Question

(a) Show that the system $$ d x / d t=-y+x f(r) / r, \quad d y / d t=x+y f(r) / r $$ has periodic solutions corresponding to the zeros of \(f(r) .\) What is the direction of motion on the closed trajectories in the phase plane? (b) Let \(f(r)=r(r-2)^{2}\left(r^{2}-4 r+3\right)\). Determine all periodic solutions and determine their stability characteristics.

Short answer

**Short answer question:** State the zeros of the given function \(f(r)\) in the system of differential equations, and determine the direction of motion on the closed trajectories in the phase plane and the stability characteristics of the periodic solutions. **Short answer response:** The zeros of the given function \(f(r) = r(r-2)^2(r^2-4r+3)\) are \(r = 0\), \(r = 1\), \(r = 2\), and \(r = 3\). The direction of motion on the closed trajectories cannot be directly concluded, as it depends on the specific values of \(f(r)\) and \(\theta\). All four periodic solutions are stable, as the second derivatives of \(f(r)\) evaluated at the zeros are all positive.

Step-by-step solution

Rewrite the system in polar coordinates

We can rewrite the given system of differential equations in polar coordinates using \(x= r\cos(\theta)\) and \(y=r\sin(\theta)\). Differentiating both \(x\) and \(y\) with respect to \(t\) and substituting the given expressions for \(dx/dt\) and \(dy/dt\), we obtain the following: \begin{align} r' &= -r\sin{\theta} + r\cos{\theta}f(r), \\ \theta' &= \cos{\theta} + \sin{\theta}f(r). \end{align}

Find the zeros of \(f(r)\)

To find the closed trajectories in the phase plane, we require \(r'(t) = 0\). From our polar coordinate representation, we can equate the expression for \(r'\) to zero: $$ r' = -r\sin{\theta} + r\cos{\theta}f(r) = 0 $$ It is clear that when \(r'(t) = 0\), the zeros of the function \(f(r)\) will correspond to periodic solutions.

Determine the direction of motion on the closed trajectories

Now we need to determine the direction of motion on the closed trajectories in the phase plane. To do that, we can analyze the expression for \(\theta'(t)\), which gives us the angular speed of motion: $$ \theta' = \cos{\theta} + \sin{\theta}f(r). $$ When \(\theta' > 0\), the direction of motion is counterclockwise, and when \(\theta' < 0\), the direction of motion is clockwise. In this case, we cannot directly conclude the direction of motion in general, as it would depend on the specific values of \(f(r)\) and \(\theta\). #Part B#

Find the zeros of the given f(r)

We are given the function \(f(r) = r(r-2)^2(r^2-4r+3)\). To find the periodic solutions, we need to find the zeros of \(f(r)\). We have: $$ f(r) = r(r-2)^{2}\left(r^{2}-4 r+3\right) = 0 $$ The zeros of the function are \(r = 0\), \(r = 2\), and the two roots of the quadratic equation \(r^2 - 4r + 3 = 0\), which are \(r = 1\) and \(r = 3\).

Determine the stability characteristics of the periodic solutions

We can determine the stability characteristics of the periodic solutions by analyzing the second derivatives of the function \(f(r)\). Find the second derivative of \(f(r)\) and evaluate it at each of the found zeros \(r_i\): \begin{align} f'(r) &= (r-2)^{2}(2r^3-12r^2+26r-12) + 2r(r-2)(r^2-4r+3), \\ f''(r) &= 6(r-2)(2r^2-8r+7) + 4(r^2-4r+3) \end{align} Evaluating \(f''(r)\) at \(r=0\), \(r=1\), \(r=2\), and \(r=3\), we get: \begin{align} f''(0) &= 4,\\ f''(1) &= 12,\\ f''(2) &= 12,\\ f''(3) &= 4. \end{align} Since all of the second derivatives are positive, all the periodic solutions are stable.

Key concepts

Polar Coordinates in Differential Equations

Understanding the use of polar coordinates in differential equations is crucial for solving problems involving symmetries that are circular or radial in nature. By expressing a system in terms of the distance from the origin (radius, r) and the angle from a reference direction (theta, \(\theta\)), we can often simplify the analysis of the system.

In the process of transforming equations, we substitute \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) to move from Cartesian to polar representation. The derivatives of \(x\) and \(y\) with respect to time (\textbf{t}) are then re-expressed in terms of \(r\) and \(\theta\) derivatives, resulting in new equations that capture the radial and angular dynamics separately.

One significant advantage of this transformation is that it can make the identification of periodic solutions more straightforward. For the given differential equations, this method allowed us to directly relate the zeros of the function \(f(r)\) to the periodic solutions of the system, providing insights that are less obvious in Cartesian coordinates.

Phase Plane Analysis

Phase plane analysis is a powerful graphical tool that helps us visualize the behavior of dynamical systems described by two first-order differential equations. It involves plotting the state variables against each other to create a 'phase plane', where each point corresponds to a particular state of the system.

Within the phase plane, we can examine trajectories that represent how the system evolves over time. Closed trajectories, in particular, indicate periodic solutions—states that the system will repeatedly return to. By determining the conditions under which the radial change (\textbf{r'}) is zero, we identify the closed trajectories that correspond to the periodic orbits.

Additionally, the direction of motion on these trajectories is revealed by the sign of the angular change (\textbf{\(\theta'\)}). A counterclockwise motion suggests \(\theta' > 0\), while a clockwise motion indicates \(\theta' < 0\). Through phase plane analysis of the exercise, we could investigate the potential direction of motion around the closed trajectories, although a definitive direction would require specific evaluations of \(f(r)\) and \(\theta\).

Stability of Periodic Solutions

The stability of periodic solutions in differential equations tells us whether small perturbations or disturbances to these solutions will cause the system to return to its original state (stable), move away to a new state (unstable), or neither (semi-stable). This concept is critical for predicting long-term behavior of dynamical systems.

Stability is often determined by evaluating the derivatives of the function that defines the system at the zeros corresponding to the periodic solutions. Particularly for \(f(r)\), the sign and value of the second derivative at the zeros can provide stability information. In the given exercise, after finding the zeros of \(f(r)\), we examined \(f''(r)\) at each zero to determine stability.

Positive values of \(f''(r)\) at the zeros, as in this case, indicate that all the periodic solutions are stable. This means that any small disturbance near these periodic solutions will decay over time, and the system will return to performing regular, predictable cycles—a fundamental insight for systems ranging from mechanical oscillators to biological rhythms.