Problem 8 - Advanced Engineering Mathematics

Question

Show that the given plane autonomous system (or second-order differential equation) has no periodic solutions. $$ x^{\prime \prime}+x=\left[\frac{1}{2}+3\left(x^{\prime}\right)^{2}\right] x^{\prime}-x^{2} $$

Step-by-Step Solution

Answer

The system has no periodic solutions by the Bendixson-Dulac criterion.

Step 1: Rewrite as a System of First Order Equations

Introduce a substitution to rewrite the second-order differential equation as a system of first-order equations. Let \( y = x' \) so that our system becomes:\[\begin{align*}x' &= y, \y' &= -x + \left[\frac{1}{2} + 3 y^2\right]y - x^2.\end{align*}\]These are our first-order equations.

Step 2: Examine the System's Vector Field

Identify the vector field defined by the system of equations:\[\mathbf{F}(x, y) = \begin{pmatrix} y \ -x + \left[\frac{1}{2} + 3y^2\right]y - x^2 \end{pmatrix}.\]The behavior of solutions (e.g., existence of periodic solutions) can be understood by analyzing this field.

Step 3: Determine Potential Energy Function

Check if there is a potential energy function or other conserved quantity that the system obeys. For conservative systems, solutions lie on level sets of a potential function. However, in our situation, the system is not explicitly conservative due to the presence of \( y^3 \) and \( x^2 \) terms.

Step 4: Analyze the Bendixson-Dulac Criterion

Apply the Bendixson-Dulac criterion, which states that if there exists a continuously differentiable function \( B(x, y) \) such that the divergence \[\frac{\partial}{\partial x}(B \cdot y) + \frac{\partial}{\partial y}\left(B \cdot \left( -x + \left[\frac{1}{2} + 3y^2\right]y - x^2 \right) \right)\]has a constant sign in a simply connected region, then the system has no periodic solutions in that region. Calculate and confirm that you can find such \( B \).

Step 5: Computing the Divergence with \( B = 1 \)

Choose \( B(x, y) = 1 \) for simplicity. Then, compute \[\frac{\partial}{\partial x}(y) + \frac{\partial}{\partial y}\left( -x + \left[\frac{1}{2} + 3y^2\right]y - x^2 \right) = 0 - 1 + (\frac{1}{2} + 9y^2).\]This simplifies to \[\frac{1}{2} + 9y^2 - 1 = 9y^2 - \frac{1}{2},\]which is non-zero and positive for any \( y eq 0 \), indicating no periodic solutions.

Key Concepts

Vector Field Analysis

In studying autonomous systems of differential equations, vector field analysis is a valuable tool. A vector field assigns a vector to every point in a region of space, visualizing how solutions of a differential equation flow. This can reveal behaviors like fixed points, periodic solutions, or attractors.
When analyzing a vector field from a system of first-order equations, each point \( (x, y) \) corresponds to a vector \( \mathbf{F}(x, y) \). In the presented exercise, we have:

• \( x' = y \)
• \( y' = -x + \left[\frac{1}{2} + 3y^2\right]y - x^2 \)

To understand the system's behavior, each vector shows the direction and speed at which the system evolves over time.
This snapshot of the vector field helps identify whether the solutions will return to the same state (indicative of periodic solutions), converge, or diverge over time. For instance, if vectors tend to circle around a point, this might suggest periodic behavior.

Bendixson-Dulac Criterion

The Bendixson-Dulac criterion is a mathematical tool that helps determine the absence of periodic solutions in a dynamical system. It's especially useful when dealing with two-dimensional, continuous, and differentiable autonomous systems.
This criterion states that if there exists a continuously differentiable function \( B(x, y) \), and the divergence \[ \frac{\partial}{\partial x}(B \cdot y) + \frac{\partial}{\partial y}\left(B \cdot \left( -x + \left[\frac{1}{2} + 3y^2\right]y - x^2 \right) \right)\]
has a constant sign throughout a simply connected region, then there are no periodic solutions there.
In this problem, setting \( B(x, y) = 1 \) leads to a divergence that's consistently non-zero positive, confirming no periodic solutions exist in the system.
The simplicity of choosing \( B = 1 \) helps ease calculations while fully leveraging the Bendixson-Dulac criterion.

Potential Energy Function

Potential energy functions are often used to understand conservative systems, where they can simplify analysis by focusing on energy conservation. Systems that exhibit conservation of energy have solutions resting on the level sets of the potential energy function.
However, the given equation includes non-linear terms in \( y^{3} \) and \( x^{2} \), indicating dynamics that aren't purely conservative. This complicates the search for a potential energy function.
While potential functions are helpful for smooth autonomous systems, they are not applicable here due to the inherently non-conservative nature of the system's terms.
Thus, other approaches, such as vector field analysis and the Bendixson-Dulac criterion, offer more suitable insights into the presence (or absence) of periodic solutions in this context.

First-Order Differential Equations

First-order differential equations involve the rate of change of a variable. They are often simpler to solve and analyze than higher-order equations. To transform a higher-order differential equation into a system of first-order equations, substitutions are made to introduce new variables for each derivative.
In our exercise, beginning with substitution \( y = x' \) simplifies our understanding by converting the second-order equation:

• \( x'' + x = \left[\frac{1}{2} + 3(x')^{2}\right]x' - x^{2} \)

To a more manageable system of first-order equations:

• \( x' = y \)
• \( y' = -x + \left[\frac{1}{2} + 3y^2\right]y - x^2 \)

These equations maintain the same qualitative behavior while simplifying the analysis process.
This step is crucial for utilizing techniques like vector field analysis and the Bendixson-Dulac criterion to assess the presence of periodic solutions.