Problem 11 - Advanced Engineering Mathematics

Question

Show that the plane autonomous system $$ \begin{aligned} &x^{\prime}=x\left(1-x^{2}-3 y^{2}\right) \\ &y^{\prime}=y\left(3-x^{2}-3 y^{2}\right) \end{aligned} $$ has no periodic solutions in an elliptical region about the origin.

Step-by-Step Solution

Answer

The system has no periodic solutions in the elliptical region as \( \frac{dV}{dt} \leq 0 \).

Step 1: Identify the equations

Let's first identify the system of equations given: \( x' = x(1 - x^2 - 3y^2) \) and \( y' = y(3 - x^2 - 3y^2) \). This defines the behavior of the autonomous system in terms of \( x' \) and \( y' \).

Step 2: Propose a candidate Lyapunov function

We'll use a Lyapunov function to help determine the stability of solutions in the system. Consider the function \( V(x, y) = x^2 + 3y^2 \), which is elliptic and covers the entire possible phase space we are interested in.

Step 3: Compute the derivative of the Lyapunov function

We calculate the time derivative of the Lyapunov function along the trajectories of the system: \( \frac{dV}{dt} = \frac{d}{dt}(x^2 + 3y^2) = 2x x' + 6y y' \).

Step 4: Substitute the system's differential equations

Substitute \( x' = x(1 - x^2 - 3y^2) \) and \( y' = y(3 - x^2 - 3y^2) \) into \( \frac{dV}{dt} \). This gives: \( \frac{dV}{dt} = 2x [x(1 - x^2 - 3y^2)] + 6y [y(3 - x^2 - 3y^2)] \).

Step 5: Simplify the expression

Simplifying this expression, we get \( \frac{dV}{dt} = 2x^2(1 - x^2 - 3y^2) + 6y^2(3 - x^2 - 3y^2) \). This further simplifies to \( \frac{dV}{dt} = 2x^2 - 2x^4 - 6x^2y^2 + 18y^2 - 6y^2x^2 - 18y^4 \).

Step 6: Factorize and analyze \( \frac{dV}{dt} \)

Group and simplify terms to show the nonpositivity: \( \frac{dV}{dt} = 2(x^2 - x^4 - 3x^2y^2 + 9y^2 - 9y^4) \). Notice that \( x^4 + 3x^2y^2 + 9y^4 \) are positive definite, indicating \( \frac{dV}{dt} \leq 0 \).

Step 7: Conclude the region has no periodic solutions

Since the derivative \( \frac{dV}{dt} \leq 0 \) and is not identically zero except at equilibrium points, the autonomous system has no periodic orbits within the region since a Lyapunov function without any increase implies stability.

Key Concepts

Understanding Autonomous Systems

Autonomous systems are dynamical systems governed by differential equations that do not explicitly depend on time. Here, the term "autonomous" means that the rules governing the behavior of the system stay consistent over time.
In our exercise, we have the systems of equations:

• \( x' = x(1 - x^2 - 3y^2) \)
• \( y' = y(3 - x^2 - 3y^2) \)

This is an example of an autonomous system, as the functions \( x' \) and \( y' \) evolve due to the existing state \( (x, y) \), independent of the time variable. Autonomous systems are significant because they generally model stable phenomena in real-life applications, such as the predator-prey models in ecology or circuits in engineering.
The focus in an autonomous system is on understanding how these state changes, described by \( x' \) and \( y' \), propagate over time and whether they stabilize, diverge, or form repeating cycles.

Unraveling Differential Equations

Differential equations describe the relationships involving the rates of change with respect to variables. Simply put, they tell us how things change over time or space. The system of equations in our exercise allows us to look into how the variables \( x \) and \( y \) change.
The two equations are:

• \( x' = x(1 - x^2 - 3y^2) \)
• \( y' = y(3 - x^2 - 3y^2) \)

These equations specify how \( x \) and \( y \) not only change but how they influence each other's changes. For example, when the value of \( x \) or \( y \) is large, the terms such as \( -x^2 \) or \( -3y^2 \) may cause \( x' \) and \( y' \) to reduce, indicating a sort of self-regulation in the system. Exploring these equations in detail helps us determine behaviors such as equilibria – where \( x' = 0 \) and \( y' = 0 \) – and the conditions under which solutions might repeat or settle into a particular pattern.

Exploring Periodic Solutions

Periodic solutions refer to solutions of a system that repeat after certain intervals, much like a swinging pendulum or the cycles of the moon. In terms of differential equations, such solutions imply consistent, repeating behavior in the outputs of the system.
For the given autonomous system:

• \( x' = x(1 - x^2 - 3y^2) \)
• \( y' = y(3 - x^2 - 3y^2) \)

We analyzed the potential for periodic solutions to understand if the system could have repeating cycles. By using a Lyapunov function, which was determined as \( V(x, y) = x^2 + 3y^2 \), the exercise showed that the derivative \( \frac{dV}{dt} \) is non-positive in the region of interest.
This derivative outcome implies that the system does not exhibit periodic solutions in the elliptical region because there is no net 'gain' that would typically lend itself to a repeating cycle. In essence, the stability inferred by the Lyapunov function means the system either stabilizes or heads towards equilibrium, not periodicity.