Question

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

Short answer

$$ There is an equilibrium point at (0,0). However, due to the eigenvalues of the Jacobian matrix being non-hyperbolic, we cannot determine its stability using the linearization technique. Additionally, we have not been able to identify any limit cycles analytically. As a result, we would need to rely on other methods, such as numerical simulations, to study this system.

Step-by-step solution

Rewrite the system in Cartesian coordinates

To rewrite the system in Cartesian coordinates, remember that \((x,y)\) and \((r,\theta)\) are related by the equations \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Let's find the relationship between \(d r/d t\) and \(d x/d t\), \(d y/d t\), as well as \(d \theta /d t\): $$ d x / d t=dr \cos (\theta)-r \sin (\theta) d \theta, \quad d y / d t=dr \sin (\theta)+r \cos (\theta) d \theta. $$ We also know $$ dr/dt = \sin(\pi r), \quad d\theta/dt = 1. $$ Therefore, we can substitute these into the expressions for \(dx/dt\) and \(dy/dt\): $$ \begin{aligned} dx/dt &= \sin(\pi r) \cos(\theta) - r \sin(\theta), \\ dy/dt &= \sin(\pi r) \sin(\theta) + r \cos(\theta). \end{aligned} $$ This gives us the system in Cartesian coordinates.

Identify equilibrium points

Equilibrium points occur when both \(dx/dt\) and \(dy/dt\) are zero. From the system, we have: $$ \begin{aligned} 0 &= \sin(\pi r) \cos(\theta) - r \sin(\theta), \\ 0 &= \sin(\pi r) \sin(\theta) + r \cos(\theta). \end{aligned} $$ Inspecting the equations, we find that \((r,\theta) = (0,0)\) is an equilibrium point.

Linearize the system near the equilibrium point

To analyze the stability of the equilibrium point, we linearize the system near the equilibrium point by calculating the Jacobian matrix: $$ J(r,\theta) = \begin{bmatrix} \frac{\partial}{\partial r} (\sin(\pi r) \cos(\theta) - r \sin(\theta)) & \frac{\partial}{\partial \theta} (\sin(\pi r) \cos(\theta) - r \sin(\theta)) \\ \frac{\partial}{\partial r} (\sin(\pi r) \sin(\theta) + r \cos(\theta)) & \frac{\partial}{\partial \theta} (\sin(\pi r) \sin(\theta) + r \cos(\theta)) \end{bmatrix}. $$ Simplifying, we obtain $$ J(r,\theta) = \begin{bmatrix} \pi r \cos(\pi r) \cos(\theta) - \sin(\theta) & -\sin(\pi r) \sin(\theta) - r \cos(\theta) \\ \pi r \cos(\pi r) \sin(\theta) + \cos(\theta) & \sin(\pi r) \cos(\theta) - r \sin(\theta) \end{bmatrix}. $$ Evaluate the Jacobian at the equilibrium point \((r,\theta) = (0,0)\): $$ J(0,0) = \begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix}. $$

Determine stability

The eigenvalues of the Jacobian matrix at the equilibrium point \((0,0)\) will help us determine the stability. The eigenvalues are found by solving the following equation: $$ \text{det}(J(0,0) - \lambda I) = \begin{vmatrix} -\lambda & 0 \\ 1 & -\lambda \end{vmatrix} = \lambda^2 = 0. $$ The eigenvalues are both zero. Therefore, the equilibrium point \((0,0)\) is non-hyperbolic, and we cannot determine its stability using the linearization technique. We may need to apply other methods, such as Lyapunov's direct method.

Check for limit cycles

In polar coordinates, our system is given by $$ dr/dt = \sin(\pi r), \quad d\theta/dt = 1. $$ To check for limit cycles, we need to find a function \(V(r)\) such that its derivative along the system trajectories is positive, indicating a stable limit cycle. Let's consider the candidate Lyapunov function: $$ V(r) = 1- r^2. $$ Now, let's compute the time derivative of \(V\) along the system trajectories: $$ \frac{dV}{dt} = \frac{\partial V}{\partial r}\frac{dr}{dt} = -2r\sin(\pi r). $$ Since the derivative of \(V\) is not positive definite (for example, when \(r \neq 0\), \(\frac{dV}{dt} = 0\)), we cannot conclude the existence of stable limit cycles using Lyapunov's direct method. The existence of limit cycles and stability characteristics cannot be determined analytically, and we may need to rely on numerical simulations or other methods to study this system.

Key concepts

Polar Coordinates in Autonomous Systems

In autonomous systems, representing complex dynamics using polar coordinates can be very useful. Polar coordinates

• consist of two components: the radial distance \(r\) and the angular position \(\theta\).
• offer a way to express motion and trajectories in a plane using these two parameters.

The provided system has the equations \( dr/dt = \sin(\pi r) \) and \( d\theta/dt = 1 \). Here,

• \( r \) changes based on the sine function, oscillating as \( r \) changes, while
• \( \theta \) changes linearly with time, steadily increasing.

This setup helps in transforming complex Cartesian behaviors into simpler radial and angular descriptions, making analysis and computations more manageable.

Understanding Periodic Solutions

A periodic solution in the context of dynamical systems is a path or trajectory that repeats itself after a fixed time interval. In the given autonomous system, examining \( dr/dt = \sin(\pi r) \) reveals its periodicity.

• The sign of \( \sin(\pi r) \) dictates whether the radius grows or shrinks over time.
• Given the periodic nature of the sine function, \( r \) will exhibit repeating intervals of growth and decay.

For our case, \( d\theta/dt = 1 \) implies \( \theta \) changes uniformly, contributing to a regular periodic angular motion. Combined, these periodic changes in \( r \) and \( \theta \) describe circular orbits, which repeat over time, forming the basis of periodic solutions in this system.

Limit Cycles and Their Characteristics

Limit cycles are isolated closed trajectories that do not spiral outward or inward but form a loop. They showcase stable, self-sustaining periodic behavior in autonomous systems. In our system with \( dr/dt = \sin(\pi r) \),

• we seek curves where the radial component \( r \) stabilizes to a repeated cycle,
• forming a closed loop as a circular path when paired with the continuous \( \theta \).

Checking for limit cycles involves exploring if changes in \( r \) over time continuously maintain the sine wave's periodic behaviors. However, in this case, finding analytical solutions for existence is complex, typically requiring numerical methods or simulations to observe stable or unstable limit cycles.

Stability Analysis in Polar Form

Stability analysis helps determine how systems behave when perturbed from equilibrium. In polar coordinates,

• we begin by detecting equilibrium points where changes in both \( r \) and \( \theta \) are zero, but
• analyse near those points using tools like Jacobian matrices, as done in the initial solution.

The calculation shows non-hyperbolic behavior at \((0, 0)\), evidenced by zero eigenvalues. This requires approaches such as Lyapunov's method for deeper insights. Without definitive stability characteristics from linearization, simulations or alternative analysis techniques are often needed to probe specific long-term behaviors or potential oscillatory patterns efficiently in autonomous systems.