OK, let me explain this problem in full. First I consider a bit more general problemâ€”with coefficient $\alpha\ge 1$ instead of $1$ or $2$ (respectively) and correspondingly with

\begin{equation}

H_\alpha=\alpha \cos (x) + \cos(y).

\label{P}

\end{equation}

Then

(i) As $\alpha\ne 0$ there are the same stationary points $(\pi m, \pi n)$ with $\sin(x)=\sin(y)=0$;

(ii) As $\alpha>0$ points with $\cos(x)=\cos(y)=\pm 1$ are extremums (maxima for $+$ and minima for $-$) aka centers and points with $\cos(x)=-\cos(y)=\pm 1$ are saddle points;

(iii) As $\alpha=1$ (problem (a)) all saddle points are on $H_1(x,y)=0$ and these lines are just strait lines with slopes $\pm 1$ (which are separatrices). All other trajectories are periodic (so picture consists of "whirlwinds" around centers, separated by separatrices);

(iv) As $\alpha > 1$ (or $0<\alpha <1$) saddles with $\cos(x)=-\sin(y)=1$ are on $\{H_\alpha =(\alpha-1)\}$ and saddles with $\cos(x)=-\sin(y)=-1$ are on $\{H_\alpha =-(\alpha-1)\}$ and therefore they are on different curves; in this case separatrices from $(\pi m, \pi n)$ (with $m$ and $n$ having different parities) cannot go to "neighbouring" saddle, but only either to $(\pi m\pm 2\pi, \pi n)$ or to $(\pi m, \pi n\pm 2\pi)$ and from (vi) follows that for $\alpha>1$ it will be the latter case (and for $0<\alpha <1$ one can see that it will be the former one);

(v) Consider what happens as $\alpha>1$ and trajectory starts from point where $| H_\alpha (x,y)|<\alpha (1-\epsilon) -1$ with any $\epsilon >0$. Since $H_\alpha$ is constant along trajectory, $|\cos(x)|$ is confined between $-1+\epsilon$ and $1-\epsilon$ and therefore $\sin(x)$ cannot cross $0$ and remains either in $\{\sin (x) > \epsilon_1=(1-(1-\epsilon)^2)^{1/2}\}$ or in $\{\sin (x) <-\epsilon_1\}$. In the former case $y'>\epsilon_1$ along the whole trajectory (and in the latter $y'<-\epsilon_1$).

(vi) Therefore as $\alpha >1$ we have "whirlwinds" covering zones $\{|H_\alpha |>\alpha-1\}$ and the whirlwinds with the centers at $(\pi m,\pi n)$ ($m$ and $n$ have the same parity) and $(\pi m,\pi n+2\pi)$ are separated just by a saddle at $(\pi m,\pi n+\pi)$. However whirlwinds with the centers at $(\pi m,\pi n)$ and whirlwinds with the centers at $(\pi m+2\pi,\pi n)$ are separated by a vertical "river" floating either up or down (depending on $m$). Sure "shores" of "rivers" are not straight and at saddles the jumper between rivers has width $0$.

(vii) Similarly as $0<\alpha<1$ rivers are horizontal.

**Remark.** Note the similarity to pendulum (see f.e.

http://weyl.math.toronto.edu/MAT244-2011S-forum/index.php?topic=126.msg450#msg450) where there are "normal" oscillations and fast rotations when pendulum goes over top point.