Abstract. A 3-D **novel** **double**-**convection** **chaotic** system with three nonlinearities is proposed in this research work. The dynamical properties of the new **chaotic** system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, stability analysis of equilibria, etc. **Adaptive** **control** and synchronization of the new **chaotic** system with unknown parameters are achieved via nonlinear controllers and the results are established using Lyapunov stability theory. Furthermore, an electronic **circuit** realization of the new 3-D **novel** **chaotic** system is presented in detail. Finally, the **circuit** experimental results of the 3-D **novel** **chaotic** **attractor** show agreement with the numerical simulations.

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For practical implementation of **chaotic** systems, it is important to design suitable electronic **circuit** design of **chaotic** systems [31-35]. Sambas et al. [31] discussed the **circuit** design of a six-term **novel** **chaotic** system with hidden **attractor**. Sambas et al. [32] derived a **circuit** design for a new 4-D **chaotic** system with hidden **attractor**. Sambas et al. [33] discussed the numerical **simulation** and **circuit** implementation for a Sprott **chaotic** system with one hyperbolic sinusoidal nonlinearity. Vaidyanathan et al. [34] presented a new 4-D **chaotic** hyperjerk system, and discussed **its** synchronization, **circuit** design and applications in RNG, image encryption and chaos-based steganography. Vaidyanathan et al. [35] reported a new **chaotic** **attractor** with two quadratic nonlinearities and discussed **its** synchronization via **adaptive** **control** and **circuit** implementation.

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Abstract—This paper reports the finding of a new 2-scroll **chaotic** system with four quadratic nonlinear terms. We also discover interesting properties of the new **chaotic** system such as equilibrium points, bifurcation and Lyapunov exponents. For a comparative study, we apply four different **control** methods for the global synchronization of the new **chaotic** system with itself and present a comparative analysis of the results obtained with the four **control** methods, viz. (A) **Adaptive** **Control**, (B) Backstepping **Control**, (C) Passive **Control** and (D) Integral Sliding Mode **Control**. Our results show that integral sliding mode **control** gives the best results for achieving synchronization of the new **chaotic** system with itself. An electronic **circuit** **simulation** of the new 2-scroll **chaotic** system is shown using MultiSIM to check the feasibility of the model.

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This work described a new two-scroll **chaotic** system with three quadratic nonlinearities. First, the qualitative properties of the new two-scroll **chaotic** system are detailed. Dynamical behaviors of the new two-scroll **chaotic** system with three quadratic nonlinearities are investigated through equilibrium points, projections of **chaotic** attractors, Lyapunov exponents and Kaplan–Yorke dimension. In addition, the **adaptive** **control** scheme of the new two-scroll **chaotic** system is shown via **adaptive** **control** approach. Furthermore, an electronic **circuit** realization of the new two-scroll **chaotic** system using the electronic **simulation** package MultiSIM confirmed the feasibility of the theoretical model.

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. These ac- curacy issues where inevitable due to the limited accuracy of floating point storage. In **chaotic** time series, rounding errors be- come magnified, and this is likely to result in different Lyapunov values in different machines or implementations. The Lyapunov values obtained will always be approximate as this observation is supported in literature. The table 4 below is a subset of the data generated on one of those computers. According to the table above, the model experiences Chaos for values of sensitivity(β) equals to 2.4 and 2.3 . At these values, the rate of divergence(λ) is approximately equal to 0.601171 and 0.472365 respectively. The time series observed at certain values of β

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We must point out that the technical part of the successful use of this method is Re- mark 1 (i.e., Lemma 5 in [32] or Lemma 7 in [33]) which is indeed questionable. In this paper, we proposed some corrections and veriﬁed that the revised theory can still work in **control** of **chaotic** systems. Our contributions are as follows:

