\begin{aligned}&x + xy = 0 \&-y - xy = 0\end{aligned}\]

Step 2: Solve the First Equation

From the first equation, \(x + xy = 0\), factor out \(x\) to obtain:\[ x(1 + y) = 0 \]This implies two possibilities: \(x = 0\) or \(1 + y = 0\).

Step 3: Solve the Second Equation

From the second equation, \(-y - xy = 0\), factor out \(-y\) to obtain:\[ -y(1 + x) = 0 \]This implies two possibilities: \(y = 0\) or \(1 + x = 0\).

Step 4: Determine Critical Points by Combining Solutions

Combine the solutions from both equations:- If \(x = 0\), substitute into the second equation \(-y - 0 \cdot y = 0\) to get \(y = 0\). This gives one critical point: \((0, 0)\).- If \(1 + y = 0\) implies \(y = -1\), substitute into the second equation \(-(-1) - x(-1) = 0\) giving \(1 + x = 0\), thus \(x = -1\). This gives another critical point: \((-1, -1)\).

Step 5: List the Critical Points

The critical points found are \((0, 0)\) and \((-1, -1)\). These are the points where the flow of the system is stationary.

Key Concepts

Phase Plane Analysis

Phase plane analysis is a valuable tool used to study systems of differential equations. It involves plotting the trajectories of solutions in a two-dimensional plane where each axis corresponds to one of the variables. This visual representation can give us insights into the nature of the system. By examining the paths taken by solutions over time, we can identify patterns or behaviors, such as periodic orbits or chaotic behavior.

For autonomous systems, where the equations do not explicitly depend on time, phase plane diagrams are especially useful. These diagrams can reveal critical points, or equilibria, which are positions in the plane where the system remains stationary. By studying the locations and nature of these points, we gain a better understanding of the system's dynamics.

Differential Equations

Differential equations describe how a quantity changes over time or space. In the given system, we have two differential equations representing the rates of change for variables \( x \) and \( y \).

The primary goal in analyzing such systems is to understand how these variables evolve over time. The system is defined as:

• \( x' = x + xy \)
• \( y' = -y - xy \)

These equations are first-order, meaning they involve the first derivative of the variables with respect to time.
Solving the differential equations helps us find critical points, or values where these rates of change are zero, indicating the system's stationary states.

Dynamical Systems

A dynamical system is a mathematical model used to describe the evolution of variables over time. The behavior of these systems can be linear or nonlinear, altering the complexity of their analysis.

In our exercise, the system is autonomous, meaning it does not depend on an external time-dependent force. Autonomous dynamical systems are governed solely by their internal structure. They evolve naturally based on their initial conditions.

Critical points play a key role in understanding the long-term behavior of dynamical systems. These are points where the system exhibits no changes, and in some systems, all other paths lead toward or away from these points. Identifying and analyzing the critical points of a system can help predict the system's future behavior.

Stability Analysis

Stability analysis is crucial for determining the nature of critical points in a system of differential equations. It tells us whether small perturbations around a critical point will die out or amplify over time.

The stability of a critical point can be categorized as:

• Stable: Solutions stay near the critical point when disturbed slightly.
• Unstable: Solutions diverge from the critical point when slightly disturbed.
• Semi-stable or neutrally stable: Solutions neither approach nor diverge significantly from the critical point.

Analyzing the stability often involves linearizing the system around the critical points and studying the resulting linear system. Essentially, stability analysis helps predict how the system will react over time to small changes, which is vital for understanding long-term dynamics.