Synchronization and Electronic Circuit Application of Hidden Hyperchaos in a Four-Dimensional Self-Exciting Homopolar Disc Dynamo without Equilibria

1Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin 537000, China 2School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China 3Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK 4Department of Computer Technologies, Vocational School of Karacabey, Uludağ University, Karacabey, 16700 Bursa, Turkey 5Department of Electrical and Electronic Engineering, Faculty of Technology, Sakarya University, Adapazarı, Turkey


Introduction
Hyperchaos is a feature of a chaotic system having more than one positive Lyapunov exponent [1].Because of potential theoretical and practical applications in technology, such as secure communications, lasers, nonlinear circuits, neural networks, generation, control, and synchronization, hyperchaos has featured as an important research area in nonlinear science [2][3][4][5].The theory about hidden hyperchaos with either only stable or no equilibrium states is still in its infancy and has only recently been understood by mathematicians [6][7][8][9].
In 1979, Moffatt identified inconsistencies in the modeling of a simple self-exciting homopolar disc dynamo because of the neglect of induced azimuthal eddy currents, which can be resolved by introducing a segmented disc dynamo [10].
Here we investigate hidden hyperchaos, synchronization, and electronic circuit realization for a higher-dimensional version of the self-exciting homopolar disc dynamo, which was not yet completely well understood.
Since Pecora and Carroll [11] investigated synchronization in chaotic systems in 1990, such behavior has become an important research area in nonlinear science, not only for understanding the complicated phenomena in various fields but also for its potential applications especially in secure communication and image encryption.Two indistinguishable chaotic systems, starting from nonidentical initial values, would evolve in time to completely different trajectories because of the sensitive dependence of chaotic systems to their initial values.The aim of synchronizing chaos is to ensure that the states track the desired trajectory.Many effective methods exist to deal with synchronization of chaotic and 2 Complexity hyperchaotic systems.These include active control [12][13][14][15][16][17][18][19][20], passive control [21], sliding mode control [22][23][24][25][26][27][28][29][30][31], adaptive control [32], and backstepping design [33].Of these, active control is an important simple method used in the synchronization of nonlinear systems.It maintains asymptotic stability at zero error by eliminating the nonlinear terms and making all the eigenvalues have negative real parts.The other commonly preferred method, sliding mode control, maintains the synchronization by enforcing the error system to stay on a constructed sliding surface.
The first model with a simple electronic application was realized by Chua et al. [34].In the following years, many electronic circuit applications such as simple RLC, RC circuits [35][36][37], oscillators [38,39], power circuits [40,41], and capacitor circuits which show chaotic features were improved upon.On the one hand, numerous electronic circuit realizations with interesting features, which mimic novel chaotic and hyperchaotic systems, have been proposed in recent years [3,[42][43][44][45].
Current interest in hidden hyperchaotic attractors motivates us to study an extension about the self-exciting homopolar disc dynamo [10] to 4D homopolar dynamo without equilibria.The existence of hidden hyperchaotic attractors in this new disc dynamo is confirmed.Synchronization of two such coupled 4D self-exciting homopolar disc dynamo systems is analyzed with active and sliding mode control methods.Moreover, we have designed an electronic circuit and have used an oscilloscope to view the hyperchaotic rescaled dynamo without equilibria, implemented in real time.

Model and Hidden Hyperchaos of 4D Self-Exciting Homopolar Disc Dynamo System without Equilibria
Dynamo models have been the object of much interest in order to understand both the generation of magnetic fields and their reversals in astrophysics.Moffatt [10] extended the simplest self-exciting Bullard dynamo to include radial magnetic diffusion, to produce the disc dynamo model, written nondimensionally as where , , and  are state variables and , , and  are positive parameters. and  represent the magnetic fluxes due to radial and azimuthal current distributions, respectively. denotes the angular velocity of the disc. is the applied torque;  and  are the constants which depend on the inductances and the electrical resistance of the dynamo.By modifying the characteristics of the segmented disc dynamo (1), hidden chaotic or hyperchaotic spiral attractors have been observed numerically under special initial conditions with two symmetric stable node-foci.This leads to the interesting and striking observation of multiple attractors for a broad range of parameters.As in many nonlinear dynamical systems, the occurrence of multiple attractors implies the existence of multistability in the self-exciting dynamo, with the long-term behavior being fundamentally different depending on which basin of attraction the initial conditions belong.Now, we introduce a dislocated feedback controller to system (1) as a new state  and translate  to  −  to result in following 4D system: where  is a positive parameter.Although system (2) is similar to the algebraic forms of hyperchaotic Lorenz, Chen, Lü, and unified systems, they are not topologically equivalent [46][47][48][49][50][51][52].The proposed system (2) will be a way of understanding the generation of magnetic fields and their reversals in the Earth, the Sun, and other astrophysical bodies.Figure 1 shows a bifurcation diagram exhibiting a period-doubling route to chaos of the peak of  (max) of system (2) versus the parameters  ∈ [0.5, 1],  = 8,  = 35, and  = 3.There are some periodic windows in the chaotic region.Plots of the Lyapunov exponents about  ∈ [0.5, 1.5] are shown in Figure 2. Figure 3 indicates that system (2) is indeed hidden hyperchaotic for initial states (1.13, 0.5, 0.8, and 1.5) and parameter  = 0.5.Its Lyapunov exponents are 0.4113, 0.2233, 0.0000, and −10.1345) and Kaplan-Yorke dimension is  KY = 3.0626.

