The graphs on the right does not make any sense: when the load is zero (P=0), the displacement is non zero (2 and 4)

Denis Benasciutti, It's a reaction force vs. displacement graph, not applied force.I didn't mention there properly.

Let me first explain what I am getting from the figure: Two trusses connected by pin joint. On increasing the force, compression in trusses is increasing and bars are shortening in length. After a load value, structure snaps through and new equilibrium stablishes. Now bars are in tension and direction of reaction reverses.

If the above explained understanding is correct you may try solving as a deflection control problem rather than load controlled. Although it may be difficult to achieve convergence.