q-deformations of functions and distributions have been used in the literature to explain several experimental observations. In this work, we study the dynamics of the Tinkerbell map under -deformations. The system exhibits a rich variety of dynamical behavior as varies, including occurrences of interior crises, paired cascades, simultaneous occurrence of Neimark-Sacker and reverse Neimark-Sacker bifurcations, and co-existence of attractors and multistability. Numerical analysis reveals the existence of 3 limit cycles occurring simultaneously in a certain parameter regime. An appropriate choice of initial conditions enables one to choose a desired attractor for the system among other co-existing ones, thus switching the system between different dynamical states. We demonstrate the possibility of secure encryption and decryption of messages with the -deformed Tinkerbell map. The system’s sensitivity to the initial conditions and to the deformation parameter makes the cryptic message secure, and decrypting the original message difficult. We propose the use of the -deformed map as a novel method for transmission of messages securely.
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