Delayed feedback control of fractional-order chaotic systems

Abstract

We study the possibility to stabilize unstable steady states and unstable periodic orbits in chaotic fractional-order dynamical systems by the time-delayed feedback method. By performing a linear stability analysis, we establish the parameter ranges for successful stabilization of unstable equilibria in the plane parametrizad by the feedback gain and the time delay. An insight into the control mechanism is gained by analyzing the characteristic equation of the controlled system, showing that the control scheme fails to control unstable equilibria having an odd number of positive real eigenvalues. We demonstrate that the method can also stabilize unstable periodic orbits for a suitable choice of the feedback gain, providing that the time delay is chosen to coincide with the period of the target orbit. In addition, it is shown numerically that delayed feedback control with a sinusoidally modulated time delay significantly enlarges the stability region of the steady states in comparison to the classical time-delayed feedback scheme with a constant delay.

Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay

“…It is noted that, in such a combination, an integer order delayed system is presented when the fractional order is restricted to be one; and the fractional order models can be obtained by removing feedback gains. A growing body of research activities on fractional delay systems has recently been attracted, such as the existence and uniqueness of solutions 4 [47], stability analysis [48][49][50], dynamic behaviors [51][52][53][54], vibrational resonance [55], fractional order controls [56] and etc. Nevertheless, to our knowledge, no efforts were made to predict periodic solutions and analyze associated bifurcations of fractional delay systems.…”

Desynchronization of systems of coupled Hindmarsh-Rose oscillators

“…to suppress the mean field X(t) by applying the feedback force F 2 (t) and making a suitable choice of the control parameters. The motivation to use variable time-delay comes from previous studies of ours [10] with varying delay applied to problems of stabilization of unstable steady states in various systems [11][12][13]. The results were very favourable leading to significant enlargement of the domains in the parameter space for which stabilization is achievable.…”

Geometric Structures of Fractional Dynamical Systems in Non‐Riemannian Space: Applications to Mechanical and Electromechanical Systems

“…This non‐local property in space and time of fractional calculus has been used to analyze the behavior of a viscoelastic body, the deformation field near a fault zone and the memory effect for economic variables . A wide range of dynamical systems can be expressed as fractional differential equations such as a delayed‐feedback control of a fractional‐order chaotic system, a fractional‐order chaotic Lorenz system, a fractional‐order Rössler system, dynamical processes of a complex system, behavior of a fractional oscillator, and fractional‐order analytical mechanics . In particular, synchronization for the class of chaotic fractional‐order systems has been investigated extensively …”