Problem 24 - 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). $$ \begin{aligned} &x^{\prime}=y+x\left(x^{2}+y^{2}\right) \\ &y^{\prime}=-x+y\left(x^{2}+y^{2}\right), \mathbf{X}(0)=(4,0) \end{aligned} $$

Step-by-Step Solution

Answer

The system spirals inward towards the origin from (4,0) with increasing angle \( \theta(t) = t \).

Step 1: Convert to Polar Coordinates

The given system is in terms of Cartesian coordinates \((x, y)\). To convert to polar coordinates, we use the transformations: \( x = r \cos \theta \) and \( y = r \sin \theta \). The system equations become \( x' = r \sin \theta + r^3 \cos^3 \theta + r^3 \cos \theta \sin^2 \theta \) and \( y' = -r \cos \theta + r^3 \sin^3 \theta + r^3 \sin \theta \cos^2 \theta \).

Step 2: Derive Polar Coordinate Equations

In polar coordinates, the equations for \( r \) and \( \theta \) are derived from \( x' \) and \( y' \). Calculate the derivatives of \( x = r \cos \theta \) and \( y = r \sin \theta \) to express \( x' \) and \( y' \) in terms of \( r' \) and \( \theta' \). This gives:\[ x' = r' \cos \theta - r \sin \theta \theta', \quad y' = r' \sin \theta + r \cos \theta \theta' \]Equate these with the polar-transformed \( x' \) and \( y' \) from Step 1.

Step 3: Simplify the Polar System

Set the polar system equal to zero to isolate \( r' \) and \( \theta' \). After substitution and simplification, the simplified polar system is:\[ r' = r(r^2), \quad \theta' = 1 \]This suggests that \( r' = r^3 \) and \( \theta' = 1 \), meaning \( r \) changes with \( r^3 \) and \( \theta \) changes linearly with time.

Step 4: Solve for \( r(t) \) and \( \theta(t) \)

The equation \( r' = r^3 \) is a separable differential equation \( \frac{dr}{r^3} = dt \). Solving this yields:\[ -\frac{1}{2r^2} = t + C \]Given the initial condition \( r(0) = 4 \), solve for \( C \) to get the full expression for \( r(t) \). For \( \theta' = 1 \), we integrate to get \( \theta(t) = t + \theta_0 \) where \( \theta_0 = 0 \) with the initial condition.

Step 5: Analyze the Geometric Behavior

The solution indicates that \( r(t) \) decreases over time, suggesting the trajectory spirals inward as \( t \to \infty \). The angle \( \theta(t) = t \) shows a uniform rotation. Thus, the trajectory is a spiraling inward path approaching the origin. Given \( \mathbf{X}(0)=(4,0) \), the path starts at point 4 units away along the positive x-axis and spirals towards the origin.

Key Concepts

Geometric Behavior

When solving differential equations in polar coordinates, understanding geometric behavior is essential to visualize how solutions change over time. In this context, the term "geometric behavior" refers to the path or trajectory that the solution takes in the plane. For the nonlinear system given, the polar coordinate transformation reveals the equation for radius, \( r(t) \), and the angle, \( \theta(t) \).

The derived polar coordinate system, \( r' = r^3 \) and \( \theta' = 1 \), provides a clear image: the system defines a spiraling inward motion. As \( r(t) \) decreases, the trajectory moves closer to the origin, forming a spiral pattern. Meanwhile, \( \theta(t) = t \) implies that the motion rotates uniformly over time.

This type of geometric behavior shows a combination of a decreasing radius and a constant angular velocity, leading to a spiraling path. It's crucial to understand this behavior in terms of the given initial conditions, which determine where the spiral starts and how rapidly it progresses. In this case, starting at \( (4,0) \) indicates an initial radius of 4 units, along the positive x-axis.

Nonlinear System

A nonlinear system is characterized by equations that are not just linear combinations of variables or their derivatives. This makes them inherently complex and often less predictable. In the exercise, the nonlinear system is captured by the equations \(x' = y + x(x^2 + y^2)\) and \(y' = -x + y(x^2 + y^2)\).

These equations depict a relationship that involves terms like \(x^3\) and \(y^3\), which are nonlinear in nature. Nonlinear terms often lead to more intricate solutions, such as spirals or cycles, as opposed to simple linear growth or decay. This requires more intricate methods to solve and analyze them, such as converting to polar coordinates, as shown in the exercise.

The transformation to polar coordinates is particularly useful for simplifying such nonlinear systems, allowing us to express complex dynamics (like spirals) using more straightforward radial and angular components. This provides insight into the role of nonlinear terms in shaping the behavior of the system. Recognizing the nonlinear nature is pivotal in fully understanding the unique behavior exhibited by the solutions.

Initial Conditions

Initial conditions are values used to determine a unique solution to a differential equation. These conditions specify the starting point of the system in terms of position and time. For the given problem, the initial condition is \(\mathbf{X}(0) = (4, 0)\), specifying that at time \(t = 0\), the position is at the point (4, 0) in Cartesian coordinates.

In polar coordinates, this translates to \(r(0) = 4\) and \(\theta(0) = 0\), setting the initial radius and angle. Understanding initial conditions is crucial because nonlinear dynamical systems can have a wide range of behaviors, depending on where they begin.

The initial conditions influence the system's trajectory and are essential in determining the specific solution out of a family of possible paths. Without these initial conditions, we could not uniquely calculate the constants in our solution, like \(C\) in the solution for \(r(t)\). This allows us to predict how the system evolves from its starting point, leading to a clearer understanding of its long-term behavior.