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.

This paper provides a contribution to the topic of full state hybrid projective synchronization (FSHPS) by introducing an observer-based approach that enables synchronization to be achieved via a scalar synchronizing signal. The method is based on a theorem that assures dead-beat synchronization (i.e., exact synchronization in finite time) to a wide class of discrete-time chaotic (hyperchaotic) systems. Two examples, involving the hyperchaotic Grassi–Miller map and the hyperchaotic double scroll map, show that FSHPS can be effectively achieved in finite time using a scalar synchronizing signal only

A new fractional-order chaotic system with no equilibria is presented. The proposed system can be considered elegant in the sense given by Sprott (2010), since the corresponding system equations contain very few terms and the system parameters have a minimum of digits. The chaotic dynamics are analyzed using the predictor-corrector algorithm when the fractional-order of the derivative is 0.98. Finally, the presence of chaos is validated by applying different numerical methods.

In 1695, G. Leibniz laid the foundations of fractional calculus, but mathematicians revived it only 300 years later. In 1971, L.O. Chua postulated the existence of a fourth circuit element, called memristor, but Williams’s group of HP Labs realized it only 37 years later. By looking at these interdisciplinary and promising research areas, in this paper, a novel fractional-order system including a memristor is introduced. In particular, chaotic behaviors in the simplest fractional-order memristor-based system are shown. Numerical integrations (via a predictor–corrector method) and stability analysis of the system equilibria are carried out, with the aim to show that chaos can be found when the order of the derivative is 0.965. Finally, the presence of chaos is confirmed by the application of the recently introduced 0-1 test.

This paper introduces a novel type of synchronization, where two chaotic systems synchronize up to an arbitrary scaling matrix. In particular, each drive system state synchronizes with a linear combination of response system states by using a single synchronizing signal. The proposed observer-based method exploits a theorem that assures asymptotic synchronization for a wide class of continuous-time chaotic (hyperchaotic) systems. Two examples, involving Rössler’s system and a hyperchaotic oscillator, show that the proposed technique is a general framework to achieve any type of synchronization defined to date.