Tinkerbell map

Abstract: Shadowing is a method of backward error analysis that plays a important role in hyperbolic dynamics. In this paper, the shadowing by containment framework is revisited, including a new shadowing theorem. This new theorem has several advantages with respect to existing shadowing theorems: It does not require injectivity or differentiability, and its hypothesis can be easily verified using interval arithmetic. As an application of this new theorem, shadowing by containment is shown to be applicable to infinite length orbits and is used to provide a computer assisted proof of the presence of chaos in the well-known noninjective Tinkerbell map.

Abstract: In this paper, the dynamical behaviors of the Tinkerbell map are investigated in detail. Conditions for the existence of fold bifurcation, flip bifurcation and Hopf bifurcation are derived, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations include bifurcation diagrams in two- and three-dimensional spaces, phase portraits, and the maximum Lyapunov exponent and fractal dimension, as well as the distribution of dynamics in the parameter plane, which exhibit new and interesting dynamical behaviors. More specifically, this paper reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2, 7, 8, 9, 10, 13, 17, 19, 23, 26 and so on, and suddenly appearing or disappearing chaos, convergence of an invariant circle to a period-one orbit, symmetry-breaking of periodic orbits, interlocking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2, 10, 13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on, all in the Tinkerbell map. In particular, it is found that there is no obvious road from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invariant circle and then to transient chaos as the parameters are varied. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding of the Tinkerbell map is obtained.

Abstract: This paper proposes a novel pseudo-random bit generation algorithm combining Chebyshev polynomial and Tinkerbell map. We calculated the key space of the proposed scheme. The output zero-one bits are statistically tested with three packages: NIST, DIEHARD and ENT. The experimental results show that the data are uniformly distributed with sufficient enough statistical properties to disturb brute-force attacks.

Abstract: image encryption schemes for secure transmission and storage are increasingly needed for a number of applications like medical, military, satellite etc. In this paper, a novel image encryption algorithm based on Logistic and Tinkerbell map is proposed.The proposed method uses two 1-D Logistic maps with different keys and one 2-D Tinkerbell map. The chaotic sequence generated is mixed sequence from the and sequences of Tinkerbell map depending on the chaotic sequences of two logistic maps. The main advantage of such a scheme is complex chaotic behavior of the generated chaotic sequences. The security and performance of the proposed method is analyzed thoroughly by using key-sensitivity, key- space, statistical, entropy, differential and performance analysis. The proposed approach achieves the required level of security with only one round of encryption operation. Hence the proposed method is computationally efficient.