You can get bisection in a model without plasticity. It is not necessarily linked.

I will oversimplify this to describe it briefly. In a nonlinear analysis, a small portion of the total load, say 20% is first attempted, which is called a substep. The solver inverts the stiffness matrix to solve for the unknown displacements, then plugs them back in to see if the force is within a small tolerance of being in equilibrium. That is called an iteration. If the tolerance was exceeded, the solver updates the stiffness matrix and inverts it to solve for revised unknown displacements and repeats the check for being in equilibrium. In your image above, it took 12 iterations for the force error to drop below the tolerance, which is called the convergence criterion and we say the substep has converged. Having successfully solved the first 20% of the load, it attempts to solve the second 20% of the load (the next substep), and the whole iteration process repeats. The second substep took only 3 iterations.

The software has rules built-in to prevent it from iterating too many times, trying to find a set of displacements within the force equilibrium tolerance. For example, it might take 49 iterations to converge on a 20% load increment, but instead, after 10 iterations, the software changes to a 10% load increment and starts a new set of iterations. That is called a bisection. It cuts the load increment in half. Then it uses 3 iterations to find equilibrium for the first 10% load increment, and it increments another 10% and uses another 3 iterations to find equilibrium for that substep. So in this example, it took 16 iterations to attempt 20%, give up, then succeed at 10% twice in 3 iterations each. This is better than taking 49 iterations to get 20%, but if you could have told it to take 10% in the first place, it would have only taken 6 iterations.

This discussion has more info.