Bi-cal{PT} symmetry in nonlinearly damped dynamical systems and tailoring cal{PT} regions with position dependent loss-gain profiles

We investigate the remarkable role of position dependent damping in determining the parametric regions of symmetry breaking in nonlinear cal{PT}-symmetric systems. We illustrate the nature of cal{PT}-symmetry preservation and breaking with reference to a remarkable integrable scalar nonlinear system. In the two dimensional cases of such position dependent damped systems, we unveil the existence of a class of novel bi-cal{PT}-symmetric systems which have two fold cal{PT} symmetries. We analyze the dynamics of these systems and show how symmetry breaking occurs, that is whether the symmetry breaking of the two cal{PT} symmetries occurs in pair or occurs one by one. The addition of linear damping in these nonlinearly damped systems induces competition between the two types of damping. This competition results in a cal{PT} phase transition in which the cal{PT} symmetry is broken for lower loss/gain strength and is restored by increasing the loss/gain strength. We also show that by properly designing the form of the position dependent damping, we can tailor the cal{PT}-symmetric regions of the system.