Control and stability on chaotic convection in porous media with time delayed fractional orders
- Khaled Moaddy^{1}Email author
https://doi.org/10.1186/s13662-017-1372-2
© The Author(s) 2017
Received: 5 January 2017
Accepted: 21 September 2017
Published: 4 October 2017
Abstract
In this paper, we study the effect of time delay and the scaled Rayleigh number on chaotic convection in porous media with fractional order. The stability analysis for different fractional-order cases is investigated and the effective chaotic range of the fractional order is determined by a general synchronization of nonidentical chaotic systems based on the active control technique. The numerical results demonstrate that the effect of various values of the scaled Rayleigh number R, time delay and fractional orders changes the chaotic convection behavior to limit cycles or stable system in porous media.
Keywords
1 Introduction
There is a great need to control or obtain accurate numerical results for chaotic convection as chaos theory plays an important role in industrial applications, particularly in chemical reactions, biological systems [1], information processing, secure communications [2], electronics [3, 4], and with memristors [5]. Recently, control and synchronization of fractional chaotic systems has raised up some problems [6–8]. The consistency of the improvement of models based on fractional order differential structure has had increased reputation in the research of dynamical systems [9–11]. Chaos synchronization plays an important role in secure communication, digital cryptography, image encryption, signal and control processing [12–19]. Much attention has been devoted to the search for better and more efficient methods for the control or determination of a solution, approximate or analytical, of chaotic systems [9, 10, 13, 20, 21]. Yu et al. studied the synchronization of three chaotic fractional-order Lorenz systems with bidirectional coupling [20]; Odibat et al. [21] investigated the chaos synchronization of two identical systems via linear control; Bhalekar and Daftardar-Gejji [22] demonstrated that two different fractional-order chaotic systems can be synchronized using active control. Zhen et al. [23] implemented the delayed fractional-order chaotic logistic system in a simple chaotic masking method to illustrate the security enhancement. Daftardaret et al. investigated the effect of delay on chaos in the fractional-order Chen system [24].
More recently, chaotic behavior in a fluid-saturated porous medium has attracted interest due to its wide application in such fields as geothermal energy utilization, oil reservoir modeling, catalytic packed beds filtration, thermal insulation and nuclear waste disposal [25]. It has been observed in many natural systems, such as the time evolution of the magnetic field of celestial bodies, molecular vibrations, the dynamics of satellite in the solar system, the weather, in ecology and in neurons [26, 27]. Moaddy et al. [28] studied the effect of the fractional-order chaotic behavior of nanofluids in a fluid layer heated from below; Jawdat et al. [29] investigated the effects of a uniform internal heat generation on chaotic behavior in thermal convection in a fluid-saturated porous layer subject to gravity and heated from below for low Prandtl number.
The stability analysis of fractional differential equations is very important according to the required application behavior. Each behavior is related to the location of the system’s poles with respect to the equilibrium points. In this paper we will study the eigenvalue problem for the fractional-order case for different values of Rayleigh number R, while the stability of integer-order case was studied in [29].
Motivated by the above discussions, in this paper we have four aims, where the first aim is to extend the work of Jawdat et al. [29] to study the effect of the fractional order on chaotic behavior in a fluid-saturated porous layer subject to gravity and heated from below for low Prandtl number. The second aim is to study the property of time delay with fractional-order range which exhibits chaotic behavior for the chaotic system of fractional orders. Several cases are investigated for different fractional orders changing only a single system parameter. Stable, periodic and chaotic responses are shown for each system parameter but with different fractional-order ranges. The third aim is to discuss the stability analysis of the fractional-order system for different order and different values of Rayleigh number R. The last aim is to investigate the synchronization of different fractional-order chaotic systems; the numerical solutions of the master, slave and error systems using Adams-Bashforth-Moulton predictor corrector algorithm are proposed.
2 The proposed system
3 Stability analysis
One of the methods to study the stability of fractional-order systems is the W-plane method [15] which can be used for different fractional orders inside the same system. There are two planes in the fractional-order case where the first is the so-called W-plane which has all poles either in physical or nonphysical planes. The second is the s-plane where the physical poles for different values of R in the s-plane will be introduced.
