Analysis, Stabilization, and DSP-Based Implementation of a Chaotic System with Nonhyperbolic Equilibrium

This paper reports an autonomous dynamical system, and it finds that one nonhyperbolic zero equilibrium and two hyperbolic nonzero equilibria coexist in this system. Thus, it is difficult to demonstrate the existence of chaos by Šil’nikov theorem. Consequently, the topological horseshoe theory is adopted to rigorously prove the chaotic behaviors of the system in the phase space of Poincaré map. Then, a single control scheme is designed to stabilize the dynamical system to its zero-equilibrium point. Besides, to verify the theoretical analyses physically, the attractor and stabilization scheme are further realized via DSP-based technique.

Uniform Hyperbolicity of a Scattering Map with Lorentzian Potential

We show that a two-dimensional area-preserving map with Lorentzian potential is a topological horseshoe and uniformly hyperbolic in a certain parameter region. In particular, we closely examine the so-called sector condition, which is known to be a sufficient condition leading to the uniformly hyperbolicity of the system. The map will be suitable for testing the fractal Weyl law as it is ideally chaotic yet free from any discontinuities which necessarily invokes a serious effect in quantum mechanics such as diffraction or nonclassical effects. In addition, the map satisfies a reasonable physical boundary condition at infinity, thus it can be a good model describing the ionization process of atoms and molecules.

Global exponential stabilization for chaotic brushless DC motor with simpler controllers

In this paper, computer assisted proofs in dynamics (CAPD) group and conventional dynamic analysis methods are applied to find the parameters that make the brushless DC motor (BLDCM) system chaotic. For the purpose to make the BLDCM system exponentially stable, four simple controllers with just one state variable are proposed, which means that it only needs one sensor instead of two or more in practice. Numerical simulations show that the motor with the new controllers can converge efficiently. Finally, the topological horseshoe theory is introduced to verify the existence of chaos in BLDCM system.

Topological Horseshoe Analysis, Ultimate Boundary Estimations of a New 4D Hyperchaotic System and Its FPGA Implementation

This paper constructs a new four-dimensional (4D) hyperchaotic system. Firstly, the influence of parameter variation on the dynamic behavior of the system is analyzed in detail using Lyapunov exponents and the bifurcation diagram. Additionally, the topological horseshoe finding algorithm is based on three-dimensional (3D) hyperchaotic mapping. Through searching for the 3D topological horseshoe with two-dimensional stretching on the Poincaré section, the existence of the 4D hyperchaotic system is proved in the mathematical sense. Next, Lyapunov stability theory and optimization method are used to further analyze the ultimate boundary of the proposed 4D hyperchaotic system. Thus, the 3D ellipsoidal boundary of the hyperchaotic system is found. Finally, this paper also takes the hyperchaotic system as an example and presents the experimental results of generated hyperchaotic attractors by FPGA technology. The experimental results show that the phase diagram of hyperchaotic system is consistent for the simulated results. Due to the more complex dynamic behavior, the proposed system is suitable for engineering application, such as in chaotic secure communications.

Topological Horseshoe Analysis and FPGA Implementation of the Fractional-order Liu System

This paper first discusses a fractional-order Liu system of order as low as 2.7 and shows its chaotic characteristics by carrying out numerical simulations such as Lyapunov exponents, bifurcation diagrams and phase portraits. Then, by using the topological horseshoe theory and computer-assisted proof, the existence of chaos in the system is verified theoretically. Finally, the fractional-order system is implemented on a Field Programmable Gate Array (FPGA) and the results obtained show that the fractional-order Liu system is indeed chaotic.