Question

an autonomous system is expressed in polar coordinates. Determine all periodic solutions, all limit cycles, and determine their stability characteristics. $$ d r / d t=r|r-2|(r-3), \quad d \theta / d t=-1 $$

Short answer

Based on the given autonomous system of differential equations in polar coordinates, we analyzed the radial and angular differential equations to find all periodic solutions, limit cycles, and their stability characteristics. We found that there are two equilibrium points (\(r=2\) and \(r=3\)) that correspond to stable and unstable limit cycles, respectively. We determined that the periodic solutions have a period of \(-2\pi\). The limit cycles are stable for \(r=2\) and unstable for \(r=3\).

Step-by-step solution

Analyze the radial differential equation

The radial differential equation is given as: $$ \frac{dr}{dt} = r|r-2|(r-3) $$ To find the equilibrium points, we set \(\frac{dr}{dt} = 0\) and solve for \(r\) in terms of the equilibrium points: $$ r|r-2|(r-3) = 0 $$ This equation has three solutions for \(r\): \(r = 0\), \(r = 2\), and \(r = 3\). These correspond to the equilibrium points of the system.

Analyze the angular differential equation

The angular differential equation is given as: $$ \frac{d\theta}{dt} = -1 $$ As there are no equilibrium points for this equation, the angular dynamics are a simple rotation with a constant angular velocity \(-1\). Now combine this information with the radial dynamics for periodic solutions.

Discuss the limit cycles and their stability characteristics

For \(r=0\), the system collapses to the origin and does not show any oscillatory behavior or limit cycles. Hence, it is asymptotically stable. For \(r=2\) and \(r=3\), these equilibria can be considered as periodic solutions since they correspond to circles moving in the angular direction with constant angular velocity \(-1\). Their period can be computed as follows: $$ T(\theta) = \frac{d\theta}{dt} = \frac{\Delta\theta}{-1} = \frac{2\pi}{-1} = -2\pi $$ Thus, we have two periodic solutions with period \(-2\pi\). For limit cycles, we need to look at the behavior of the radial dynamics in the vicinity of the equilibrium points. - Near \(r=2\), the dynamics is given as \(\frac{dr}{dt}=-r(r-1)(2-r)\), indicating a stable limit cycle with a radius of \(r = 2\). - Near \(r=3\), the dynamics is given as \(\frac{dr}{dt}=r(3-r)(r-1)\), indicating an unstable limit cycle with the radius of \(r = 3\). In conclusion, we have found the periodic solutions, limit cycles (r=2, stable; r=3, unstable), and characterized their stability.