Problem 25 - Advanced Engineering Mathematics

Question

In Problems, solve the given nonlinear plane autonomous system by changing to polar coordinates. Describe the geometric behavior of the solution that satisfies the given initial condition(s). $$ \begin{aligned} &x^{\prime}=-y+x\left(1-x^{2}-y^{2}\right) \\ &y^{\prime}=x+y\left(1-x^{2}-y^{2}\right), \mathbf{X}(0)=(1,0) ; \mathbf{X}(0)=(2,0) \end{aligned} $$

Step-by-Step Solution

Answer

The solutions spiral towards a circle of radius 1; they move in circular paths.

Step 1: Understand the System

The problem provides us with a system of nonlinear differential equations describing a plane autonomous system. We're given two pairs of initial conditions which will be used later in the solution.

Step 2: Convert to Polar Coordinates

To convert the system to polar coordinates, use the transformations: - Let \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). - Thus, \( x^2 + y^2 = r^2 \). - Calculate derivatives: \( x' = \frac{dr}{dt}\cos(\theta) - r\sin(\theta)\frac{d\theta}{dt} \) and similarly for \( y'. \)

Step 3: Substitute and Simplify

Substituting \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \) into the differential equations yields: \[ r' = r(1 - r^2) \]\[ \theta' = 1 \]These simplifications indicate that the radius \( r \) changes according to the equation while \( \theta \) increases linearly with time.

Step 4: Solve the Radial Equation

Focus on solving the radial equation: \[ r' = r(1 - r^2) \]This is separable. Rewriting gives: \[ \frac{dr}{r(1 - r^2)} = dt \]Integrate both sides: the left-hand side with respect to \( r \) and the right-hand side with respect to \( t \).

Step 5: Integrate the Radial Equation

Integrate the left-hand side using partial fraction decomposition: \[ \frac{1}{r(1-r^2)} = \frac{1}{r} + \frac{1}{2(1+r)} + \frac{1}{2(1-r)}\]Integrate these separately.

Step 6: Analyze the Solution

Solve the resulting equations from Step 5 to find \( r(t) = \pm 1 \text{ or } r(t) = ce^t / \sqrt{1+ce^{2t}} \). Given initial conditions, \( r(t) \to 1 \) for large \( t \).

Step 7: Solve the Angular Equation

Solve \( \theta' = 1 \) to find \( \theta(t) = t + \theta_0 \). The system circles around the origin with increasing time.

Step 8: Discuss the Geometrical Behavior

The solution trajectories lie on circles of radius 1, spiraling outward if outside the circle, inward if inside until they reach the circle. With initial conditions at \( (1,0) \), solution reaches circular equilibrium immediately. For \( (2,0) \), they spiral towards radius 1.

Key Concepts

Polar Coordinates

When working with differential equations on a plane, converting to polar coordinates is a powerful technique. Polar coordinates express a point in terms of its distance from a central point (the radius, \( r \)) and the angle (\( \theta \)) it makes with a reference line. This conversion simplifies the analysis of circular motion, especially when the equations are nonlinear.
To transform a system into polar coordinates, use:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

This transformation helps us leverage the circular symmetry, as \( x^2 + y^2 = r^2 \) simplifies many radial computations. Once converted, the system often reduces to simpler forms, like separating radial and angular components, making analysis straightforward.

Plane Autonomous System

A plane autonomous system features differential equations that do not depend explicitly on the independent variable (usually time \( t \)). Such systems are defined purely by the dependent variables, here \( x \) and \( y \). In the given exercise, this means the derivatives \( x' \) and \( y' \) only rely on \( x \) and \( y \), not directly on time.
This offers benefits:

• Predicts system behavior solely from state variables.
• Allows analysis of trajectories over time without recomputing with every time-step.

Studying these systems involves examining how points move on the plane, which are determined entirely by their current positions, fostering a predictable dynamic like cycling or spiraling patterns.

Geometric Behavior

Geometric behavior in the context of differential equations describes how solutions evolve visually over time. Here, when the system is expressed in polar coordinates, its geometric nature becomes clearer.
The transformation yielded two main equations:

• \( r' = r(1 - r^2) \)
• \( \theta' = 1 \)

The radial equation shows that as time progresses, the radius \( r \) stabilizes at 1. This means solutions either spiral outward from within a circle of radius 1 or spiral inward from outside until reaching this stable circle. The angular equation indicates a constant circular motion around the origin, reinforcing this spiraling behavior.
Understanding geometric behavior allows prediction of how systems evolve, such as approaching a limit cycle or orbit.

Initial Conditions

Initial conditions set the starting point of a solution for differential equations. They are crucial because they determine the specific trajectory a solution will follow over time.
In this specific system, two initial conditions are given:

• \( \mathbf{X}(0) = (1,0) \)
• \( \mathbf{X}(0) = (2,0) \)

When \( r = 1 \), the initial condition lies directly on the stable circle, so it remains on this circle as \( t \) increases. Conversely, \( r = 2 \) starts outside the stable circle and spirals inward until it reaches \( r = 1 \). This influence of initial conditions highlights the importance of starting points in predicting long-term behavior of solutions.