Question

an autonomous system is expressed in polar coordinates. Determine all periodic solutions, all limit cycles, and determine their stability characteristics. $$ d r / d t=r^{2}\left(1-r^{2}\right), \quad d \theta / d t=1 $$

Short answer

Based on the given autonomous system in polar coordinates: $$ \frac{dr}{dt} = r^2\left(1-r^2\right), \quad \frac{d\theta}{dt} = 1 $$ There are no periodic solutions. However, there is a stable limit cycle with a radius of 1. This means that the trajectories inside the circle with radius 1 will spiral outward, and those outside will spiral inward, eventually settling onto the circle of radius 1.

Step-by-step solution

Analyze the given system

Start by carefully examining the given system of equations: $$ \frac{dr}{dt} = r^2\left(1-r^2\right), \quad \frac{d\theta}{dt} = 1 $$ The first equation represents the behavior of the radial coordinate r with respect to time. The second equation indicates that the angular coordinate \(\theta\) is increasing linearly with respect to time, which means that the trajectories will always spiral in the clockwise or counterclockwise direction.

Identify nullclines

A nullcline is a curve in the phase plane along which either \(\frac{dr}{dt}\) or \(\frac{d\theta}{dt}\) is zero. For our problem, we need to find the r-nullcline: $$ r^2\left(1-r^2\right) = 0 $$ This equation has three solutions: \(r=0\), \(r=1\), and \(r=-1\). Since r represents the radial coordinate in a polar system, only positive values are meaningful. Thus, we have two nullclines, \(r=0\) and \(r=1\).

Determine periodic orbits

To find the periodic solutions, we need to analyze the behavior of the system near the nullclines. In this case, there are no continuous closed orbits on the nullclines. Therefore, there are no periodic solutions.

Find limit cycles

Limit cycles are the closed orbits along which neighboring trajectories either converge or diverge as time progresses. In our system, there are no continuous closed orbits present, and thus there are no limit cycles.

Determine the stability

To analyze the stability of the system, we will focus on the behavior of the neighboring trajectories near the nullclines. For \(r = 0\) (origin): Since the angular coordinate \(\theta\) increases linearly with time, the trajectories will spiral away from the origin. Therefore, the origin is an unstable fixed point. For \(r = 1\): The trajectories will spiral around the circle of radius 1. However, the \(r\) component of our system has a positive value inside the circle (\(r < 1\)) and a negative value outside the circle (\(r > 1\)). This means that all the neighboring trajectories inside the circle will spiral outward, and those outside will spiral inward, eventually settling onto the circle of radius 1. Therefore, the circle with radius 1 forms a stable limit cycle. In conclusion, there are no periodic solutions for the given system, but there is a limit cycle with radius 1, which is stable.

Key concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system used to represent points in a plane. Instead of using the typical x and y coordinates, polar coordinates use a radius and an angle:

• The radius, denoted as \( r \), measures the distance from the origin to a point in the plane.
• The angle, denoted as \( \theta \), specifies the direction of the radius from a fixed direction, typically the positive x-axis.

In the context of autonomous systems, polar coordinates can vastly simplify the analysis of systems that exhibit rotational symmetry or periodic behavior. The given system of equations uses polar coordinates to describe its state:

• \( \frac{dr}{dt} = r^2(1-r^2) \)
• \( \frac{d\theta}{dt} = 1 \)

Here, \( \frac{dr}{dt} \) represents the rate of change of the radius, showing how the system grows or shrinks over time. \( \frac{d\theta}{dt} = 1 \) illustrates how the angular component changes, implying continuous rotation.

Periodic Solutions

Periodic solutions are solutions to differential equations that repeat over time. They describe cyclic behaviors where the system returns to its initial state after some period.
In our system, a periodic trajectory would mean a closed path in the phase space, such that:

• The radius \( r \) and angle \( \theta \) repeat their values cyclically.

Analyzing the given equations, \( \frac{dr}{dt} = r^2(1 - r^2) \) tells us that the radius will grow or shrink depending on its value:

• For \( r = 1 \), the radial component stops changing, suggesting a stable circle of radius 1.
• There are no other values of \( r \) that create continuously repeated behavior over time.

This implies there are no other periodic solutions since no other trajectories form closed loops or repeat continuously except the described limit cycle.

Limit Cycles

Limit cycles are specific types of periodic orbits in an autonomous system where neighboring trajectories converge to form a closed path. They signal robust behavioral patterns as time progresses.
For our system, \( \frac{dr}{dt} = r^2(1 - r^2) \) and \( \frac{d\theta}{dt} = 1 \), we find the limit cycle at \( r = 1 \):

• Inside the circle of radius 1, trajectories expand outwards.
• Outside this circle, trajectories contract inwards.

This creates a boundary, specifically the circle of radius 1, where neighboring paths settle and form a stable limit cycle. Limit cycles are crucial in dynamical systems as they indicate stable, repeating behavior over time and often represent real-life, self-sustaining systems.

Stability Analysis

Stability analysis involves understanding how a system behaves when slightly perturbed. In autonomous systems, it helps determine whether solutions are stable, unstable, or semi-stable.

• An equilibrium is stable if small disturbances do not lead to significant changes in behavior.
• An equilibrium is unstable if any small disturbance causes large changes.

For our system, the stability characteristics are determined at specific points:

• At \( r = 0 \): Trajectories spiral away from the origin, indicating the origin is an unstable point.
• At \( r = 1 \): Trajectories approach the circle of radius 1, both from inside and outside. This implies a stable boundary, forming a stable limit cycle.

In conclusion, stability analysis reveals that aside from the origin being unstable, the system stabilizes around the limit cycle, describing the system's long-term behavior efficiently.