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On another research frontier, **adaptive** **control** method [] is an eﬀective way to es- timate the unknown parameters due to **its** advantages on witnessed rapid and impres- sive developments leading to global stability and tracking results for nonlinear systems. It has been successfully applied to synchronize **chaotic** systems with unknown parame- ters, and many important results have been presented. For example, Park studied adap- tive synchronization of a uniﬁed **chaotic** systems with an unknown parameter [, ]. Zhang et al. proposed the **adaptive** controllers and **adaptive** laws to synchronize two dif- ferent **chaotic** systems with unknown parameters []. In [], the **adaptive** complete syn- chronization between **chaotic** systems with fully uncertain parameters were realized. Li et al. gave a deeply research on **adaptive** impulsive synchronization for fractional-order **chaotic** systems with unknown parameters []. In [], **adaptive** synchronization of two diﬀerent **chaotic** systems was addressed by considering the time varying unknown param- eters. **Adaptive** added-order and reduced-order anti-synchronization of **chaotic** systems were investigated in [, ], respectively. He et al. made a thorough inquiry about syn- chronization of hyperchaotic systems with multiple unknown parameters []. Zhao et al. presented a discussion of chaos synchronization between the coupled systems on net- work with unknown parameters based on **adaptive** **control** method []. Liu developed **adaptive** anti-synchronization of **chaotic** complex nonlinear systems with unknown pa- rameters []. Wu and Yang achieved the **adaptive** synchronization of coupled nonidenti- cal **chaotic** systems with complex variables and stochastic perturbations []. However, all of these works only deal with the synchronization problems between two **chaotic** systems with unknown parameters. Up to now, no related results have been established for the synchronization of multiple **chaotic** systems with unknown parameters, which is another motivation of this paper.

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In this case, the **chaotic** **control** methods designed by using neural networks have made some achievements [5-12]. On the other hand, the studies in [13] show that the neural network with the orthogonal polynomial function has global approximation properties for approaching continuous function on any compact set with arbitrary precision. Particularly, when the orthogonal polynomial function is taken as Chebyshev polynomial, the performance of the designed neural networks is optimal. The reason is that the connection weights of Chebyshev neural networks (CNNs) is determined by the unidirectional gradient method, which is easy to make the objective function into local optimal impacting the efficiency of such neural network. Additionally, the particle swarm optimization (POS) adopts the speed-displacement search model, where the computational complexity is low, and the optimal solution is obtained by the cooperation and competition between particles. In this sense, the weights of the neural networks (NNs) are trained by using POS, which can give full play to the global optimization capability and rapid local convergence advantages for the PSO. Moreover, the PSO algorithm can also improve the generalization and learning capability of neural network [14]. These advantages of the PSO algorithm and CNNs motivate us to develop a **control** approach based on the PSO and CNNs for **chaotic** systems. Furthermore, to be best of the author’s knowledge, few results have been reported on this issue.

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Abstract The **attractor**-repeller pairs are binary systems which are stable systems in termo-dynamic equilibrium, and one of the leaders we have in Nature to understand Tornadoes. A mathematical model of a tornado is presented, within a model of chaos theory, where two complementary fractals are combined to understand this natural phenomenon. It is a thermodynamic state where the wind formed by warm air rises (the repeller) while the swirling cold (**attractor**) wind descends joining together and creating a tornado, which is an equilibrium system. The mathematical modeling we present here is based on algorithms and it has been performed with Matlab code.

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Motivated by the above analysis, we aim to present the stability analysis of a FOCH **chaotic** system with unknown external disturbances by using the backstepping technique. Firstly, the system uncertainties are approximated by NNs; secondly, based on the frac- tional Lyapunov stability theory, a fractional adaptation law that ensures that the states of a system converge to a small region of zero is designed; ﬁnally, the NN backstepping controller is implemented step by step. The main contribution of the proposed method in- cludes the following: (1) **Adaptive** NN **control** together with backstepping **control** method is proposed for fractional-order **chaotic** systems, and the prior knowledge of the system model is not needed. (2) In each step, a fractional-order signal is constructed to cancel the approximation error of the unknown nonlinear function. That is to say, the aforemen- tioned problem of [15] is solved in this paper.

