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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 q -deformations. The system exhibits a rich variety of dynamical behavior as q 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 q -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 q -deformed map as a novel method for transmission of messages securely.

We study the dynamics of a discrete-time tritrophic model which mimics the observed periodicity in the population cycles of the larch budmoth insect which causes widespread defoliation of larch forests at high altitudes periodically. Our model employs q-deformation of numbers to model the system comprising the budmoth, one or more parasitoid species, and larch trees. Incorporating climate parameters, we introduce additional parasitoid species and show that their introduction increases the periodicity of the budmoth cycles as observed experimentally. The presence of these additional species also produces other interesting dynamical effects such as periodic bursting and oscillation quenching via oscillation death, amplitude death, and partial oscillation death which are also seen in nature. We suggest that introducing additional parasitoid species provides an alternative explanation for the collapse of the nine year budmoth outbreak cycles observed in the Swiss Alps after 1981. A detailed exploration of the parameter space of the system is performed with movies of bifurcation diagrams which enable variation of two parameters at a time. Limit cycles emerge through a Neimark-Sacker bifurcation with respect to all parameters in all the five and higher dimensional models we have studied.