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A quasi-periodic gravity modulation to suppress chaos in a Lorenz system

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Abstract

This work is devoted to study the influence of quasi-periodic gravitational modulation on suppress chaos in a two-dimensional rectangular Newtonian fluid heated from below. The model consists of the heat equation coupled with the Navier–Stokes equation under the Boussinesq approximation. The problem is reduced to an autonomous system of ordinary differential equations by using the approximation on some Fourier modes. We solved these system by using the fourth-order Runge–Kutta method. A transition from periodic oscillatory convection to chaotic convection is identified for certain value of Rayleigh number and shape parameter in 3D Lorenz model. The results also show that the chaos can be suppressed by applied to a medium a quasi-periodic gravitational modulation for high and low Prandtl number in both Lorenz models 3D and 5D.

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References

  1. Gresho PM, Sani RL (1970) The effects of gravity modulation on the stability of a heated fluid layer. J Fluid Mech 40:783. https://doi.org/10.1017/S0022112070000447

    Article  MATH  Google Scholar 

  2. Christov CI, Homsy GM (2001) Nonlinear dynamics of two-dimensional convection in a vertically stratified slot with and without gravity modulation. J Fluid Mech 430:335–360. https://doi.org/10.1017/S0022112000002986

    Article  MATH  Google Scholar 

  3. Boulal T, Aniss S, Belhaq M, Rand R (2007) Effect of quasiperiodic gravitational modulation on the stability of a heated fluid layer. Phys Rev E 76(52):56320. https://doi.org/10.1103/PhysRevE.76.056320

    Article  Google Scholar 

  4. Govender S (2004) Stability of convection in a gravity modulated porous layer heated from below. Transp Porous Media 57:113. https://doi.org/10.1023/B:TIPM.0000032739.39927.af

    Article  Google Scholar 

  5. Govender S (2005) Linear stability and convection in a gravity modulated porous layer heated from below-transition from synchronous to subharmonic solutions. Transp Porous Media 59(2):227–238. https://doi.org/10.1007/s11242-004-1369-7

    Article  MathSciNet  Google Scholar 

  6. Vadasz JJ, Meyer JP, Govender S (2013) Vibration efects on weak turbulent natural convection in a porous layer heated from below. Int Commun Heat Mass Transf 45:100–110. https://doi.org/10.1016/j.icheatmasstransfer.2013.04.012

    Article  Google Scholar 

  7. Vadasz JJ, Meyer JP, Govender S (2014) Chaotic and periodic natural convection for moderate and high Prandtl numbers in a porous layer subject to vibrations. Transp Porous Media 103(2):279–294. https://doi.org/10.1007/s11242-014-0301-z

    Article  MathSciNet  Google Scholar 

  8. Allali K (2018) Suppression of chaos in porous media convection under multifrequency gravitational modulation. Adv Math Phys. https://doi.org/10.1155/2018/1764182

    Article  MathSciNet  MATH  Google Scholar 

  9. Malashetty MS, Swamy MS (2011) Effect of gravity modulation on the onset of thermal convection in rotating fluid and porous layer. Phys Fluids 23(6):064108. https://doi.org/10.1063/1.3593468

    Article  MATH  Google Scholar 

  10. Siddheshwar PG, Sekhar GN, Jayalatha G (2010) Effect of time-periodic vertical oscillations of the Rayleigh–Bénard system on non-linear convection in viscoelastic liquids. J Non-Newton Fluid Mech 165:1412–1418. https://doi.org/10.1016/j.jnnfm.2010.07.008

    Article  MATH  Google Scholar 

  11. Sara F-N, Shen BW (2018) Quasi-periodic orbits in the five-dimensional nondissipative Lorenz model: the role of the extended nonlinear feedback loop. Int J Bifurc Chaos 28(06):1850072. https://doi.org/10.1142/S0218127418500724

    Article  MathSciNet  MATH  Google Scholar 

  12. Allali K, Joundy Y, Taik A, Volpert V (2017) Influence of natural convection on the heat explosion in porous media. Combust Explos Shock Waves 53(2):134–139. https://doi.org/10.1134/S0010508217020022

    Article  Google Scholar 

  13. Wang Y, Singer J, Bau HH (1992) Controlling chaos in a thermal convection loop. J Fluid Mech 237:479–498. https://doi.org/10.1017/S0022112092003501

    Article  MathSciNet  Google Scholar 

  14. Allali K, Joundy Y, Taik A, Volpert V (2020) Dynamics of convective thermal explosion in porous media. Int J Bifurc Chaos 30(06):2050081. https://doi.org/10.1142/S0218127420500819