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In this paper, a two-dimensional **double** diffusive natural **convection** in a porous cavity filled with viscoplastic fluids is sim- ulated. The dimensional and non-dimensional macroscopic equations are presented, employing the Papanastasiou model for viscoplastic fluids and the Darcy–Brinkman–Forchheimer model for porous media. An innovative approach based on a modifica- tion of the lattice Boltzmann method is explained and validated with previous studies. The effects of the pertinent dimensionless parameters are studied in different ranges. The extensive results of streamlines, isotherms, and isoconcentration contours, yielded/unyielded regions, and local and average Nusselt and Sherwood numbers are presented and discussed.

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popularly adopted to deepen our insights into the fundamentals of heat and mass transfer within ﬂuid-saturated porous media. Bejan et al. [5] and Pop et al. [6] perhaps are the pioneers in this ﬁeld and they reviewed the progress of the early stage of this ﬁeld. As the public’s concern on sustainable de- velopment and environment protection has been soaring, research on **double** diﬀusive **convection** in ﬂuid-saturated porous media has received increasing attention again. The eﬀects of the Rayleigh and Darcy numbers on **double** d- iﬀusive **convection** in a porous cavity were discussed in Ref.[7]. Through their numerical **simulation** one can observe that the behaviors of buoyant ﬂow in a cavity ﬁlled by porous media were quite diﬀerent from **its** non-porous counter- part [8]. Mondal and Sibanda [9] conducted a numerical study on inﬂuences of buoyancy ratio on unsteady **double** diﬀusive natural **convection** in a porous cavity. The authors claimed the patterns of heat and mass transfer in porous media would change signiﬁcantly. **Double** diﬀusive **convection** of nanoﬂuid in a porous enclosure was simulated in Ref.[10]. It was reported that heat transfer was reduced by increasing the bulk volume fraction of nanoparticles. Turbulent **double** diﬀusive **convection** in porous media was also investigated numerical- ly[11]. The work implied new numerical models are desired to deeply reveal the fundamentals of **double** diﬀusive **convection** in ﬂuid-saturated porous me- dia. The amount of publications on this topic is so huge and only a few can be cited here. In almost all research on this topic, the REV (representative elementary volume) scale mathematic description is adopted to model dou- ble diﬀusive **convection** in ﬂuid-saturated porous media as a REV scale model can not only provide acceptable accuracy for engineering applications but also save computational cost for industrial-scale **simulation** [12,13].

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White Rose Consortium ePrints Repository http://eprints.whiterose.ac.uk/ This is an author produced version of a paper published in Journal of Fluid Mechanics.. This paper has been peer-[r]

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For illustration it is assumed that a frequency 5 % above or below the optimum of 50% is achieved, and in Figure 3 lines (a) and (b) show the population fertility W [r]

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We test the PID controller to adjust the time-step based on the error generated during the Newton-Raphson iteration. In case a solution is not within the defined tolerance, the PID controller would adjust the time-step such that the next step has a better chance of convergence to a solution within tolerance. In order to test this controller, we had to tighten the requirement for convergence to force steps to be rejected. As a result, we lower the maximum number of N.R iteration per time-step from 15 to 3. This change forces the system to reject most of the iterations and use the error **control** PID algorithm. The basic controller in this case consist of cutting the value of the previous time-step by half until the solution is within tolerance. We compare this algorithm with a PID

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Directional **control** unit consists of a logic **circuit** implementing the logic equations described at Eq. 1- 2. This unit determines the up or downward direction of the lift at an instant. The outputs of this unit, Y0 and Y1 are used as the enablers of the buffers used to pass the level positioning data to the central bus. These buffers are used to avoid the race condition (two logic levels at the same node) since at each instant Y0 and Y1 values are updated according to logic Eq. 1-2 and only one of them should send level positioning data to bus. This **control** unit has been presented in Fig. 4.