When \(m = 10\), the s-plane is represented by tenth, the W-plane as shown in Figures 4, 5, 6 (W-plane) by the outer dotted lines, while the inner dotted lines represent the equivalent \(\pm j\omega\) axis in the s-plane. Thus any poles between the inner and outer dotted lines are equivalent to the left half of the s-plane (stable poles). All possible roots in the W-plane in the case when \(R\in[0.1, 30]\) are illustrated in Figure 4 (s-plane) with step 0.001. However, Figure 4 (s-plane) shows only the poles in the physical s-plane through the transformation \(s = W^{10}\) for poles between the outer dotted lines in Figure 4 (W-plane). In addition, Figure 4 shows the pole locations versus the Rayleigh number R, when R spans the range \([0.1, 30]\). From this figure, the poles at \(R=30\) are located in the right-half plane producing unstable performance. However, as R decreases, the real pole goes into the negative direction entering the left half plane. The clear changes of the Rayleigh number R from \(G=0.1\) (weak heating) to \(G=10\) (strong heating) affect the stability at the same points as shown in Figures 5-6. Moreover, Figures 5 and 6 show the pole locations versus the Rayleigh number R, when R spans the range \([1, 26.3]\) and \([1,20]\), respectively.
4 Active control technique
Assume that we have two different chaotic systems, one of them is the master system and the other is the slave one. We need to change the response of the slave system to synchronize with the master chaotic system via active control functions. These functions affect only the slave system without making any loading on the master chaotic response.
Then the eigenvalues of the linear system (36) are equal \((-k, -k, -k)\), which is enough to satisfy the necessary and sufficient condition (37) for all fractional orders \({\alpha_{1}}, {\alpha _{2}}, {\alpha_{3}}<2\). In the following cases, we take \(k = 1\) for simplicity.
5 Results and discussion
In this section, we present some numerical simulations of the fractional chaotic system (4)-(6) for the time domain \(0 \leq t \leq40\). All calculations were done using Matlab with step size 0.001, fixing the values \(\delta=5, \gamma=0.5\) and taking the initial conditions \(X(0)=Y(0)=Z(0)=0.9\).
6 Conclusions
In this paper, the effect of time delay with fractional order in a fluid saturated porous layer subjected to gravity and heated from below under the effect of various values of the Rayleigh number R has been studied. Several cases of fractional derivatives are introduced with stability analysis. Static and dynamic synchronization has been obtained using the active control by changing the switching parameters. Numerical simulations allowed us to observe that the clear changes of that chaotic behavior get stabilized for some values of time delay, Rayleigh number and the fractional order changes.
Declarations
Acknowledgements
The author expresses his thanks to unknown referees for the careful reading and helpful comments.
Author’s contributions
All authors read and approved the final manuscript.
Competing interests
The author declares that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Moaddy, K, Radwan, A, Salama, K, Momani, S, Hashim, I: The fractional-order modeling and synchronization of electricall coupled neuron systems. Comput. Math. Appl. 64(10), 3329-3339 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Feki, M: An adaptive chaos synchronization cheme applied to secure communication. Chaos Solitons Fractals 18(1), 141-148 (2003) MathSciNetView ArticleMATHGoogle Scholar
- Radwan, A, Soliman, A, El-Sedeek, A: An inductorless cmos realization of Chua’s circuit. Chaos Solitons Fractals 18, 149-158 (2003) View ArticleGoogle Scholar
- Radwan, A, Soliman, A, El-Sedeek, A: Mos realization of the modified Lorenz chaotic system. Chaos Solitons Fractals 21, 553-561 (2004) View ArticleMATHGoogle Scholar
- Radwan, A, Moaddy, K, Momani, S: Stability and nonstandard finite difference method of the generalized Chua’s circuit. Comput. Math. Appl. 62, 961-970 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Pecora, L, Carrol, T: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821-824 (1990) MathSciNetView ArticleMATHGoogle Scholar
- Pecora, L, Carrol, T: Driving systems with chatic signals. Phys. Rev. A 44(4), 2374-2383 (1991) View ArticleGoogle Scholar
