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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Vibrations in a Growing Nonlinear Chain

Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 445--459 | DOI:10.5890/DNC.2021.09.008

S.A. Surulere , M.Y. Shatalov, A.V. Mkolesia, I.A. Fedotov

Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria, P/Bag X380, South Africa

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A one-dimensional chain describing the linear statistical increment of growing homogeneous atoms was arbitrarily built and investigated using an energy potential function. The analytic form of the considered potential has two exponential terms which describes chaotic behavior when the chain was excited. In order to investigate the dynamics of statistical attachment of individual atoms in the slender gold chain, the total energy of the entire system was changed by increasing the kinetic energy upon increment of homogeneous atoms in the chain. This resulted in a corresponding increase of the total energy in the system. On the other hand, the potential energy of the system on increment of homogeneous atoms equals zero, because the distance between corresponding atoms equals to the molecular distance (minimum potential distance). We considered the dynamical system with linear damping and without linear damping. \\ Different initial points were investigated to obtain trends of vibration that includes chaotic and regular oscillations. At some initial point(s), the attached atom experiences an infinite jump which means it falls off the nonlinear slender chain and the chain was broken. The interpretation of this phenomenon means the gold chain will result into an unstable nanostructure. We compared the numerical simulation of the system with different built-in ordinary differential equation solvers of various computer algebra software. Numerical simulation were carried out by plotting the system of growing atoms' displacement against time. The system of linearly attached atoms were numerically simulated and inferences were stated from the study. In all cases considered, we inferred that amplitude of oscillation significantly increased at the end of the chain (terminal point) as compared to the initial point the oscillation started.


  1. [1]  Rieth, M. (2003), { Nano-engineering in Science and Technology: An introduction to the world of Nano-design}, Vol.6, World Scientific Publishing Ltd: Singapore.
  2. [2]  Sanjay, S. and Pandey, A. (2017), A brief manifestation of Nanotechnology, { In EMR/ESR/EPR Spectroscopy for Characterization of Nanomaterials}, Springer, New Delhi, 47-63.
  3. [3]  Mitin, V.V., Sementsov, D.I., and Vagidov, N.Z. (2010), { Quantum Mechanics for Nanostructures}, Cambridge University Press: New York.
  4. [4]  Feynman, R.P. (1959), Theres plenty of room at the bottom, { Miniaturization}, 282-296.
  5. [5]  Surulere, S.A. (2018), { Investigation of the vibrations of linearly growing nanostructures}, Masters thesis, Tshwane University of Technology, South Africa.
  6. [6]  Boscovich, R.G. (1763), { Theory of natural philosophy, reduced to a single law of forces existing in nature}, { Remondiniana publisher}.
  7. [7]  Birajdar, S., Kadam, C., Shelke, R., and Behere, S. (2005), Potential function for diatomic molecules, { Indian Journal of Pure and Applied Physics}, {\bf 43}, 427-431.
  8. [8]  Kaplan, I.G. (2006), { Intermolecular Interactions: Physical Picture, Computational Methods and Model Potentials}, John Wiley \& Sons: New York.
  9. [9]  Rafii-Tabar, H. and Mansoori, G. (2004), Interatomic potential models for Nanostructures, In { Encyclopedia of Nanoscience and Nanotechnology, American Scientific Publishers}, {\bf 4}, 231-247.
  10. [10]  Varshni, Y.P. (1957), Comparative study of potential energy functions for diatomic molecules, { Reviews of Modern Physics}, {\bf 29}(4), 664.
  11. [11]  Zhang, G.D., Liu, J.Y., Zhang, L.H., Zhou, W., and Jia, C.S. (2012), Modified Rosen-Morse potential-energy model for diatomic molecules, { Physical Review A}, {\bf 86}(6), 062510.
  12. [12]  Morse, P.M. (1929), Diatomic molecules according to the wave mechanics. II. Vibrational levels, { Physical Review}, {\bf 34}(1), 57.
  13. [13]  Barakat, T., Abodayeh, K., and Al-Dossary, O. (2006), Exact solutions for vibrational levels of the Morse potential via the asymptotic iteration method, { Czechoslovak Journal of Physics}, {\bf 56}(6), 583-590.
  14. [14]  Toda, M. (2012), { Theory of nonlinear lattices}, Vol.20, Springer Verlag: New York.
  15. [15]  Surulere, S.A., Mkolesia, A.C., Shatalov, M.Y., and Fedotov, I. (2018), An investigation of vibrations of linearly growing discrete chain of atoms in Nano-structures, { International Journal of Applied Engineering Research}, {\bf 13}(18), 13596-13602.
  16. [16]  Surulere, S.A., Shatalov, M.Y., Mkolesia, A.C., Malange, T., and Adeniji, A.A. (2020), The integral-differential and integral approach for the exact solution of the hybrid functional forms for Morse potential, International Journal of Applied Mathematics - IAENG, 50(2), 242-250.
  17. [17]  Surulere, S.A., Malange, T., Shatalov, M.Y., and Mkolesia, A.C. (2020), Parameter estimation of potentials that are solutions of some second order ordinary differential equation, Discontinuity, Nonlinearity and Complexity, \underline{under review}.
  18. [18]  Surulere, S.A., Shatalov, M.Y., Mkolesia, A.C., and Fedotov, I. (2020), A modern approach for the Identification of the Classical and Modified Generalized Morse potential, { Nanoscience $\&$ Nanotechnology - Asia} {\bf 10}(2), 142-151.
  19. [19]  Surulere, S.A., Shatalov, M.Y., Mkolesia, A.C., and Adeniji, A.A. (2020), A comparative investigation of complex conjugate eigenvalues for Generalized Morse and Classical Lennard-Jones potential, { Nanoscience \& Nanotechnology - Asia}, {\bf 10}(3), 356-363.
  20. [20]  Olsson, P.A. (2010), Transverse resonant properties of strained gold nanowires, { Journal of Applied Physics}, {\bf 108}(3), 034318.
  21. [21]  Baruh, H. (1999), Analytical Dynamics, WCB/McGraw-Hill: Boston