A 3-D spherical chaotic attractor

Abstract: A simple smooth chaotic system, which showed a 3-layer sphere chaotic attractor, is investigated. It is found that this chaotic attractor is a limit cycle instead of chaotic attractor. This situation was caused by the simulation time which is too short to reach its real status. It also shows that it is not reliable to construct chaotic system based only on the Šhilnikov criterion without finding the exact homoclinic orbits. Then a chaotic system with the real sphere shape is proposed. This proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, bifurcation diagrams and Poincaré section

Adaptive Sliding Mode Controller Design for Projective Synchronization of Different Chaotic Systems with Uncertain Terms and External Bounded Disturbances

Synchronization is very useful in many science and engineering areas. In practical application, it is general that there are unknown parameters, uncertain terms, and bounded external disturbances in the response system. In this paper, an adaptive sliding mode controller is proposed to realize the projective synchronization of two different dynamical systems with fully unknown parameters, uncertain terms, and bounded external disturbances. Based on the Lyapunov stability theory, it is proven that the proposed control scheme can make two different systems (driving system and response system) be globally asymptotically synchronized. The adaptive global projective synchronization of the Lorenz system and the Lü system is taken as an illustrative example to show the effectiveness of this proposed control method

A hyperchaotic system without equilibrium

Abstract: This article introduces a new chaotic system of 4-D autonomous ordinary differential equations, which has no equilibrium. This system shows a hyper-chaotic attractor. There is no sink in this system as there is no equilibrium. The proposed system is investigated through numerical simulations and analyses including time phase portraits, Lyapunov exponents, and Poincaré maps. There is little difference between this chaotic system and other chaotic systems with one or several equilibria shown by phase portraits, Lyapunov exponents and time series methods, but the Poincaré maps show this system is a chaotic system with more complicated dynamics. Moreover, the circuit realization is also presented

A new conservative system with isolated invariant tori and six-cluster chaotic flows

Based on the matrix differential equation of the Sprott-A system, this paper presents a class of rare 3D conservative systems by adjusting its skew-symmetric state matrix and Hamiltonian. Then, an example system is reported to show the conservative dynamical behaviors. For given parameters and initial conditions, the example system can generate six isolated invariant tori and six-cluster conservative chaotic flows. Numerical results show that the six isolated invariant tori are located in six isolated isosurfaces, while the six-cluster conservative chaotic flows approximately run on an interconnected isosurface. Moreover, it is found that the shape and number of invariant tori and conservative chaotic flows relay on the Hamiltonian of the example system

A New No-Equilibrium Chaotic System and Its Topological Horseshoe Chaos

A new no-equilibrium chaotic system is reported in this paper. Numerical simulation techniques, including phase portraits and Lyapunov exponents, are used to investigate its basic dynamical behavior. To confirm the chaotic behavior of this system, the existence of topological horseshoe is proven via the Poincaré map and topological horseshoe theory

A Strange Double-Deck Butterfly Chaotic Attractor from a Permanent Magnet Synchronous Motor with Smooth Air Gap: Numerical Analysis and Experimental Observation

A permanent magnet synchronous motor (PMSM) model with smooth air gap and an exogenous periodic input is introduced and analyzed in this paper. With a simple mathematical transformation, a new nonautonomous Lorenz-like system is derived from this PMSM model, and this new three-dimensional system can display the complicated dynamics such as the chaotic attractor and the multiperiodic orbits by adjusting the frequency and amplitude of the exogenous periodic inputs. Moreover, this new system shows a double-deck chaotic attractor that is completely different from the four-wing chaotic attractors on topological structures, although the phase portrait shapes of the new attractor and the four-wing chaotic attractors are similar. The exotic phenomenon has been well demonstrated and investigated by numerical simulations, bifurcation analysis, and electronic circuit implementation

Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points

Chaotic dynamics exists in many natural systems, such as weather and climate, and there are many applications in different disciplines. However, there are few research results about chaotic conservative systems especially the smooth hyperchaotic conservative system in both theory and application. This paper proposes a five-dimensional (5D) smooth autonomous hyperchaotic system with nonhyperbolic fixed points. Although the proposed system includes four linear terms and four quadratic terms, the new system shows complicated dynamics which has been proven by the theoretical analysis. Several notable properties related to conservative systems and the existence of perpetual points are investigated for the proposed system. Moreover, its conservative hyperchaotic behavior is illustrated by numerical techniques including phase portraits and Lyapunov exponents