Synchronization of the 4D Self-Exciting Homopolar Disc Dynamo System
For synchronization, two self-exciting homopolar disc dynamo hyperchaotic systems are coupled together with different initial values.The driver system, , controls the response system, .They are given, respectively, by where  1 and  2 in system (5) are the control functions to be determined.The synchronization errors are obtained by subtracting the driver and response systems.Thus, they are defined as   =   −  (for  = 1, 2, 3, 4) and the error dynamics become as follows: Our objective is to make system (6) asymptotically stable at the zero error state.

Active Control.
The nonlinear terms in system (6) can be eliminated by defining the control functions  1 and  2 as in the following: Then, substituting (7) into system (6) gives This implies that the equations in system (8) are linear.Provided that the proper choices of control inputs V 1 and V 2 stabilize the error system (8), then  1 ,  2 ,  3 , and  4 will converge to zero as  → +∞.Then, the synchronization of two identical self-exciting homopolar disc dynamo hyperchaotic systems will be achieved.A number of choices are possible for control functions V 1 and V 2 .They are simply taken as where  1 and  2 are positive control gains.Substituting (9) to system (8) gives the following synchronization error dynamics: The associated characteristic matrix of system (10) is For the particular choice of control functions in (9), the closed loop system (10) has all of its eigenspectrum in the negative half plane since all of the parameters , ,  1 , and  2 are positive.So, this choice leads to a stable system where the error states  1 ,  2 ,  3 , and  4 tend to zero as time  tends to infinity.Synchronization of two identical hyperchaotic selfexciting homopolar disc dynamos is therefore completed with the active control method.

Sliding Mode Control.
It can be seen from system (6) that when  2 and  3 become zero, ė4 will be zero and then ė1 = − 1 .Therefore, when time goes to infinite,  1 will converge to zero, too.Appropriate sliding surfaces for  2 and  3 can be, respectively, designed as where  3 and  4 are positive control gains.The attainability condition for sliding mode is  ṡ < 0. Provided that this condition is satisfied, the sliding mode control functions are  where  5 and  6 are positive control gains.Large values of  5 and  6 decrease the time  taken to reach the sliding surface but lead to chattering; small values of  1 and  2 reduce chattering but increase the time to reach the sliding surface.
Here "sign" means the signum function.
The designed control functions (13) guarantee that system (6) is held on the sliding surface  = 0.The time derivations of sliding surfaces are For the Lyapunov function the time derivative of  becomes ) where  5,6 ≥ 0. These conditions guarantee that the constructed sliding surfaces  1 and  2 would asymptotically stabilize to the zero synchronization error state, and we obtain synchronization between the two identical hyperchaotic selfexciting homopolar disc dynamos via the sliding mode control method.

Numerical Simulations.
We now perform some numerical experiments to show that synchronization occurs.We use an ode45 integration solver function with a variable step size.We take  = 8,  = 0.5,  = 35, and  = 3 to ensure that hyperchaotic behavior occurs.The gains of the active controllers are taken to be  1 = 1 and  2 = 1.The gains of the sliding mode controllers are taken to be  3 = 1,  4 = 1,  5 = 1,  6 = 1,  1 = 0.5, and  2 = 0.5.We choose the initial conditions for the driver and response systems to be (1.13,0.5, 0.8, 1.5) and (1.3, 2, 11.1, 1.4), respectively.The controllers are activated when  = 10, and we plot the synchronization simulations in Figure 4, while the synchronization errors are plotted in Figure 5.
Figure 4 shows that both active and sliding mode controllers achieve the synchronization of the four-dimensional self-exciting homopolar disc dynamo hyperchaotic system.Figure 5 also shows that, after the activation of controllers at  = 10, the synchronization errors approach zero asymptotically, thereby validating the theoretical analyses.Synchronization is complete for  ≥ 16 with active control and for  ≥ 14 with sliding mode control.Activation of control at different times gives similar synchronization performances.The comparisons point out that the sliding mode control scheme has the advantage of a faster synchronization time.However, when the results are viewed more closely, as in Figure 6, the chattering problem of sliding mode controllers is evident.A solution is to switch to active control when the mean squared error is less than 0.00001.The new results from switching controllers are presented in Figure 7.It has no chattering and the synchronization performance is similar to that with the sliding mode control scheme.
Outputs from the ORCAD-PSpice simulation and oscilloscope phase portraits for the scaled hyperchaotic system (17) with parameters  = 8,  = 0.5,  = 35, and  = 3 are given in Figures 10 and 11, respectively.The outputs verify those of the hyperchaotic system, which was modeled using MATLAB.

Conclusion
In this paper, we propose a novel four-dimensional selfexciting homopolar disc dynamo without equilibria, but exhibiting hidden hyperchaos.Furthermore, active control and sliding mode control methods are applied to synchronize two coupled four-dimensional dynamo systems.The feasibility of active controllers is ensured by requiring that the spectrum of eigenvalues of the synchronized error system falls in the left half plane.A Lyapunov function is proposed to guarantee the asymptotic stability of sliding surfaces and convergence to zero synchronization error.Numerical integrations are presented to compare the performances of the two controllers.Sliding mode control scheme gives better results but has chattering.Solution is provided by a switch to active controllers when chattering starts.An electronic circuit for the rescaled system is implemented via the ORCAD-PSpice program.Numerical simulations validated the theoretical analyses.This physical example will form the basis of more systematic studies of hyperchaos without equilibria in a future study.

Figure 4 :
Figure 4: Time series of driver and response four-dimensional self-exciting homopolar disc dynamo hyperchaotic systems with the controllers are activated at  = 10.

Figure 5 :Figure 6 :Figure 7 :Figure 8 :
Figure 5: Synchronization errors between driver and response four-dimensional self-exciting homopolar disc dynamo hyperchaotic systems with the controllers are activated at  = 10: (a) active controllers and (b) sliding mode controllers.

Figure 9 :
Figure 9: The experimental circuit of the hyperchaotic circuit.