Question

an autonomous system is expressed in polar coordinates. Determine all periodic solutions, all limit cycles, and determine their stability characteristics. $$ d r / d t=r(r-1)(r-3), \quad d \theta / d t=1 $$

Short answer

Based on the analysis and solution of the autonomous system in polar coordinates, identify the stability characteristics of the limit cycle found in the system. Solution: The stability characteristics of the limit cycle found in the system is that it is a stable limit cycle, as it attracts trajectories in its neighborhood. The radius of this limit cycle is 1.

Step-by-step solution

Equilibria of the radial variable

First, we examine the equilibria of the radial variable, by setting \(\frac{dr}{dt}=0\) and solving for \(r\): $$ 0 = r(r-1)(r-3) $$ This gives us three equilibria: \(r_1 = 0\), \(r_2 = 1\), and \(r_3 = 3\).

Stability of the equilibria

To analyze the stability of the equilibrium points, we examine the behavior of \(\frac{dr}{dt}\) near each equilibrium. If the rate is positive, it is unstable, and if negative, it is stable. - Around \(r_1=0\), \(r\) is increasing, meaning the system is moving away from this equilibrium. Thus, it is unstable. - Around \(r_2=1\), \(r\) is decreasing, meaning the system is moving towards this equilibrium. Thus, it is stable. - Around \(r_3=3\), \(r\) is increasing, meaning the system is moving away from this equilibrium. Thus, it is unstable.

Periodicity and limit cycles

To find periodic solutions for the system, we look at the angular variable \(\theta\). Since the rate of change of \(\theta\) is constant, the solution will be periodic with a period of \(2\pi\). Now let's consider limit cycles. These can exist only on the equilibria of the radial variable. Since \(r_2 = 1\) is a stable equilibrium, it forms a stable limit cycle (circle) with radius \(r_2 = 1\).

Stability characteristics of limit cycles

Now that we've identified the limit cycle, we will determine its stability characteristics. The circle with radius \(r_2 = 1\) is a stable limit cycle, as it attracts trajectories in its neighborhood. In conclusion, the given autonomous system has one stable equilibrium at \(r=1\), and a stable limit cycle with a radius of 1. The system is periodic with a period of \(2\pi\) in the angular variable \(\theta\).

Key concepts

Polar Coordinates

To understand autonomous differential systems, it's crucial to know what polar coordinates are. Unlike the more conventional Cartesian coordinate system which uses a grid of x and y axes, polar coordinates represent points in a plane by a distance and an angle. The distance from the origin to the point is denoted as 'r' (radius), and the angle is referred to as 'θ' (theta), which is measured from the positive x-axis.

In differential equations, expressing a system in polar coordinates can simplify the problem and make it easier to analyze, especially if the system's behavior is naturally circular or rotational. For example, in the exercise provided, the behavior of the system is defined in terms of the radial distance 'r' and the angular component 'θ'. This is particularly useful for visualizing solutions that revolve around a certain point or follow a circular path.

Periodic Solutions

A periodic solution in a differential system is one that repeats after a certain interval, which is referred to as the period. Think of it like a clock's hands that circle back to the same position every 12 hours. In the given autonomous system, the angular component \(d\theta/dt=1\) implies a constant rotation with no dependence on 'r', meaning that the angle \(\theta\) will complete a cycle after every \(2\pi\) interval.

Importance of Periodic Solutions

Periodic solutions help in predicting the long-term behavior of systems. They are often sought after in physics and engineering to understand oscillations and waves, such as the motion of pendulums or the propagation of light.

Limit Cycles

Limit cycles are closed trajectories in a phase space that solutions can spiral towards or away from, but never intersect. They represent a state of stable or unstable periodic behavior in a system. Limit cycles can be pinpointed by analyzing the equilibria of the radial variable, as shown in the exercise. Specifically, the stable equilibrium at \(r_2=1\) suggests that the system behaves like a heart beating in a stable rhythm: drawing nearby states towards it in an endless loop.

In practical terms, imagine a limit cycle as the track on which a race car must stay, continuously looping without veering off. The stability identifies if the car is naturally guided onto the track (stable limit cycle) or off the track (unstable limit cycle).

Equilibrium Stability

Equilibrium points are the 'resting states' of a system where no changes occur as time passes. In the context of the given differential system, these points occur where the change in 'r' is zero. Stability of these points is critical; it dictates whether the system, when perturbed, will return to the equilibrium (stable), move away (unstable), or neither (semi-stable).

Analogy with a Marble in a Bowl

Imagine placing a marble in a bowl. If the marble rolls back to the bottom when nudged, the bottom point is a 'stable' equilibrium. However, if you turn the bowl upside down and place the marble on the peak, any nudge will cause the marble to roll off — an 'unstable' equilibrium. In our example, the stable equilibrium at \(r=1\) acts as the bowl's bottom, gently guiding the system back to a state of balance.