Charging batteries at different rates and the associated side-effects?
Look closely at the top plot and you can see it is just applying a simple maximum current OR voltage scheme.  Up to about .9 hour the maximum current (looks like about 700 mA) causes lower than the maximum voltage, so the battery is charged with that current.  When the battery reaches the 4.2 V limit, the charger holds the voltage there and the current then decreases.  Eventually the battery gets so full that the charging current tapers off to nearly 0.

The charger puts out either 700 mA or 4.2 V, whichever results in lower voltage (or current).  The battery then decides the other of the two.