A quasi-periodic gravity modulation to suppress chaos in a Lorenz system
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A quasi-periodic gravity modulation to suppress chaos in a Lorenz system Youssef Joundy1 · Hamza Rouah1
· Ahmed Taik1,2
Received: 22 April 2020 / Revised: 25 July 2020 / Accepted: 20 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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. Keywords Chaos · Gravitational modulation · Lyapunov exponents · Runge–Kutta method
1 Introduction We consider in this work a horizontal layer of a Newtonian fluid subjected to a temperature gradient ∇θ parallel to the gravitational acceleration g. There are several works that have been devoted to study the effect of periodic gravitational modulation on convective instability in a fluid layer. Among them, we begin with Gresho and Sani [1] who studied the effect of a gravitational modulation on the stability of a horizontal layer of fluid heated from above or below. The authors demonstrated that gravity modulation can affect the stability limits of a heated fluid layer. Christov and Homsy [2] have shown that the modulation of gravity leads to a parametric instability of the plane parallel to the one-dimensional solutions, for
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Hamza Rouah [email protected] Youssef Joundy [email protected] Ahmed Taik [email protected]; [email protected]
1
Laboratory of Mathematics and Applications, FST-Mohammadia, University Hassan II of Casablanca, PO Box 146, 20650 Mohammadia, Morocco
2
Modeling Simulation and Data Analysis, Mohammed VI Polytehnic University (UM6P), Ben Guerir, Morocco
a modest number of Prandtl a Floquet bifurcation in subharmonic or isochronous mode takes place, however for a large number of Prandtl, the only type of bifurcation that the authors found is the isochronous mode (for more details see [2]). Boulal et al. [3] investigated the effect of quasiperiodic gravitational modulation on the stability of a heated fluid layer. The authors showed that the modulation with two incommensurate frequencies (i.e their ratio is an irrational number) has a stabilizing or a destabilizing effect depending on the frequencies ratio, therefore this parameter plays an important role in the control of the convection threshold. Govender [4] studied the effect of gravitational modulation on convection in a porous homogene
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