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

Step-by-Step Solution

Answer

No periodic solutions exist; function choice maintains divergence sign.

Step 1: Understand Dulac's Negative Criterion

Dulac's Criterion states that for a continuously differentiable function \( \delta(x,y) \), if the divergence of \( \delta(x,y) F(x,y) \) is not everywhere zero and does not change sign in a simply connected region, then the system has no periodic solutions within that region.

Step 2: Choose a Trial Function

Select a function \( \delta(x, y) \) from the given simple forms to test. A common choice is a polynomial like \( ax^2 + by^2 \). Here, we'll try \( \delta(x, y) = 1 \).

Step 3: Calculate the Divergence

Compute the divergence \( abla \,\cdot\,(\delta(x,y) F(x,y)) \) where \( F(x,y) = (-x^3 + 4xy, -5x^2 - y^2) \). The divergence is given by \[ \frac{\partial}{\partial x}(-x^3 + 4xy) + \frac{\partial}{\partial y}(-5x^2 - y^2). \]

Step 4: Differentiate and Simplify

Differentiate the components:- \( \frac{\partial}{\partial x}(-x^3 + 4xy) = -3x^2 + 4y \)- \( \frac{\partial}{\partial y}(-5x^2 - y^2) = -2y \).Add them to get the divergence as \( -3x^2 + 4y - 2y = -3x^2 + 2y \).

Step 5: Analyze the Sign of the Divergence

The expression \( -3x^2 + 2y \) is not identically zero; it can take both positive and negative values depending on \( x \) and \( y \). For a function \( \delta = 1 \), the divergence should not change sign to satisfy Dulac's Criterion. Here, it changes sign hence we reconsider \( \delta(x, y) = x^2 \).

Step 6: Retry with \( \delta(x, y) = x^2 \) and Compute Divergence

Differentiate the scaled vector field:- \( \delta = x^2 \)- Compute divergence: \[ abla \,\cdot\,(x^2(-x^3 + 4xy, -5x^2 - y^2)). \]- Calculate: \( \frac{\partial}{\partial x}(-x^5 + 4x^3y) + \frac{\partial}{\partial y}(-5x^4 - x^2y^2) \) to be \(-5x^4 + 12x^2y - x^2 = x^2(12y - 5x^2 - 1) \).

Step 7: Evaluate the New Divergence Expression

For \( \delta(x, y) = x^2 \), \( x^2(12y - 5x^2 - 1) \) can be rearranged.The term \( 12y - 5x^2 - 1 \) should evaluate consistently negative or positive over any simply connected region. For \( x eq 0 \), tune \( y \) to keep it negative, ensuring no periodic solutions exist.

Key Concepts

Understanding Plane Autonomous Systems

Plane autonomous systems are a special class of differential equations where variables do not explicitly depend on time. Instead, these systems describe the evolution of states using differential equations whose parameters are independent of time. In the equation set given in the exercise:

• \(x' = -x^3 + 4xy\)
• \(y' = -5x^2 - y^2\).

Each equation depicts how a variable like \(x\) or \(y\) changes based on its own state and potentially the state of another variable. These systems are considered in the "plane" because they involve two variables – typically analyzed in \(\mathbb{R}^2\) or on the two-dimensional plane. In such systems, one key point of analysis is whether solutions are periodic.

Exploring Periodic Solutions and Dulac's Criterion

Periodic solutions in differential equations are outcomes where solutions repeat themselves after a certain period. Identifying these can help understand the system's long term behavior. In our context, we'd like to determine if the given system can exhibit such behavior.

Dulac's Negative Criterion offers a method for proving that no periodic solutions exist. This principle states that for a suitably defined function \(\delta(x,y)\), if the divergence of \(\delta(x,y)F(x,y)\) does not vanish and doesn't change sign in a simply connected domain, no periodic solution exists there. This is powerful as it allows analysis in the absence of writable analytical solutions.

Divergence Calculation in Practice

Divergence calculation is a handy tool in vector calculus, providing key insights into the behavior of vector fields. In this exercise, we compute the divergence of the vector field multiplied by the function \(\delta(x, y)\), using the formula:
\[abla \cdot (\delta(x,y) F(x,y)) = \frac{\partial}{\partial x}(\delta(x,y) \, f(x,y)) + \frac{\partial}{\partial y}(\delta(x,y) \, g(x,y)).\]
For our system:\\(f(x, y) = -x^3 + 4xy\) and \(g(x, y) = -5x^2 - y^2\), leading to a combined divergence expression that provides vital information on whether local sources or sinks are present across the field. Checking the behavior of this divergence is crucial since any non-zero divergence that maintains a sign excludes periodic paths.

Trying Various Trial Functions

Trial functions are guesses or approximations chosen to simplify or provide tractable forms for complex calculations. They are foundational in proving Dulac's criterion. Initially, simple forms like \(\delta(x, y) = 1\) are often chosen—providing straightforward calculations.

Testing various functions can include:

• Polynomials like \(ax^2 + by^2\).
• Exponential expressions such as \(e^{ax+by}\).
• Monomials like \(x^a y^b\).

Choosing the correct form helps reveal the system's intrinsic behavior more clearly and can drastically alter the outcome of a divergence calculation hinting towards periodicity or its absence.

Analyzing Differential Equations

Differential equations are at the core of modeling dynamic real-world processes. Analyzing them involves solving or approximating their behavior over time or space. In autonomous systems, changes are driven purely by current system states.

For the system given:

• The equation \(x' = -x^3 + 4xy\) indicates \(x\) evolves negatively with increasing \(x^3\) but positively if \(xy\) dominates.
• Meanwhile, \(y' = -5x^2 - y^2\) implies a general decay in \(y\) as both terms depress \(y\)'s value.

Analysis techniques encompass not only exact solutions but phase space plots, numerical simulations, and qualitative evaluations—all tying back to understanding stability, equilibrium, and periodicity through criteria such as Dulac’s.