Chaos in a new fractional-order system without equilibrium points

• New fractional-order chaotic system with no equilibrium points.
• “Elegant” system: equations with few terms and parameters with minimum of digits.
• Fractional dynamics analyzed using the predictor–corrector algorithm.
• Presence of chaos validated by applying three different numerical methods.
• Application of observer-based synchronization to chaotic fractional system.

Chaotic systems without equilibrium points represent an almost unexplored field of research, since they can have neither homoclinic nor heteroclinic orbits and the Shilnikov method cannot be used to demonstrate the presence of chaos. In this paper a new fractional-order chaotic system with no equilibrium points is presented. The proposed system can be considered “elegant” in the sense given by Sprott, since the corresponding system equations contain very few terms and the system parameters have a minimum of digits. When the system order is as low as 2.94, the dynamic behavior is analyzed using the predictor–corrector algorithm and the presence of chaos in the absence of equilibria is validated by applying three different methods. Finally, an example of observer-based synchronization applied to the proposed chaotic fractional-order system is illustrated.