A comparative study on the co-existing attractors in the Gaussian map and its q-deformed version

In a recent study Jaganathan and Sinha [Jaganathan R, Sinha S. A q-deformed nonlinear map. Phys Lett A 2005;338:277–87] have introduced a scheme for the q-deformation of nonlinear maps using the logistic map as an example and shown that the q-logistic map exhibits a wide spectrum of dynamical behaviours including the co-existence of attractors (which is a rare phenomenon in one-dimensional maps). In this paper, we aim to analyze another famous one-dimensional map – the Gaussian map (a known one-dimensional map exhibiting co-existing attractors) subject to the same q-deformation scheme. We compare the dynamical behaviour of the Gaussian map and q-deformed Gaussian map with a special attention on the regions of the parameter space, where these maps exhibit co-existing attractors. An important conclusion of the present study is that the appearance of co-existing attractors for a particular choice of system parameters can be understood as a consequence of the presence of multiple fixed points in one-dimensional nonlinear maps; however the converse is not always true.