Question

show that the given system has no periodic solutions other than constant solutions. $$ d x / d t=-2 x-3 y-x y^{2}, \quad d y / d t=y+x^{3}-x^{2} y $$

Short answer

Based on the given system of differential equations, use Bendixson's Criterion to determine if there are any periodic solutions other than constant solutions. Given system of differential equations: $$ \begin{cases} \frac{dx}{dt} = -2x - 3y - xy^2\\ \frac{dy}{dt} = y + x^3 - x^2y \end{cases} $$ Bendixson's Criterion states that if the divergence of the vector field is non-zero and does not change its sign in a simply connected region, then there are no periodic orbits in that region. Divergence of the vector field: $$ div(\vec{F}) = -3 - x^2 - y^2 $$ This is always negative for all values of \(x\) and \(y\). Therefore, based on Bendixson's Criterion, there are no periodic solutions other than constant solutions for the given system of differential equations.

Step-by-step solution

Write down the given system of differential equations

Given system is: $$ \begin{cases} \frac{dx}{dt} = -2x - 3y - xy^2\\ \frac{dy}{dt} = y + x^3 - x^2y \end{cases} $$

Calculate the divergence of the vector field associated with the system

To apply Bendixson's Criterion, we need to calculate the divergence of the vector field associated with the given differential equations. The divergence is defined as the sum of the partial derivatives of each component with respect to the corresponding variable. In this case, the divergence is: $$ div(\vec{F}) = \frac{\partial}{\partial x}(-2x - 3y - xy^2) + \frac{\partial}{\partial y}(y + x^3 - x^2y) $$

Compute the partial derivatives and sum them up

Compute the partial derivatives and sum them up to find the divergence: $$ div(\vec{F}) = (-2 - y^2) + (-x^2 - 1) = -3 - x^2 - y^2 $$

Apply Bendixson's Criterion

According to Bendixson's Criterion, if the divergence of the vector field is non-zero and does not change its sign in a simply connected region, then there is no periodic orbit in that region. In this case, we have: $$ div(\vec{F}) = -3 - x^2 - y^2 $$ which is always negative for all \(x\) and \(y\). Since the divergence is nonzero and does not change its sign, the given system has no periodic solutions other than constant solutions according to Bendixson's Criterion.

Key concepts

Bendixson's Criterion

When studying non-periodic solutions in differential equations, Bendixson's Criterion is a fundamental concept. It's a qualitative tool that helps predict the absence of periodic solutions—or closed orbits—in a two-dimensional system. For students tackling such systems, Bendixson's Criterion can often be a lifeline, providing a clear-cut method to determine the presence or absence of these indirect pathways.

According to this criterion, if the divergence of a vector field associated with a differential equation does not change sign and is never zero within a certain region, then there are no non-constant periodic solutions within that region. This implies that a system’s trajectory does not loop back on itself in any repeating fashion within the area of study, excluding equilibrium points where the system's state remains constant over time.

The practical application involves calculating the divergence of the associated vector field and examining its sign over the region in question. If the divergence maintains one sign (always positive or always negative) and is not zero, this criterion confirms that there is no possibility of a periodic orbit lurking within that slice of the phase plane.

Differential Equations

Differential equations are mathematical beasts that describe relationships between functions and their derivatives. In essence, they are equations that incorporate not just values of a function itself, but the rates at which those values change. They crop up in a multitude of scientific fields, often serving as the linchpin for understanding phenomena in physics, engineering, economics, and biology, to name a few.

At the heart of a differential equation is the unknown: a function that presumably exists and satisfies the relationship imposed by the differential equation. Solving these equations involves finding that function or, at least, characterizing its behavior. Solutions can be as varied as simple constants or as complex as swirling patterns over time and space. The solutions to such equations are the keys to predicting and describing real-world dynamics in both time-steady and ever-changing scenarios.

Divergence of Vector Field

The divergence of a vector field is a scalar measure that provides insight into the field's behavior; think of it as the amount of 'spreading out' or 'converging in' that occurs. Put simply, it tells you whether the vector field is acting as a source, creating outflow in every direction, or a sink, drawing everything in.

Determining the divergence involves taking partial derivatives of the vector field components with respect to their own variables and summing them up. If we picture the vector field as representing fluid flow, a non-zero divergence indicates that fluid is either being created or destroyed at a point, resulting in the absence of a cyclic pattern. This concept ties directly into Bendixson's Criterion for the analysis of differential equations since it can highlight whether or not a system exhibits periodic solutions without having to actually solve the entire, possibly complex, differential equation.

Partial Derivatives

Grasping the idea of partial derivatives is crucial for anyone diving into multivariable calculus or any field dealing with functions of several variables. When we deal with functions that depend on multiple factors, partial derivatives give us the rate of change of the function with respect to one variable, keeping all other variables constant.

A real-world analogy would be assessing the influence of temperature on an ice cream's melting rate without considering the wind speed. In solving differential equations, particularly when applying Bendixson's Criterion, partial derivatives help in computing the divergence of the vector field, which is pivotal in determining the presence of periodic solutions. Understanding how to find and interpret partial derivatives is an essential skill that allows students to analyze multifaceted dynamic systems and predict their behaviors under different conditions.