Generation of fractals as Duffing equation orbits

Akhmet, Marat
Alejaily, Ejaily Milad
Dynamics are constructed for fractals utilizing the motion associated with Duffing equation. Using the paradigm of Fatou-Julia iteration, we develop iterations to map fractals accompanied with a criterion to ensure that the image is again a fractal. Because of the close link between mappings, differential equations and dynamical systems, one can introduce dynamics for fractals through differential equations such that they become points of the solution trajectory. There is no doubt that the differential equations have a distinct role for studying chaos. Therefore, characterization of fractals as trajectory points is an important step toward a better understanding of the link between chaos and fractal geometry. Moreover, it would be helpful to enhance and widen the scope of their applications in physics and engineering.


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We present analytically the exact energy bound-states solutions of the Schrodinger equation in D dimensions for a recently proposed modified Kratzer plus ring-shaped potential by means of the Nikiforov-Uvarov method. We obtain an explicit solution of the wave functions in terms of hyper-geometric functions (Laguerre polynomials). The results obtained in this work are more general and true for any dimension which can be reduced to the well-known three-dimensional forms given by other works.
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The effective mass Klein-Gordon equation in one dimension for the Woods-Saxon potential is solved by using the Nikiforov-Uvarov method. Energy eigenvalues and the corresponding eigenfunctions are computed. Results are also given for the constant mass case.
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Utilizing an appropriate ansatz to the wave function, we reproduce the exact bound-state solutions of the radial Schrodinger equation to various exactly solvable sextic an-harmonic oscillator and confining perturbed Coulomb models in D-dimensions. We show that the perturbed Coulomb problem with eigenvalue E can be transformed to a sextic anharmonic oscillator problem with eigenvalue P. We also check the explicit relevance of these two related problems in higher-space dimensions. It is shown that exact solut...
Citation Formats
M. Akhmet and E. M. Alejaily, “Generation of fractals as Duffing equation orbits,” CHAOS, pp. 0–0, 2019, Accessed: 00, 2020. [Online]. Available: