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A nonlinear autonomous system is a mathematical setup where the evolution of the system's state variables does not explicitly depend on time. Instead, it depends only on the current state of the system. This self-contained behavior makes these systems especially intriguing and can lead to complex dynamic behaviors despite having relatively simple defining equations.
For the given problem, the system is expressed as:

• \(x' = y - \frac{x}{\sqrt{x^2+y^2}}(4-x^2-y^2)\)
• \(y' = -x - \frac{y}{\sqrt{x^2+y^2}}(4-x^2-y^2)\)

Here, the derivatives \(x'\) and \(y'\) depend only on \(x\) and \(y\), not on any external time variable. This adds a level of complexity but also stability, since the behavior remains consistent over time. As a result, solutions can be analyzed for long-term behavior, which is heavily dependent on their initial conditions and intrinsic properties like nonlinearity.

Critical points in a system like this identify where changes slow down or completely halt. In the context of dynamical systems, they are points where the derivative (or rate of change) of the system is zero, which often indicates a state of equilibrium.
For the exercise at hand, we determine the critical points by setting the radial component of our polar-transformed system to zero:

• \(r' = r(1-r^2) = 0\)

Solving \(r(1-r^2) = 0\), we find:

• \(r = 0\) or \(r = 1\)

In polar coordinates:

• \(r = 0\) corresponds to the origin, a stable equilibrium.
• \(r = 1\) represents a unit circle, indicating a cyclical behavior around the origin.

Identifying these critical points is key in predicting the system’s long-term behavior and understanding its geometry.

The geometric behavior of a dynamical system helps describe how solutions move through or transform within their mathematical space over time. By understanding geometric behavior, we can predict how the system evolves.
For the exercise, the given polar system:

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

reveals specific geometric patterns:

• When \(r = 1\), the solution traces a unit circle, denoting a stable orbit.
• If \(r > 1\), the trajectories will spiral inward towards the unit circle due to the term \(1-r^2\), which attains negative values, thereby reducing \(r\).
• The angle \(\theta\) decreases steadily for all \(r\), indicating a continuous counterclockwise rotation.

The overall behavior illustrates how the trajectories evolve over time, providing a valuable visual to understand long-term dynamics.

Coordinate transformation refers to the change from one coordinate system to another, making it easier to analyze different aspects of a system. In our exercise, we convert from Cartesian to polar coordinates:

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

This transformation simplifies the evaluation of circular or rotational aspects of the system.
By substituting these into the original equations, derivatives transform as:

• \(x' = r' \cos(\theta) - r \sin(\theta) \theta'\)
• \(y' = r' \sin(\theta) + r \cos(\theta) \theta'\)

These allow us to express and solve the system in terms of \(r\) and \(\theta\), exposing the intrinsic circular symmetry and aiding in identifying things like critical points and long-term behaviors effortlessly.