A duck is swimming in a circle in the counterclockwise direction in a pond at a speed of v. Let O be the center of the duck's circular path. Let $\phi$ be the angle between the ray from O to the duck, and the ray between O and some fixed landmark outside the pond. The radius of duck's circular path is 1 unit, and hence the position D of the duck is given by $D = e^{i\phi}$.
An alligator starts swimming towards the duck (its position is A), and R is the distance from A to D (so that R=|D-A| at each moment). The alligator is always swimming directly towards the duck, at a constant speed of $w$. Finally, let $k=w/v$.
Let $k<1$ so that the alligator is slower than the duck. Consider the critical point $(\theta_0, R_0)$ in the region $-\pi\le\theta<\pi$, $0<R<2$. Find the linearised system $x'=Ax$ near the critical point by writing down the matrix A in terms of k. (Here, the first component of $x$ approximates the deviation from $\theta_0$ and the second component approximates the deviation from $R_0$.
I am given that $\theta ' = cos(\theta)/R-1$ and $R' = \sin(\theta) - k$ (from a previous problem).
I derived a matrix using the partial derivatives of $\theta'$ and $R'$. However, none of them contains $k$, but they're supposed to be entries for the matrix. Or am I supposed to have $k$'s? But how would I still have $k$s when $k$ is only a constant in $R'$ equation? Derivatives in terms of theta or R will just render $-k$ term as 0.
(I am pretty sure that my equations for $R'$ and $\theta'$ are right, but please let me know if nothing else explains the difficulty I'm having at the moment.)