Diodes Circuit Analysis
I'll offer a visual approach to the solution.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Three cases. The arrows are the direction of the current.


(Not shown in Figure 1.) When Vi = Vb - Vd (1.3 V using a typical diode) there is current around the D1 D2 loop but no current in the outer loop. This is where you curve crosses the X-axis.
a. The easiest case: When Vi is high D4 is reverse biased and so that branch is out of circuit. That gives right-side portion of your graph.
b. When Vi &lt; Vb - Vd some current will flow in the D1 D2 loop but current will also flow in the outer loop. This gives the left-side portion of your graph.
c. In the in-between cases Vo will rise from 0 to the upper line.


Sorry, this needs more work. Saving incomplete solution for later editing.