Target and Chaser

Same Apoapsis smaller Periapsis

The chaser is in an elliptical orbit, with the apoapsis on the same radius as the target. On the Apoapsis the chaser has a lower speed as the target since the radius is the same, but the total energy of the orbit is lower. So the relative motions makes the small loop-the-loop.

Smaller Apoapsis

Still slower at Apoapsis

The radius of the chaser is always lower than that of the target, but the chaser is still slower at the apoapsis.

Same velocity at Apoapsis

There is a edge case that the chaser has the same speed at the apoapsis as the target. This is when the following condition is met:

\begin{aligned} v_{a p o}^{2} & = v_{c i r c}^{2} \\ μ (\frac{2}{r_{a}} - \frac{1}{a}) & = \frac{μ}{R} \end{aligned}

With a given periapsis, the needed apoapsis can be calculated.

Faster at Apoapsis

If the apoapsis is low enough the chaser is always faster than the target.

Periods

Same Period

The period of an orbit is only dependent on the semi-major axis. This means if the periapsis difference is the same as the apoapsis difference, the semi-major axis is the same and both orbits have the same period.

Faster Period

If the difference of the apoapsis is smaller than the one of the periapsis, the orbit of the chaser is faster than the one of the target.

Slower Period

If the difference of the apoapsis is bigger than the one of the periapsis, the orbit of the chaser is slower than the one of the target.