    Article  MathSciNet  MATH  Google Scholar 

  15. Smith JM, Cohen RJ (1984) Simple finite-element model accounts for wide range of cardiac dysrhythmias. Proc Natl Acad Sci 81(1):233–237. https://doi.org/10.1073/pnas.81.1.233

    Article  Google Scholar 

  16. Garfinkel A, Spano ML, Ditto WL, Weiss JN (1992) Controlling cardiac chaos. Science 257:12305. https://doi.org/10.1126/science.1519060

    Article  Google Scholar 

  17. Babloyantz A, Destexhe A (1986) Low-dimensional chaos in an instance of epilepsy. Proc Natl Acad Sci 83(10):3513–3517. https://doi.org/10.1073/pnas.83.10.3513

    Article  Google Scholar 

  18. Iasemidis LD, Sackellares JC (1996) Review: chaos theory and epilepsy. The Neuroscientist 2(2):118–126. https://doi.org/10.1177/107385849600200213

    Article  Google Scholar 

  19. Aschenbach B (2004) Measuring mass and angular momentum of black holes with high-frequency quasi-periodic oscillations. Astron Astrophys 425(3):1075–1082. https://doi.org/10.1051/0004-6361:20041412

    Article  Google Scholar 

  20. Ringot J, Szriftgiser P, Garreau JC, Delande D (2000) Experimental evidence of dynamical localization and delocalization in a quasiperiodic driven system. Phys Rev Lett 85(13):2741. https://doi.org/10.1103/PhysRevLett.85.2741

    Article  Google Scholar 

  21. Saltzman B (1962) Finite amplitude free convection as an initial value problem. J Atmos Sci 19:329–341. https://doi.org/10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2

    Article  Google Scholar 

  22. Vadasz P, Olek S (1998) Transitions and chaos for free convection in a rotating porous layer. Int J Heat Mass Transf 41(11):1417–1435. https://doi.org/10.1016/S0017-9310(97)00265-2

    Article  MATH  Google Scholar 

  23. Roy D, Musielak ZE (2007) Generalized Lorenz models and their routes to chaos. II. Energy-conserving horizontal mode truncations. Chaos Solitons Fractals 31(3):747–756. https://doi.org/10.1016/j.chaos.2006.03.082

    Article  MathSciNet  MATH  Google Scholar 

  24. Lorenz EN (1963) Deterministic nonperiodic flow. J Atmos Sci 20(2):130–141. https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2

    Article  MathSciNet  MATH  Google Scholar 

  25. Grebogi C, Ott E, Yorke JA (1983) Crises, sudden changes in chaotic attractors, and transient chaos. Physica D 7:181–200. https://doi.org/10.1016/0167-2789(83)90126-4

    Article  MathSciNet  MATH  Google Scholar 

  26. Shen BW (2014) Nonlinear feedback in a five-dimensional Lorenz model. J Atmos Sci 71:1701–1723. https://doi.org/10.1175/JAS-D-13-0223.1

    Article  Google Scholar 

  27. Roy D, Musielak ZE (2007) Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations. Chaos Solitons Fractals 32:1038–1052. https://doi.org/10.1016/j.chaos.2006.02.013

    Article  MathSciNet  MATH  Google Scholar 

  28. Felcio CC, Rech PC (2018) On the dynamics of five- and six-dimensional Lorenz models. J Phys Commun 2:025028. https://doi.org/10.1088/2399-6528/aaa955

    Article  Google Scholar 

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Authors and Affiliations

  1. Laboratory of Mathematics and Applications, FST-Mohammadia, University Hassan II of Casablanca, PO Box 146, 20650, Mohammadia, Morocco

    Youssef Joundy, Hamza Rouah & Ahmed Taik

  2. Modeling Simulation and Data Analysis, Mohammed VI Polytehnic University (UM6P), Ben Guerir, Morocco

    Ahmed Taik

Authors
  1. Youssef Joundy
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  2. Hamza Rouah
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  3. Ahmed Taik
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Correspondence to Hamza Rouah.

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Joundy, Y., Rouah, H. & Taik, A. A quasi-periodic gravity modulation to suppress chaos in a Lorenz system. Int. J. Dynam. Control 9, 475–493 (2021). https://doi.org/10.1007/s40435-020-00679-y

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  • DOI: https://doi.org/10.1007/s40435-020-00679-y

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