Problem 25 - Advanced Engineering Mathematics

Question

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). \(x^{\prime}=-y+x\left(1-x^{2}-y^{2}\right)\) $$ y^{\prime}=x+y\left(1-x^{2}-y^{2}\right), \mathbf{X}(0)=(1,0) ; \mathbf{X}(0)=(2,0) $$

Step-by-Step Solution

Answer

Trajectories form stable limit cycles, spiral inwards to circle of radius 1; revolve around due to constant angular velocity.

Step 1: Convert to Polar Coordinates

To start, we transform the given system into polar coordinates. In polar coordinates, \(x = r \cos \theta\) and \(y = r \sin \theta\). The derivatives are \(x' = r' \cos \theta - r \sin \theta \cdot \theta'\) and \(y' = r' \sin \theta + r \cos \theta \cdot \theta'\). Substitute these expressions into the original system.

Step 2: Derive the Radial and Angular Equations

Substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) into the equations:1. \(x^{\prime} = -y + x(1-x^2-y^2)\) becomes \(-r \sin \theta + r \cos \theta (1-r^2)\).2. \(y^{\prime} = x + y(1-x^2-y^2)\) becomes \(r \cos \theta + r \sin \theta (1-r^2)\).Isolate \(r'\) and \(\theta'\) from these equations.

Step 3: Simplify the Radial and Angular Equations

Combine the transformed equations:For the radial component, \[r' = r(1 - r^2).\] For the angular component, divide by \(r\) and use trigonometric identities:\[\theta' = 1.\] These equations describe how \(r\) and \(\theta\) evolve over time.

Step 4: Analyze \(r(t)\) Evolution

The radial equation \(r' = r(1-r^2)\) indicates that \(r = 0\) and \(r = 1\) are equilibrium points.1. If \(r(0) = 1\), the circle is stable.2. If \(r(0) = 2\), \(r\) will decrease towards \(1\) as \(t\) increases due to the negative term \(-(2^2 - 1)\).

Step 5: Analyze \(\theta(t)\) Evolution

Since \(\theta' = 1\), \(\theta(t)\) evolves linearly with time: \(\theta(t) = t + \theta(0)\).The phase angle spins at a constant rate. Geometrically, this means that the trajectory simply wraps around a circle over time.

Step 6: Determine the Initial Conditions

For \(\mathbf{X}(0) = (1,0)\), this translates to \(r(0)=1\) and \(\theta(0)=0\). The solution stays on the circle \(r=1\), spinning indefinitely as \(t\) increases.For \(\mathbf{X}(0) = (2,0)\), this starts with \(r(0)=2\). Here, the trajectory spirals inwards towards \(r=1\) over time, while still rotating around the origin.

Step 7: Geometric Behavior of Solutions

With the solution \(\mathbf{X}(0) = (1,0)\), the trajectory forms a stable limit cycle, remaining on the circle of radius 1.For \(\mathbf{X}(0) = (2,0)\), it spirals inward towards this same circle, both revolve around the circle due to the constant angular velocity.

Key Concepts

Nonlinear Autonomous Systems

At its core, a nonlinear autonomous system is a system of differential equations that defines how variables change over time without external influence. In our case, this system is described by two equations:

• \(x^{\prime}=-y+x(1-x^{2}-y^{2})\)
• \(y^{\prime}=x+y(1-x^{2}-y^{2})\)

These types of systems are crucial in mathematics and physics because they model real-world phenomena such as population dynamics and mechanical systems. Autonomy in the system comes from its self-reliance, meaning the evolution of state variables like \(x\) and \(y\) don't depend explicitly on time.
Nonlinear refers to the presence of terms like \(x^2\) and \(y^2\) which make the equations more complex and the behavior of solutions more intricate. Unlike linear systems, where solutions are comparatively predictable and straightforward, nonlinear equations can show a wide array of behaviors such as periodicity, chaos, and bifurcations.
When dealing with these systems, our goal is to analyze and predict the system's behavior over time, particularly focusing on how it reaches or approaches equilibrium.

Radial and Angular Equations

To simplify the analysis of a nonlinear autonomous system, often a polar coordinate transformation is used. In polar coordinates, the position of a point is given by \(r\), the radial distance from the origin, and \(\theta\), the angular position relative to a reference direction.
In our exercise, transforming to polar coordinates simplifies the equations to:

• Radial equation: \(r' = r(1 - r^2)\)
• Angular equation: \(\theta' = 1\)

This transformation essentially decouples the system into two parts. The radial equation governs how \(r\) changes, providing insight into the distance from the origin. The angular equation indicates how \(\theta\) evolves, revealing how the position around a central point changes.
These equations allow us to understand the behavior of solutions geometrically, providing a clear view of how trajectories move in a plane without needing to directly solve the more complex cartesian equations.

Equilibrium Points

Equilibrium points, or fixed points, are where the system remains unchanged over time; these are critical in understanding the system's long-term behavior.
For our radial equation \(r' = r(1 - r^2)\), the equilibrium points occur when \(r = 0\) and \(r = 1\). At these values, the rate of change of \(r\), \(r'\), is zero, meaning there is no radial movement:

• \(r = 0\): Represents the origin, a trivial equilibrium.
• \(r = 1\): Represents a stable limit cycle, in this case, a circle centered at the origin with radius 1.

These equilibrium points define regions of stability and instability within the system. For instance, if the initial condition is at \(r = 1\), the solution will continue moving along this circle due to its inherent stability. Conversely, if \(r > 1\), it will spiral inwards toward \(r = 1\), eventually stabilizing there as \(r'\) is negative beyond \(r = 1\).

Geometric Behavior of Solutions

Understanding the geometric behavior of solutions in a polar coordinate system gives us a visual overview of how states in the system transform over time.
With radial \(r' = r(1 - r^2)\) and angular \(\theta' = 1\) equations, we can interpret the system's dynamics as a circular motion.

• For initial point \(\mathbf{X}(0) = (1,0)\), the system describes a stable limit cycle along \(r = 1\), where the trajectory remains constant, tracing a circle perpetually as \(t\) progresses. This is due to the equilibrium nature of \(r = 1\).
• Starting from \(\mathbf{X}(0) = (2,0)\), the trajectory spirals inward, gradually reducing \(r\) towards 1. Despite this radial movement, the angular motion remains consistent, spinning around the origin. Eventually, it aligns with the stable cycle at \(r = 1\).

The geometric behavior showcases the stability and directional flow influenced by both radial dampening toward equilibrium and continuous angular movement, allowing us to predict future system states from initial conditions.