Problem 9 - Advanced Engineering Mathematics

Question

Use the Dulac negative criterion to show that the given plane autonomous system has no periodic solutions. Experiment with simple functions of the form \(\delta(x, y)=a x^{2}+b y^{2}, e^{a x+b y}\), or \(x^{a} y^{b}\). $$ \begin{aligned} &x^{\prime}=-2 x+x y \\ &y^{\prime}=2 y-x^{2} \end{aligned} $$

Step-by-Step Solution

Answer

The given system has no periodic solutions by Dulac's negative criterion.

Step 1: Understand the Dulac's Criterion

The Dulac's criterion states that if there exists a continuously differentiable function \( \delta(x,y) \) such that \( abla \cdot (\delta \mathbf{F}) eq 0 \) over a simply connected open region, then the system has no periodic solutions in that region. Here, \( \mathbf{F} = (f(x,y), g(x,y)) \), where \( f(x, y) = -2x + xy \) and \( g(x, y) = 2y - x^2 \).

Step 2: Choose a Dulac Function

Experiment with simple functions of the form \( \delta(x, y) = ax^2 + by^2, e^{ax + by}, x^ay^b \). Let's choose \( \delta(x, y) = x \), a straightforward function to work with.

Step 3: Calculate \( \nabla \cdot (\delta \mathbf{F}) \)

First, compute \( \delta f = x(-2x + xy) = -2x^2 + x^2y \) and \( \delta g = x(2y - x^2) = 2xy - x^3 \). Now, compute the divergence: \( \frac{\partial}{\partial x} (\delta f) + \frac{\partial}{\partial y} (\delta g) \).

Step 4: Differentiate and Simplify

Differentiate \( \delta f = -2x^2 + x^2y \) with respect to \( x \), giving \( -4x + 2xy \). Differentiate \( \delta g = 2xy - x^3 \) with respect to \( y \), giving \( 2x \).

Step 5: Analyze the Final Expression

Combine the results: \( abla \cdot (\delta \mathbf{F}) = (-4x + 2xy) + 2x = 2xy - 2x = 2x(y - 1) \). This expression depends on \( x \) and \( y \), and as long as it is not identically zero, over a sufficiently large region, Dulac's criterion holds.

Step 6: Conclude with Dulac's Negative Criterion

Since \( 2x(y - 1) eq 0 \) for almost all \( x eq 0 \) and \( y eq 1 \), \( abla \cdot (\delta \mathbf{F}) eq 0 \) except on a set of measure zero. Thus, by Dulac's Criterion, the system has no periodic solutions in any simply connected region of the plane.

Key Concepts

Autonomous Systems

An autonomous system in mathematics refers to a system of differential equations where the variables do not explicitly depend on time. Instead, these systems depend solely on the state of the system. This is why they are called "autonomous," meaning "self-governing." For example, in the given system:

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

The variables \( x \) and \( y \) change according to these equations, but there is no explicit time variable \( t \) involved.
This simplifies analysis since we deal with state space dynamics rather than time evolution. Autonomous systems are particularly important in physics and engineering because they often model real-world phenomena like population dynamics and electrical circuits.

Periodic Solutions

Periodic solutions in differential equations describe systems that repeat their behavior after some period. In a mathematical context, this means that a solution \((x(t), y(t))\) satisfies \((x(t + T), y(t + T)) = (x(t), y(t))\) for some period \( T \).
In the given exercise, we use Dulac's Criterion to determine the non-existence of such solutions for the system:

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

By showing that no function \( \delta(x, y) \) can make the divergence \( abla \cdot (\delta \mathbf{F}) = 0 \), we demonstrate that periodic solutions cannot exist in this system. This conclusion helps understand the stability and long-term behavior of the system.

Continuously Differentiable Functions

Continuously differentiable functions are functions that possess continuous derivatives. In the context of Dulac's Criterion, we need a function \( \delta(x, y) \) that is continuously differentiable over a region to analyze the divergence.
The importance of such functions lies in their smooth behavior, ensuring no abrupt changes or discontinuities that could complicate divergence calculations.
In practice, choosing a simple \( \delta(x, y) \) function, like \( x \), helps simplify calculations:

• \( \delta(x, y) = x \) is straightforward and smoothly differentiable.
• This choice aids in applying Dulac's Criterion effectively to rule out periodic solutions.

Ultimately, the choice of \( \delta(x, y) \) is strategic to ensure the divergence is non-zero over the region analyzed.

Divergence Analysis

Divergence analysis involves computing the divergence of a vector field. It's a vital step in applying Dulac's Criterion since we evaluate \( abla \cdot (\delta \mathbf{F}) \). This divergence must not be identically zero everywhere in the analyzed region for the criterion to hold.
In our example, after computing the divergence for the chosen \( \delta(x, y) = x \), we find \( 2x(y - 1) \), which is not identically zero. Thus, Dulac's Criterion shows there's no periodic solution for most of the region.

• Differentiation of \( \delta f \) and \( \delta g \) derived sub-expressions aids in calculating this divergence accurately.
• Final calculations helped ascertain regions where zero divergence doesn't occur.

In effect, divergence analysis helps determine if closed trajectory solutions (periodic) are impossible over large enough regions.