The q-deformed Tinkerbell map

Iyengar, Sudharsana V and Balakrishnan, Janaki (2018) The q-deformed Tinkerbell map. Chaos, 28. p. 113102.

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Abstract: 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 q-deformed Tinkerbell map and show how one can fruitfully exploit some of its properties. As the deformation parameter q changes, the map undergoes Neimark-Sacker (NS) bifurcation, which creates limit cycles. In certain regimes, multiple limit cycles (up to three in the parameter range we studied) belonging to different basins of attractions can co-exist. The presence of the deformation parameter q makes the system rich in its dynamical behavior, such as occurrences of interior crises, paired cascades, co-existence of attractors, multistability, and simultaneous occurrence of Neimark-Sacker and reverse Neimark-Sacker bifurcation, etc. Improvements in computational resources and methods have also made it easier for encoded messages being hacked into and the original message extracted by unauthorized third parties. This has also led to the creation of a large body of literature on the use of chaotic regimes of dynamical systems, both flows and maps, for facilitating secure transmission of messages.1–5 See also Refs. 1–5 and references therein. Here, we propose the use of the q-deformed Tinkerbell map for the secure encryption of messages. The presence of the deformation parameter q brings in a further degree of complexity into the system and makes encryption of a message more secure. The system is prepared in a particular initial condition and with a chosen set of parameters, and the attractor in the phase space is divided into different regions that correspond to various alphabets of a language, special characters, and numerals. The system is then allowed to evolve in time. The message is encrypted as the system traverses different regions of the phase space. Decryption is possible only when not only the initial condition and the parameter values are known, but also when the exact value of the deformation parameter is chosen. Even a slight difference in the deformation parameter or initial condition lands the system on a different attractor and renders decryption impossible.
Item Type: Journal Paper
Subjects: School of Natural and Engineering Sciences > Complex Systems
Programmes > Complex Systems Programme
Divisions: Schools > Natural Sciences and Engineering
Date Deposited: 14 Nov 2018 11:17
Last Modified: 22 Dec 2020 15:11
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    Funders: S.V. Iyengar was supported by CSIR (Council for Scientific and Industrial Research, New Delhi) during a part of this work.
    Projects: UNSPECIFIED

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