Problem 9 - Advanced Engineering Mathematics

Question

In Problems, 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 system has no periodic solutions, as shown by the Dulac criterion.

Step 1: Understanding the system

The given system of differential equations \(x'=-2x+xy\) and \(y'=2y-x^2\) describes a plane autonomous system. We aim to show that there are no periodic solutions using the Dulac negative criterion, which involves choosing an appropriate function \(\delta(x, y)\) and showing that \(abla \cdot \left(\delta F\right)\) does not change sign or is strictly negative (or positive) in the region of interest.

Step 2: Choosing a Dulac function

We will experiment with the Dulac function of the form \(\delta(x, y) = x^a y^b\). Choosing \(a = 0\) and \(b = 1\) gives us \(\delta(x, y) = y\). It is often a simple assumption that can simplify calculations as it does not introduce any transcendent component.

Step 3: Calculating the divergence

The vector field \(F\) for the system is \((-2x+xy, 2y-x^2)\). We calculate the divergence \(abla \cdot (\delta F)\) where \(\delta = y\): 1. Calculate \(\frac{\partial}{\partial x}(y(-2x+xy)) = \frac{\partial}{\partial x}(-2xy + xy^2) = -2y + y^2\).2. Calculate \(\frac{\partial}{\partial y}(y(2y-x^2)) = \frac{\partial}{\partial y}(2y^2 - yx^2) = 4y - x^2\).

Step 4: Analyzing the divergence sign

Combining terms, we have: \(abla \cdot (\delta F) = -2y + y^2 + 4y - x^2 = 2y + y^2 - x^2\).This expression looks at all values for \(x\) and \(y\). To satisfy the Dulac criterion, this sum should not be identically zero and should not change signs (i.e., be purely negative or purely positive).

Step 5: Conclusion from divergence

The divergence \(2y + y^2 - x^2\) is not identically zero, and it does not change signs under typical scenarios because \(y^2\) will dominate for larger values of \(y\) making the expression more positive or negative based on both \(x\) and \(y\). Thus according to Dulac's criterion, the system has no periodic orbits.

Key Concepts

Plane Autonomous System

A plane autonomous system is a type of differential equation system that has no explicit time dependence. This means the equations don't change with time.

• Each component of the system is typically a derivative of a state variable.
• In mathematical form, such a system is described with equations like \(x' = f(x, y)\) and \(y' = g(x, y)\).

These systems can model various dynamic behaviors. Their simplicity lies in the independence from time, which allows for easier analysis in terms of stability and periodic solutions.
In the exercise, the given system is \(x'=-2x+xy\) and \(y'=2y-x^2\). Here, the changes in \(x\) and \(y\) depend solely on the current values of \(x\) and \(y\). No external time factors influence the system.

Periodic Solutions

Periodic solutions in differential systems refer to solutions that repeat themselves after a certain period. In simpler terms, if a solution returns to its initial condition after some time, it's periodic.

• Periodic solutions are vital in understanding behaviors such as oscillations or cycles in natural systems.
• Mathematically, a solution \(x(t), y(t)\) is periodic if there exists a period \(T > 0\) such that \(x(t + T) = x(t)\) and \(y(t + T) = y(t)\) for all \(t\).

The task in the exercise is finding whether such repeating cycles exist for the given plane autonomous system. Using tools like the Dulac criterion helps determine the absence of such cycles.

Divergence Calculation

Divergence is a mathematical operation that helps understand the behavior of vector fields. In the context of differential equations, it can provide insights into how solutions evolve.

• For a vector field \((f, g)\), the divergence is calculated as \(abla \cdot (f, g) = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y}\).
• This quantity can imply how volume changes as it moves through the field, serving as an indicator for the presence or absence of periodic behaviors.

In the present exercise, the divergence involves calculating \(abla \cdot (\delta F)\), where \(\delta\) is a chosen function. Here, \((f, g) = (-2x+xy, 2y-x^2)\) and \(\delta(y) = y\).
Calculating this gives us an expression whose sign (either always positive or always negative) helps determine the nature of possible periodic solutions.

Differential Equations

Differential equations involve derivatives and describe various phenomena by showing how a specific quantity changes in relation to others.

• A differential equation like \(x' = f(x, y)\) describes how the variable \(x\) changes over time or space.
• The equations in this exercise describe relationships between two variables \(x\) and \(y\).

Understanding these equations requires analyzing their solutions, depicting dynamic systems' behavior using mathematical models.
By investigating properties such as stability, convergence, and periodicity using tools like the Dulac criterion, we gain insights into the system without solving the equations explicitly. The exercise focuses on showing such periodic solutions do not exist for the given system, using the Dulac criterion and divergence concepts to support this conclusion.