- Pinto, CMA: Strange dynamics in a fractional derivative of complex-order network of chaotic oscillators. Int. J. Bifurc. Chaos 25(1), 1550003 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Yan, J, Li, C: On chaos synchronization of fractional differential equations. Chaos Solitons Fractals 32(2), 725-735 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Chua, L, Makoto, I, Kocarev, L, Eckert, K: Chaos synchronization in chuas circuit. J. Circuits Syst. Comput. 3(1), 93-108 (1993) MathSciNetView ArticleGoogle Scholar
- Pinto, CMA, Carvalho, ARM: Strange patterns in one ring of Chen oscillators coupled to a buffer cell. J. Vib. Control 22, 3267-3295 (2014) MathSciNetView ArticleGoogle Scholar
- Sun, K, Sprott, J: Bifurcations of fractional-order diffusionless Lorenz system. Electron. J. Theor. Phys. 6(22), 123-134 (2009) Google Scholar
- Podlubny, I: Fractional Differential Equations. Academic Press, New York (1999) MATHGoogle Scholar
- Luo, C, Wang, X: Chaos generated from the fractional-order complex Chen system and its application to digital secure communication. Int. J. Mod. Phys. C 24, 1350025 (2013) MathSciNetView ArticleGoogle Scholar
- Muthukumar, P, Balasubramaniam, P, Ratnavelu, K: Fast projective synchronization of fractional order chaotic and reverse chaotic systems with its application to an affine cipher using date of birth (dob). Nonlinear Dyn. 80, 1883-1897 (2015) View ArticleMATHGoogle Scholar
- Zhang, F: Lag synchronization of complex Lorenz system with applications to communication. Entropy 17, 4974-4985 (2015) View ArticleGoogle Scholar
- Xu, F, Cressman, R, Shu, X-B, Liu, X: A series of new chaotic attractors via switched linear integer order and fractional order differential equations. Int. J. Bifurc. Chaos 25(1), 1550008 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Xu, F: Integer and fractional order multiwing chaotic attractors via the Chen system and the Lü system with switching control. Int. J. Bifurc. Chaos 24, 1450029 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Xu, F, Pei, Y, Xiaoxin, L: Synchronization and stabilization of multi-scroll integer and fractional order chaotic attractors generated using trigonometric functions. Int. J. Bifurc. Chaos 23, 1350145 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Yu, Y, Li, H, Su, Y: The synchronization of three chaotic fractional-order Lorenz systems with bidirectional coupling. Int. J. Bifurc. Chaos 20(1), 1-15 (2010) View ArticleGoogle Scholar
- Odibat, ZM, Corson, N, Aziz-Alaoui, MA, Bertellec, C: Synchronization of chaotic fractional-order systems via linear control. Int. J. Bifurc. Chaos 20, 81 (2010) MathSciNetView ArticleMATHGoogle Scholar
- Bhalekar, S, Daftardar-Gejji, V: Antisynchronization of nonidentical fractional-order chaotic systems using active control. Int. J. Differ. Equ. 2011 250763 (2011) MathSciNetMATHGoogle Scholar
- Zhen, W, Xia, H, Ning, L, Xiao-Na, S: Image encryption based on a delayed fractional-order chaotic logistic system. Chin. Phys. B 21(5), 050506 (2012) View ArticleGoogle Scholar
- Daftardar-Gejji, V, Bhalekar, S, Gade, P: Dynamics of fractional-ordered Chen system with delay. Pramana 79, 61-69 (2012) View ArticleGoogle Scholar
- Sheu, L, Tam, L, Chen, J, Chen, H, Lin, K, Kang, Y: Chaotic convection of viscoelastic fluids in porous media. Chaos Solitons Fractals 37, 113-124 (2008) MathSciNetView ArticleMATHGoogle Scholar
- Vadasz, P, Olek, S: Weak turbulence and chaos for low Prandtl number gravity driven convection in porous media. Transp. Porous Media 37, 69-91 (1999) MathSciNetView ArticleGoogle Scholar
- Vadasz, P, Olek, S: Transition and chaos for free convection in a rotating porous layer. Int. J. Heat Mass Transf. 41, 1417-1435 (1974) View ArticleMATHGoogle Scholar
- Moaddy, K, Radwan, A, Hashim, I, Jawdat, J: Bifurcation behaviourand control on chaotic convection of nanofluids with fractional-orders. In: Fujita, H, Tuba, M (eds.) Recent Advances in Mathematical Methods and Computational Techniques in Modern Science, 23-25 April 2013; Japan, pp. 63-73 (2013). WESEAS Google Scholar
- Jawdat, J, Hashim, I: Low Prandtl number chaotic convection in porous media with uniform internal heat generation. Int. Commun. Heat Mass Transf. 37, 629-636 (2010) View ArticleGoogle Scholar
- Lorenz, E: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130-142 (1963) View ArticleGoogle Scholar
- Sparrow, C: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, New York (1982) View ArticleMATHGoogle Scholar
- Diethelm, K, Ford, N: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229-248 (2002) MathSciNetView ArticleMATHGoogle Scholar
- Diethelm, K, Ford, N, Freed, A: Detailed error analysis for a fractional Adams method. Numer. Algorithms 26, 31-52 (2006) MathSciNetMATHGoogle Scholar