A novel double-convection chaotic attractor, its adaptive control and circuit simulation

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.

Rucklidge chaotic system is a popular model in mechanics for nonlinear double convection [42]. When the convection takes place in a fluid layer rotating uniformly about a vertical axis and in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modeled by Rucklidge's 3-D system of differential equations, which produces chaotic solutions.
In this work, we modify the dynamics of Rucklidge system [42] and derive a new double-convection chaotic system with an absolute nonlinearity and two quadratic nonlinearities. The phase portraits of the new chaotic system are displayed in Section 2, and the dynamical properties of the new chaotic system such as dissipativity, equilibria analysis, Lyapunov exponents, Kaplan-Yorke dimension, etc. are Adaptive control and synchronization of the new chaotic system with unknown system parameters are discussed in Sections 4 and 5, respectively. The main adaptive control results derived in this work are established using Lyapunov stability theory [43]. Furthermore, an electronic circuit realization of the new chaotic system is presented in detail in Section 6. The circuit experimental results of the new chaotic attractor show agreement with the numerical simulations. Section 7 contains the conclusions.

A new nonlinear double-convection chaotic system
In fluid mechanics modeling, cases of two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints were considered by Rucklidge [42]. When the convection takes place in a fluid layer rotating uniformly about a vertical axis and in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modeled by a new thirdorder set of ordinary differential equations, which produces chaotic solutions. The Rucklidge chaotic system is described by the 3-D dynamics where 1 2 3 , , x x x are state variables and , a b are positive constants. In [42], it was established that the system (1) is chaotic for 2.2 a = and 6.7. b = In this work, we modify the dynamics of Rucklidge chaotic system (1) and obtain a new dynamics for nonlinear double convection as where 1 2 3 , , x x x are state variables and , a b are positive constants.
In this paper, we show that the system (2) is chaotic for the parameter values

Dynamical properties of the new chaotic system
In this section, we take the parameter values as in the chaotic case, i.e. 2.2 a = and 18. b =

Dissipativity
If V denotes any volume along the flow of the new chaotic system (2), then  Thus, the new chaotic system (2) is dissipative. Hence, the system limit sets are ultimately confined into a specific limit set of zero volume, and the asymptotic motion of the novel chaotic system (2) settles into a strange attractor of the system.

Equilibrium points
The equilibrium points of the system (2) are obtained by solving the system of equations We take the parameter values as in the chaotic case (3), i.e.
2.2 a = and 18. b = A simple calculation shows that the new chaotic system (2) has three equilibrium points given by To check the stability of the equilibrium points, we calculate the Jacobian of the system (2) as We find that 0 J has the eigenvalues 1 1, This shows that the equilibrium point 0 E is a saddle point. Hence, it is unstable.
Next, we find that 1 J has the eigenvalues This shows that the equilibrium point 2 E is a saddle-focus. Hence, it is unstable.

Lyapunov exponents and Kaplan-Yorke dimension
The parameters of the new system (2) are taken as The initial state of the system The Lyapunov exponents of the system (2) are calculated using Wolf's algorithm [44] (see Figure 2) as

Symmetry and invariance
The new chaotic system (2) is invariant under the coordinate transformation This shows that the chaotic system (2) has a rotation symmetry about the 3 x − axis. Hence, every non-trivial trajectory of the system (2) must have a twin trajectory. Also, the 3 x − axis is invariant under the flow of the new chaotic system (2). This invariant flow is characterized by the 1-D dynamics 3 3 x x = − ɺ which is exponentially stable.

Adaptive control of the new chaotic system
In this section, we consider the controlled new chaotic system given by where 1 2 3 , , x x x are the states and , a b are unknown system parameters.
We consider the adaptive controller defined by Substituting (17) into (16), we obtain the closed-loop system We define parameter estimation errors as follows: Using (19), the closed loop system (18) reduces to x e x e x k x Differentiating (19) with respect to , t we get Next, we consider the Lyapunov function defined by which is positive definite on 5 . R Differentiating V along the trajectories of (20) and (21), we obtain ( ) ( ) In view of (23), we take the parameter update law as Next, we prove the main theorem of this section. Theorem 1. The new chaotic system (16) with unknown parameters is globally and asymptotically stabilized by the adaptive control law (17) and the parameter update law (24), where 1 2 3 , , k k k are positive constants. (22) is quadratic and positive definite on 5 . R By substituting the parameter update law (24) into (23), we obtain the time-derivative of V as

Proof. The Lyapunov function V defined by
which is negative semi-definite on 5 . R Thus, by Barbalat's lemma [43], it follows that the closed-loop system (20) is globally asymptotically stable for all initial conditions 3 (0) . x ∈ R Hence, we conclude that the new chaotic system (16) with unknown parameters is globally and asymptotically stabilized by the adaptive control law (17) and the parameter update law (24), where 1 2 3 , , k k k are positive constants. This completes the proof. We take the initial conditions of the parameter estimates as ˆ(0) 7.5 a = and ˆ( 0) 6.8. b = Figure 3 shows the time-history of the states of the new chaotic system (16) after the implementation of the adaptive control law (17) and the parameter update law (24).

Adaptive synchronization of the new chaotic system
In this section, we use the adaptive control to synchronize a pair of identical new chaotic systems with unknown state parameters. As the master system, we consider the new chaotic system given by (26) where 1 2 3 , , x x x are the states and , a b are unknown parameters.
As the slave system, we consider the new chaotic system given by  (27) where 1 2 3 , , y y y are the states and 1 2 3 , , u u u are adaptive controls to be designed.
The synchronization error between the systems (26) and (27) where 1 2 3 , , k k k are positive gain constants.
Substituting (30) into (29), we obtain the closed-loop system In view of (36), we take the parameter update law as V k e k e k e = − − − ɺ (38) which is negative semi-definite on 5 . R Thus, by Barbalat's lemma [43], it follows that the closed-loop system (33) is globally asymptotically stable for all initial conditions 3 (0) . e ∈R Hence, we conclude that the new chaotic systems (26) and (27) with unknown parameters are globally and asymptotically stabilized by the adaptive control law (30) and the parameter update law (37), where 1 2 3 , , k k k are positive constants. This completes the proof.  Figure 4 shows the synchronization of the states of the new chaotic systems (26) and (27). Figure 5 shows the time-history of the synchronization errors 1 2 3 , , . e e e

Circuit implementation of the new double-convection chaotic system
Electronic circuit provides an alternative approach to exploring new chaotic system (2). In this section, we design and build an electronic circuit of the system (2) as shown in Figure 6. In more details, there are three integrators (U1A, U3A, U5A), which are created by the operational amplifiers. The circuit consists of simple electronic elements, such as resistors, capacitors, operational amplifiers, analog devices AD633 multipliers and two diodes (1N4148), which provide the signal |X2|. By applying Kirchhoff's laws to the circuit in Figure 6, its circuital equations are derived in the following form:   The supplies of all active devices are 15 ± volt. The proposed circuit is implemented in the electronic simulation package MultiSIM. The obtained phase portraits are shown in Figure 1. There is a good agreement between these circuital results and the theoretical ones (see Figures. 7-9).

Conclusions
This work proposed a novel three-dimensional double-convection chaotic system with three nonlinearities. The dynamical properties of the new chaotic system were discussed in detail. The qualitative properties included 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 were achieved via nonlinear controllers and Lyapunov stability theory. Furthermore, an electronic circuit realization of the new 3-D novel chaotic system was proposed and the circuit experimental results of the 3-D novel chaotic attractor showed good agreement with the numerical simulations.