MULTI-SCALE ANALYSIS of APPLIED DYNAMICAL SYSTEMS with PERIODIC COEFFICIENTS

The project belongs to an active interdisciplinary research direction - Vibrodynamics, which has been developed by the PI since 2003-4. We are going to apply the methods of Vibrodynamics to the studies of Applied Dynamical Systems (ADS). Surprisingly, the ADS with periodic coefficients (ADS-PC) have escaped almost any attention of researchers. At the same time in many practical situations, say, under the periodic changing of external condition, the coefficients in the equations must be periodic. Vibrodynamics provides an efficient method of studying of ADS-PC with the use of the multi-scale approach, averaging method, and asymptotic analysis. The main point of Vibrodynamics is the creating of an algorithm for the choosing of the slow time-scale, which is different for different equations and situations. In this project we apply Vibrodynamics to various ADS-PC. Our approach is conceptually new for many classes of ADS-PC. We will derive the averaged forms of ADS-PC including the Hamiltonian, Lagrangian equations, etc. As our main examples we will target the most popular ADS such as the equations of Classical Mechanics (vibrational transport), the Predator-Prey equations, Lorenz equations, and equations of astrophysics (including motions of planets, moons, and cosmic dust). In Astrophysics we will concentrate at the descriptions of motions in the fields of rotating dipoles and quadruples, which can have great applications. The obtained new equations will be solved numerically and results will be compared with the available experimental situations. Another direction of our activity will be a broad search among the available ADS, where we will introduce realistically justified periodic coefficients with the aim of obtaining new results in Physics, Chemistry, Engineering, and Biology.

The project belongs to an active interdisciplinary research direction - Vibrodynamics, which has been developed by the PI since 2003-4. We are going to apply the methods of Vibrodynamics to the studies of Applied Dynamical Systems (ADS). Surprisingly, the ADS with periodic coefficients (ADS-PC) have escaped almost any attention of researchers. At the same time in many practical situations, say, under the periodic changing of external condition, the coefficients in the equations must be periodic. Vibrodynamics provides an efficient method of studying of ADS-PC with the use of the multi-scale approach, averaging method, and asymptotic analysis. The main point of Vibrodynamics is the creating of an algorithm for the choosing of the slow time-scale, which is different for different equations and situations. In this project we apply Vibrodynamics to various ADS-PC. Our approach is conceptually new for many classes of ADS-PC. We will derive the averaged forms of ADS-PC including the Hamiltonian, Lagrangian equations, etc. As our main examples we will target the most popular ADS such as the equations of Classical Mechanics (vibrational transport), the Predator-Prey equations, Lorenz equations, and equations of astrophysics (including motions of planets, moons, and cosmic dust). In Astrophysics we will concentrate at the descriptions of motions in the fields of rotating dipoles and quadruples, which can have great applications. The obtained new equations will be solved numerically and results will be compared with the available experimental situations. Another direction of our activity will be a broad search among the available ADS, where we will introduce realistically justified periodic coefficients with the aim of obtaining new results in Physics, Chemistry, Engineering, and Biology.

