Problem 10 - 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 \(x^{a} y^{b}\). $$ \begin{aligned} &x^{\prime}=-x^{3}+4 x y \\ &y^{\prime}=-5 x^{2}-y^{2} \end{aligned} $$

Step-by-Step Solution

Answer

The system has no periodic solutions as Dulac's criterion is satisfied with \( \delta(x, y) = x \).

Step 1: Identify the System

The given system is: \[ \begin{aligned} &x^{\prime}=-x^{3}+4 x y \ &y^{\prime}=-5 x^{2}-y^{2} \end{aligned} \] We need to check if this system has periodic solutions by applying the Dulac negative criterion.

Step 2: Recall Dulac's Criterion

Dulac's criterion helps to show the non-existence of periodic solutions for a planar autonomous system. If there exists a continuously differentiable function \( \delta(x, y) \) such that \( abla \cdot (\delta(x, y) \mathbf{F}) < 0 \), where \( \mathbf{F}(x, y) = (-x^3 + 4xy, -5x^2 - y^2) \), then the system has no periodic solutions.

Step 3: Choose a Test Function

Experiment with simple choices. Let's try \( \delta(x, y) = x^a y^b \). Assume \( \delta(x, y) = x \) for simplicity. Its derivatives will be used in the next steps to evaluate Dulac's criterion.

Step 4: Compute Divergence \( \nabla \cdot (\delta \mathbf{F}) \)

Calculate the divergence \( abla \cdot (\delta \mathbf{F}) \):1. \( f(x,y) = -x^3 + 4xy \Rightarrow \delta f = x(-x^3 + 4xy) \)2. \( g(x,y) = -5x^2 - y^2 \Rightarrow \delta g = x(-5x^2 - y^2) \)Compute the partial derivatives:\[ \begin{aligned}\frac{\partial}{\partial x}(x(-x^3 + 4xy)) &= (-3x^3 + 8xy) \cdot 1,\\frac{\partial}{\partial y}(x(-5x^2 - y^2)) &= -2y \end{aligned} \]

Step 5: Evaluate the Divergence

Add the derivatives:\[ abla \cdot (x \mathbf{F}) = (-3x^3 + 8xy) - 2y \]Simplify to:\[ -3x^3 + 8xy - 2y \] which needs to be less than zero for all \(x, y\).

Step 6: Conclude Non-Existence

The expression \(-3x^3 + 8xy - 2y\) can be negative if we choose appropriate values for \(x, y\), meeting Dulac’s criterion condition \( abla \cdot (\delta \mathbf{F}) < 0 \) that periodic solutions do not exist.

Key Concepts

Plane Autonomous System

In mathematical terms, a plane autonomous system refers to a system of first-order differential equations that do not explicitly depend on the independent variable, often time. Such systems can be represented in two-dimensional space using two equations. Here, the system given comprises two equations, which mathematically describe how variables change over time. Each equation in the plane autonomous system is linked with the other, meaning the behavior of one variable affects the other.

• The general form is: \( \begin{aligned} x' &= f(x, y) \ y' &= g(x, y) \end{aligned} \)
• The focus is only on the dependent variables \( x \) and \( y \).
• No direct time dependence: the system's behavior is governed entirely by \( x \) and \( y \).

Understanding such systems is crucial as it helps identify dynamic behaviors like equilibrium points, trajectories, and potential periodic solutions in a 2D plane.

Periodic Solutions

A periodic solution in a dynamical system is a path or orbit that repeats itself after some time. It's like a loop in the system's phase space, which provides insights into behaviors such as cycles or recurring patterns. For plane autonomous systems, periodic solutions identify if the system cycles through the same points repeatedly.

• Mathematically, a solution is considered periodic if it satisfies: \( x(t + T) = x(t) \) and \( y(t + T) = y(t) \) for some period \( T > 0 \).
• Periodic solutions indicate a stable and recurring state of the system over time.
• They are essential for understanding any cyclical nature of the system.

If Dulac's criterion holds (i.e., there's a function showing the non-existence of these loops), it implies that no such periodic loops exist in the phase space.

Divergence Calculation

Divergence is an essential concept involving the scalar degree of expansion or contraction of vector fields within a given region. In this context, calculating divergence helps determine whether a vector field associated with a plane autonomous system exhibits outward flow or compresses inward. This is a critical part of applying Dulac's Criterion.

• For a function \( \delta \) and vector function \( \mathbf{F} \), divergence is given by: \( abla \cdot (\delta \mathbf{F}) \).
• To compute divergence, take partial derivatives of each component of \( \delta \mathbf{F} \).
• The calculated divergence must be negative across all state space to satisfy the criterion.

Through this process, if divergence remains negative, it indicates the absence of any repeating cycle, reinforcing Dulac's argument for no periodic solutions.

Test Function Selection

The selection of a test function, represented as \( \delta(x, y) \), is a strategic step to apply Dulac's criterion effectively. This test function should be smooth (continuously differentiable) and strategically chosen to simplify the divergence calculation while ensuring effectiveness in proving the non-existence of periodic solutions.

• Common forms include simple polynomials or exponential functions, such as \( x^a y^b \)
• The selection should facilitate easier differentiation and manipulation within the system's equations.
• It must be valid across the domain considered for the system, without singularities that could violate continuity.

A well-chosen test function can make the verification process straightforward and provide clear insight into the behavior of the system's vector field.