Bifurcation and Chaos in the 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.

Implementing a Chaotic Cryptosystem by Performing Parallel Computing on Embedded Systems with Multiprocessors

“…All these tests were performed using 1000 series (sequences) of stream length = 1,000,000 bit. Table 1 lists the comparative results of the success percentages obtained with Tinkerbell map [ 73 , 95 , 96 ] using high precision of = 99 significant decimals. If the proportion of success is greater than , it can be concluded that the sequences pass the NIST tests, i.e., those are random sequences.…”

“…In this work we introduce a method to encrypt/decrypt color images applying profiling techniques and parallel computing, with the aim to improve the cryptosystem efficiency to be implemented on embedded systems with multi-cores that have limited computational resources. As an example the Tinkerbell map [ 73 , 95 , 96 ] is defined by ( 1 ), which remains in a chaotic regime for the ranges of its control parameters , , and , where and , and initial conditions and that correspond to points that are within the space of the chaotic attractor, such as e [ 96 ]. …”

Bifurcation, chaos analysis and control in a discrete-time predator–prey system

“…With the definition of snap-back repeller, we adopt the iterative method [36,37]. We first provide conditions under which the fixed point z = (u, v) is a snap-back repeller.…”

Periodicity and Chaos Amidst Twisting and Folding in Two-Dimensional Maps

“…The concept of snap-back repellers is not new, but it is still the subject of intensive study. Some works that appeared in the last ten years are [Chen et al, 2016, Peng, 2007, Ren et al, 2017, Salman et al, 2016, Shi and Yu, 2008, Yuan et al, 2011, Zhao and Li, 2016. Note that differentiability of the map is an essential condition for Theorem 1, but see Gardini and Tramontana [2010a,b] for results which do not require smoothness.…”