Studying of applied ordinary differential equations (ODEs) with time-oscillating coefficients by the averaging methods has already been a central topic of research for more than 50 years. Historically, the major source of such studies was Astronomy, where the motions of stars, planets, moons, asteroids, and particles of cosmic dust are often disturbed by various periodic forces, such as periodic motions of stars or planets. Those important studies initiated the creation of the asymptotic analysis and, in particular, the celebrated Bogoliubov-Mitropolsky averaging method [1-6]. This method allows to produce the averaged form of equations for many known ODEs or for systems of ODEs. In addition, it provides a mathematical justification of the averaged solutions, which is valid on a finite time-interval. However, this method is very cumbersome and rather advanced analytically; as the first step it requires the transformation of original equations into a normal form. This transformation is usually difficult to perform and the related normal variables usually lose the physical meaning of original variables. These high technical features make the Bogolubov-Mitropolsky method accessible manly for the gurus of asymptotic methods and not popular for the ordinary users. However, over the last 10-20 years, this situation has been resolved by the rapid developing and using of multi-scale methods [1-6]. A recent and very successful and powerful approach adapting both these methods is known as Vibrodynamics [6-13]. Other important advances that could be connected to Vibrodynamics has been published by the CIs of this proposal in [14-18]. There are many dynamical systems in Physics, Chemistry, Biology, and Engineering [23-28], which requires to be considered by the methods of Vibrodynamics. References1. Kevorkian, J. and Cole, J.D. Multiple Scale and Singular Perturbation Method. Applied Mathematical Sciences, 114: Springer, 1996.2. Nayfeh, AH Perturbation Methods. NY: John Wiley and Sons. 1973.3. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I. Mathematical Aspects of Classical and Celestial Mechanics. Springer-Verlag, 1997.4. Blekhman, I.I. Vibrational Mechanics. World Scientific, 2000.5. Blekhman, I.I. (Editor) Selected Topics in Vibrational Mechanics. World Scientific, 2004.6. Vladimirov, V. A. (2016), Two-Timing Hypothesis, Distinguished Limits, Drifts, and Pseudo-Diffusion for Oscillating Flows. Studies in Applied Mathematics. doi:10.1111/ sapm.121527. Vladimirov, V.A. (2016) Stokes Drift in Dynamical Systems. E-print: arXiv:1622.03980 [physics.flu-dyn], Cornell University, USA.8. Vladimirov V.A. Vibrodynamics of pendulum and submerged solid. J. of Math. Fluid Mech. 7: S397-412, 2005.9. Vladimirov V.A. Viscous flows in a half-space caused by tangential vibrations on its boundary. Studies in Appl. Math., 121(4): 337-367, 2008.10. Vladimirov V.A. Admixture and drift in oscillating fluid flows. ArXiv: 1009, 4085v1. (physics,flu-dyn), 201011. Vladimirov V.A. MHD-drift equation: from Langmuir circulations to MHD dynamo? J.Fluid Mech. 698: 51-61, 201212. Vladimirov V.A. and Ilin K.I. An asymptotic model in acoustics: acoustic drift equations. The Journal of the Acoustical Society of America, 134(5): 3419-3424, 2013.13. Vladimirov, V.A., Nasser Al-Salti, N. (2016) Many Faces of Boussinesq Approximations.14. Goldstein R.E. and Proctor MRE (2017) Elastohydrodynamic sinchtonization of adjacent beating flagella. Phys. Rev. Fluids, in press.15. Teed, R.J. and Proctor MRE. (2016) Destruction of large-scale magnetic field in nonlinear oscillations. Month. Not. Roy. Astr. Soc. 458, 2885-2889.16. Mak, J., Griffiths, S.D. and Hughes, D.W. 2016 Shear flow instabilities in shallow-water magnetohydrodynamics. J. Fluid Mech. 788, 767-796.17. Guervilly, C., Hughes, D.W. and Jones, C.A. 2017 Large-scale-vortex dynamos in planar rotating convection. J. Fluid Mech. 81, 333-360.18. Elmojtaba, I.M. (2017) Modelling Disease Transmission in a Mixed-Species Grazing Environment. Communications in Mathematical Biology and Neuroscience, Article IG7.19. Elmojtaba, I.M. (2016) Mathematical model for the dynamics of visceral leishmaniosis-malaria co-infection. Mathematical Methods in Applied Sciences, DOI:10.1002/mma.3864.20. Elmojtaba, I.M. (2016) Using Adomian Decomposition Method for Solving a Vector-Host Model. International Journal of Applied Mathematical Research, 5(2): 107-109.21. M. Koca, M. Al-Ajmi, N. Koca, (2013) Quaterionic construction of W(F4) polytopes with their dual polytopes and branching under the subgroups W(B4) AND W(B3) ?W(A1), International Journal of Geometric Methods in Modern Physics, Vol. 10, No. 5 1350010 (24 pages), DOI: 22. G Lugones, AG Grunfeld, M Al Ajmi, (2016) Surface tension and curvature energy of quark matter in the Nambu?Jona-Lasinio model, Physical Review C 88 (4), 045803.23. Murray, J.D. Mathematical Biology. Springer-Verlag, 1991.24. Ortoleva, P.J. Nonlinear Chemical Waves. John Wiley and Sons,1992.25. Hayashi, C. Nonlinear Oscillations in Physical Systems, Princeton University Press, 1964.26. Sachdev, P.L. Nonlinear Diffusive Waves. Cambridge University Press. 1987.Large-Scale Motions in the Universe.Rubin, V.C. and Coyne, G.V. Princeton University Press. 1988.27. Courvoisier A, Hughes DW and Proctor MRE Self-consistent mean-field magnetohydrodynamics. Proc. Roy. Soc. A, 466: 583-601, 2010. E-print: arXiv: 1611.03980 [physics.flu-dyn], Cornell University, USA.28. Vladimirov, V.A. Peake N. (2016) Fluid Flows driven by Oscillating Body Force. E-print: arXiv:1608.06195 [physics.flu-dyn], Cornell University, USA.