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tion is inversely proportional to scaled Rayleigh number. Mahmud and Hasim [16] investigated the effect of magnetic field on **chaotic** **convection** in fluid layer. They found that transition from **chaotic** **convection** to steady **convection** occurs by a subcritical hopf bifurcation producing a homo- clinic explosion may be limit cycle as Hartmann number increases. The generalized Lorenz mod- els and their routes to chaos by energy-conserving horizontal mode truncations are investigated by Roy and Musielek [26]. They observed that in horizontal modes, 5D system is the lowest order generalized Lorenz model. Vadasz and Olek [28] investigated the route to chaos occurs by a pe- riod doubling sequence of bifurcations when the Prandtl number is moderate. Sheu[21] reported that the route to chaos and **its** applications of thermal non-equilibrium model tends to stabi- lize steady **convection**. Sheu et. al. [22] in- vestigated the onset of chaos through the use of an Oldrydian-fluid. The effect of feedback con- trol on chaos in porous media has been stud- ied by Mahmud and Hasim[15]. They observed that amount of feedback **control** is proportional to scaled Rayleigh number. Magyari [13] demon- strated that the structure of feedback **control** sys- tem proposed by Mahmud and Hasim[15] does not alter the original uncontrolled system but **its** effect is in changing the initial condition of the system. Gupta and Singh [5] reported the effect of anisotropic parameters on **chaotic** **convection**. They founded a proportional relation between scaled Rayleigh number and scaled anisotropic parameters. Gupta and Bhadauria [6] investi- gated the **double** diffusive **convection** in a couple stress liquid saturated porous layer with soret ef- fect using thermal non-equilibrium model. From the above paragraph, we observed that a huge amount of analysis on **chaotic** behavior has been discussed on the onset of **convection** for var- ious flow models. However, not much work has been done for couple stress liquid to analyze **its** **chaotic** behavior. Therefore, in this paper, we have intend to study, the effect of couple stress parameter on Darcy **convection** by dynamical sys- tem approach. for C = 0.1, R c2 = 26.4 for C =

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The furnace has an inner height (H) of 0.39 m (15.5 in), an inner diameter of 0.064 m (2.5 in), and is capped at both ends by irises. To **control** the vertical temperature profile along the furnace wall it is pos- sible to apply power to any combination of 11 power taps located along the vertical height of the furnace. For these experiments power was ap- plied across the mid-region of the furnace, z = −5.5 cm to z = 5.5 cm. This matches the conditions used to produce polymer fiber and results in a non-uniform temperature profile in the furnace wall. The top and bottom copper irises are each heated and independently controlled with thermocouples located on the outside surface of each iris.

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All these results are proof of the versatility that HIL offers in the implementation and emulation of plants. Reason why, this alternative to **control** nonlinear systems like **double** inverted pendulum, whose physical mechanism is difficult to build, was chosen. Among the papers found with this system, publications from [6] and [7] can be highlighted, where they **control** the angular and spatial position of the bars and mobile, on which is supported, with a LQR controller. In the same way, classic **control** systems along with neural networks have been used for estimate the controller gains and achieve the stabilization of the system [8]. Another example is the work performed in [9], where concepts related to artificial intelligence were used for controlling a similar system, achieving a real-time **control**, with more robustness and stability than using conventional methods.

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Although there have been a lot of literature works studying the synchronization problem of **chaotic** systems, we ﬁnd the following deﬁciencies: () Most of them require the nonlin- ear parts in the **chaotic** systems to satisfy the Lipschitz condition. () Each paper usually only could solve a single synchronization problem, which leads to the lack of a uniﬁed method to solve all the synchronization problems for the same master-slave model. Moti- vated by these factors, we aim in this paper to ﬁnd a kind of eﬀective method to deal with all the synchronization problems for a general master-salve **chaotic** system. The contribu- tions of this paper are as follows: (i) The lag projective synchronization which can include synchronization, projective synchronization, anti-synchronization and lag synchroniza- tion at the same time is investigated. (ii) The considered master-slave system model is diﬀerent from the systems in the literature, in which the nonlinearities only need to sat- isfy a bounded condition. Moreover, the state equations of the master system and the slave system are non-identical. (iii) The presented results are very concise and it is easy to adjust the synchronization rate by the **control** gains.

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