Controlling the Period-Bubbling Route to Chaos

We describe the period-bubbling attractors, their generation mechanism and chaos control strategy in nonlinear, unidimensional and two-parameter-dependent discrete maps. A theorem for the occurrence of sequential period-bubbling attractors leading to chaos, is presented. It delineates the role of a geometric parameter in the formation of anti-monotonic periodic sequences of each iterate of such discrete maps. The appearances of different period bubbles in two distinct Gaussian and cubic maps are demonstrated in accordance to the theorem. The delimited chaos through period-bubbling route is suppressed by employing linear density feedback with a proportional constant. The study discloses that the period-bubbling transitional route in unidimensional map cannot be suppressed by adopting constant feedback, which is in contrast to the control of period-doubling route to chaos. In Gaussian and cubic maps, the period-bubbling routes to chaos are gradually collapsed for linear density feedbacks, with the enhancements of the feedback strength parameter k. Interestingly, at [Formula: see text], the feedback-influenced Gaussian map experiences tangent bifurcation and chaos can re-enter into its dynamics through that scenario for [Formula: see text]. The tangent bifurcation in the feedback-affected Gaussian map is explained mathematically. However, no such transitional exchange has been perceived in cubic map, for augmented values of the feedback strength parameter.