To derive the equations of motion for the boat, we note that the velocity of the boat relative to the river is given by $(u, v)$. Therefore, the velocity of the boat in the $x$-direction is $u\cos\theta$, where $\theta$ is the angle between the boat’s direction and the $x$-axis. Similarly, the velocity of the boat in the $y$-direction is $u\sin\theta$. The angle $\theta$ is determined by the condition that the boat is always rowing towards the fixed point $(0,0)$ on the opposite bank. Therefore,

$\tan \theta=\frac{y}{x-u}$

Taking the time derivative of $x$ and $y$ gives

$\dot{x}=u \cos \theta-\frac{u x}{\sqrt{x^2+y^2}}, \quad \dot{y}=u \sin \theta-v-\frac{u y}{\sqrt{x^2+y^2}}$.

Using the identity $\cos^2\theta+\sin^2\theta=1$, we can write $\cos\theta=\sqrt{1-\sin^2\theta}$ and substitute for $\cos\theta$ in the expression for $\dot{x}$, giving

$\dot{x}=u \sqrt{1-\frac{y^2}{(x-u)^2}}-\frac{u x}{\sqrt{x^2+y^2}}$.

We can simplify this expression by multiplying both the numerator and denominator of the first term by $(x-u)^2$, giving

$\dot{x}=u \frac{x-u}{\sqrt{x^2+y^2}}-\frac{u x}{\sqrt{x^2+y^2}}=\frac{u}{\sqrt{x^2+y^2}}\left(-u x+\sqrt{x^2+y^2}\right)$.

Similarly, we can substitute for $\sin\theta$ in the expression for $\dot{y}$, giving

$\dot{y}=u \frac{y}{\sqrt{(x-u)^2+y^2}}-v-\frac{u y}{\sqrt{x^2+y^2}}=\frac{u}{\sqrt{x^2+y^2}}\left(-v+(x-u) \frac{y}{\sqrt{(x-u)^2+y^2}}\right)$.

Defining $r=\sqrt{x^2+y^2}$ and $\eta=y/x$, we obtain the desired equations of motion,

$\dot{r}-\frac{-u x}{\left(x^2+y^2\right)^{1 / 2}}=\frac{u}{r}(r-u), \quad \dot{\eta}-v-\frac{u \eta}{\left(x^2+y^2\right)^{1 / 2}}=\frac{u}{r} \eta\left(\frac{r-u}{x}\right)$.

To obtain the phase trajectories, we can eliminate time between these two equations by differentiating the second equation with respect to $x$ and substituting for $\dot{r}$ from the first equation. This gives

$\frac{d}{d x}\left(\frac{\dot{\eta}+v}{\eta}\right)=\frac{u}{r}\left(1-\frac{u}{r}\right)$.

Integrating this expression gives

$\ln |\eta+\dot{\eta}+v|=u \ln \left|\frac{r}{u}-1\right|+C_1$,

where $C_1$ is a constant

As an AI language model, I can provide the steps to find the limit cycles but I can’t plot the trajectories directly. However, I can help you set up the equations and provide guidance on how to find the limit cycles using a numerical method.

To begin, we can rewrite the given system as a set of first-order differential equations:

$\begin{aligned} & \dot{x}=y \ & \dot{y}=-\mu \sin (y)-x\end{aligned}$

We can use the Poincaré-Bendixson theorem to determine the existence of limit cycles. The theorem states that if a solution is bounded and does not approach a fixed point or a periodic orbit, then it must approach a limit cycle.

To find the limit cycles, we can use numerical methods such as the Runge-Kutta method to solve the system of equations and plot the trajectories. We can start by choosing different initial conditions and simulating the system for a sufficiently long time to ensure that the trajectory has converged to a limit cycle or has become unbounded.

Here are the steps to find the limit cycles:

1. Choose a set of initial conditions $(x_0, y_0)$
2. Solve the system of differential equations numerically using a numerical method such as the Runge-Kutta method.
3. Plot the trajectory $(x(t), y(t))$ for a sufficiently long time.
4. Repeat steps 1-3 for different initial conditions.
5. Look for closed trajectories that do not approach a fixed point or a periodic orbit.
6. If there are multiple closed trajectories, then there are multiple limit cycles.

Using a numerical method to find limit cycles can be computationally intensive and requires some experience with numerical methods. However, it can be a powerful tool to understand the dynamics of a system when analytical solutions are not available.

To determine the stability of the limit cycles, we can examine the trajectories in the phase plane near the limit cycles. Specifically, we can look at the behavior of nearby trajectories relative to the limit cycle.

Let’s assume that we have found a closed trajectory (i.e., a limit cycle) in the phase plane. We can then consider a nearby trajectory that starts at a point slightly displaced from the limit cycle. As time passes, this trajectory will either approach the limit cycle or move away from it.

If nearby trajectories that start on one side of the limit cycle move towards the limit cycle as time goes on, then the limit cycle is stable. On the other hand, if nearby trajectories that start on one side of the limit cycle move away from the limit cycle as time goes on, then the limit cycle is unstable.

More formally, we can use the concept of linear stability to determine the stability of the limit cycle. Linear stability analysis involves linearizing the system of equations around the limit cycle and examining the eigenvalues of the resulting linearized system. If all eigenvalues have negative real parts, then the limit cycle is stable. If at least one eigenvalue has a positive real part, then the limit cycle is unstable.

However, in the absence of an analytical solution, we can examine the trajectories in the phase plane to get an idea of the stability of the limit cycles. By examining the direction of the nearby trajectories relative to the limit cycle, we can infer whether the limit cycle is stable or unstable.

If the nearby trajectories spiral towards the limit cycle, then the limit cycle is stable. If the nearby trajectories spiral away from the limit cycle, then the limit cycle is unstable. The direction of the spiral can be determined by the sign of the eigenvalues of the linearized system.

It is important to note that this method of determining stability is not always accurate and can be subjective. However, it can provide a qualitative understanding of the stability of the limit cycles.