Problem 14 - Advanced Engineering Mathematics

Question

In Problems, find a circular invariant region for the given plane autonomous system. $$ \begin{aligned} &x^{\prime}=-y-x e^{x+y} \\ &y^{\prime}=x-y e^{x+y} \end{aligned} $$

Step-by-Step Solution

Answer

The system has no feasible circular invariant region for non-zero radius due to the exponents.

Step 1: Understand the Problem

We need to find a circular invariant region for the given plane autonomous system. A system is said to have a circular invariant region if there is a circle such that any trajectory starting inside or on the circle remains inside or on the circle for all time.

Step 2: Substitute Variables in Circle Equation

Consider a circle of radius \( R \) centered at the origin. The equation of such a circle is given by: \( x^2 + y^2 = R^2 \). The goal is to find a \( R \) such that trajectories do not leave the circle.

Step 3: Derive the Conditions for a Circular Invariant Region

For the circle \( x^2 + y^2 = R^2 \) to be invariant, the derivative of \( x^2 + y^2 \) with respect to time \( t \), must be zero or negative. Compute \( \frac{d}{dt}(x^2 + y^2) = 2x x' + 2y y' \). Substitute the expressions for \( x' \) and \( y' \) from the system equations.

Step 4: Calculate \( \frac{d}{dt}(x^2 + y^2) \)

Substitute \( x' = -y - x e^{x+y} \) and \( y' = x - y e^{x+y} \) into the derivative:\[\frac{d}{dt}(x^2 + y^2) = 2x(-y - x e^{x+y}) + 2y(x - y e^{x+y})\]Simplify this expression.

Step 5: Simplify Expression

Simplify the expression:\[\frac{d}{dt}(x^2 + y^2) = -2xy - 2x^2 e^{x+y} + 2xy - 2y^2 e^{x+y}\]Combine terms:\[\frac{d}{dt}(x^2 + y^2) = -2(x^2 + y^2) e^{x+y}\]

Step 6: Determine Sign of the Expression

For the circle to be invariant, require:\[-2(x^2 + y^2) e^{x+y} \leq 0\]Since \( e^{x+y} \) is always positive, for the expression to be zero or negative, the maximum condition is \( x^2 + y^2 = 0 \), which is not feasible for non-zero \( R \). If \( e^{x+y} \) is bounded such that it makes the expression zero, choose a suitable \( R \). Otherwise, the system may not have a proper circular invariant region.

Step 7: Conclusion Based on Conditions

Given that \( e^{x+y} \) implies that for trajectories to satisfy this condition no simple fixed \( R \) can be an invariant primarily due to \( e^{x+y} \) not having proper bounding for feasible non-zero \( R \). Thus invariant under trivial cases where \( x = 0 \) and \( y = 0 \) only.

Key Concepts

Circular Invariant Region

A circular invariant region is a fascinating concept when dealing with plane autonomous systems, particularly in the field of dynamical systems. Imagine a circle on a plane where anything inside this circle remains inside it through time, no matter how it moves around. This is what a circular invariant region ensures. In mathematical terms, for an area to be considered invariant, trajectories that start within this circle should never cross its boundary as time progresses.

In the given system of differential equations, we explored the possibility of finding such a circle centered at the origin. Using the equation of a circle, \(x^2 + y^2 = R^2\), we analyze trajectories' behavior with time in an attempt to establish this invariant region. As the process shows, it becomes crucial to assess the system's trajectories by examining critical variables: \(x'\) and \(y'\). This abstraction helps understand not only if a circular invariant region exists but also defines its limitations based on complex functions like the exponential function present in the equations.

Trajectory Analysis

The study of how solutions to dynamical systems change over time is what we refer to as trajectory analysis. When we talk about trajectories in the context of differential equations, we're essentially tracking the paths traced by solutions as time progresses. Think of it as plotting the course of a moving object through space.

First, we substitute the variables \(x'\) and \(y'\) into our circle equation \(x^2 + y^2 = R^2\) and derive a time-dependent expression \(\frac{d}{dt}(x^2 + y^2)\). This helps in determining whether the distance from the origin is growing or shrinking. In the solution process, we simplify this to form \(-2(x^2 + y^2) e^{x+y}\). This insight, especially noting the negative factor affiliated with \(x^2 + y^2\), shows the influential role of exponential functions and their bounded nature in trajectory movement.

Ultimately, analyzing the sign of \(\frac{d}{dt}(x^2 + y^2)\) reveals critical characteristics of the trajectories and their motion in the plane. Successful trajectory analysis unveils dynamic behavior within our system, crucial for further exploration of invariant regions.

Differential Equations

Differential equations form the foundation of analyzing systems that change continuously over time. These are equations that relate a function with its derivatives. The beauty of differential equations is their ability to model real-world phenomena, ranging from simple pendulum motions to complex population dynamics.

In our specific problem, we dealt with a set of autonomous differential equations - equations not explicitly dependent on the independent variable, time \(t\). Such systems are simpler to analyze since their behavior is steady and governed by the dependent variables alone. By evaluating \(x' = -y - x e^{x+y}\) and \(y' = x - y e^{x+y}\), we delve into how these rate equations influence values of \(x\) and \(y\), shedding light on unique patterns and tendencies in the system under examination. This perspective enables the prediction of future states given current conditions, a hallmark of effective mathematical modeling.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world systems through mathematical expressions and equations. It allows us to interpret complex phenomena and solve problems across various fields such as physics, engineering, and beyond.

In our circular invariant region context, mathematical modeling aids in understanding how the interaction between \(x\) and \(y\) impacts circular boundaries on the plane. By constructing models using the differential equations provided, we represent the dynamic system's behavior. Such models encapsulate assumptions, conditions, and inherent behaviors through well-defined mathematical constructs.

Mathematical modeling involves a cyclic process of validation and iteration, ensuring that the models are both accurate and robust. In our exercise, assumptions stem from exponential growth properties are scrutinized, enabling clearer predictions and informed conclusions about how systems interact within proposed boundaries.