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We introduce and investigate a four-dimensional hidden hyperchaotic system without equilibria, which is obtained by augmenting the three-dimensional self-exciting homopolar disc dynamo due to Moffatt with an additional control variable. Synchronization of two such coupled disc dynamo models is investigated by active control and sliding mode control methods. Numerical integrations show that sliding mode control provides a better synchronization in time but causes chattering. The solution is obtained by switching to active control when the synchronization errors become very small. In addition, the electronic circuit of the four-dimensional hyperchaotic system has been realized in ORCAD-PSpice and on the oscilloscope by amplitude values, verifying the results from the numerical experiments.

Hyperchaos is a feature of a chaotic system having more than one positive Lyapunov exponent [

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 [

Since Pecora and Carroll [

The first model with a simple electronic application was realized by Chua et al. [

Current interest in hidden hyperchaotic attractors motivates us to study an extension about the self-exciting homopolar disc dynamo [

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 [

By modifying the characteristics of the segmented disc dynamo (

Bifurcation diagram of system (

Lyapunov exponents

Hyperchaotic attractor of four-dimensional self-exciting homopolar disc dynamo system (

To find the equilibrium states of system (

There seems to be two equilibria:

For synchronization, two self-exciting homopolar disc dynamo hyperchaotic systems are coupled together with different initial values. The driver system,

The nonlinear terms in system (

It can be seen from system (

The attainability condition for sliding mode is

The designed control functions (

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

Time series of driver and response four-dimensional self-exciting homopolar disc dynamo hyperchaotic systems with the controllers are activated at

Synchronization errors between driver and response four-dimensional self-exciting homopolar disc dynamo hyperchaotic systems with the controllers are activated at

Figure

Synchronization errors between

Synchronization errors with the proposed switching controllers are activated: (a)

Because the values for

We can now design an electronic circuit for the scaled hyperchaotic model (

The electronic circuit schematic of the scaled hyperchaotic system (

The experimental circuit of the hyperchaotic circuit.

Outputs from the ORCAD-PSpice simulation and oscilloscope phase portraits for the scaled hyperchaotic system (

The phase portraits of scaled hyperchaotic system (

The phase portraits of scaled hyperchaotic system (

In this paper, we propose a novel four-dimensional self-exciting 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.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant no. 11561069), the Guangxi Natural Science Foundation of China (Grant no. 2016GXNSFBA380170), the Scientific Research Foundation of the Higher Education Institutions of Guangxi Province of China (Grant no. KY2016YB364), the Open Foundation for Guangxi Colleges and Universities Key Lab of Complex System Optimization and Big Data Processing (nos. 2016CSOBDP0003 and 2016CSOBDP0202), and Sakarya University Scientific Research Projects Unit (no. 201609-00-008).