Abstract
Based on existing feedback control methods such as OGY and Pyragas, alternative new schemes are proposed for stabilization of unstable periodic orbits of chaotic and hyperchaotic dynamical systems by suitable modulation of a control parameter. Their performances are improved with respect to: (i) robustness, (ii) rate of convergences, (iii) reduction of waiting time, (iv) reduction of noise sensitivity. These features are analytically investigated, the achievements are rigorously proved and supported by numerical simulations. The proposed methods result successful for stabilizing unstable periodic orbits in some classical discrete maps like 1-D logistic and standard 2-D Hénon, but also in the hyperchaotic generalized n-D Hénon-like maps.
Highlights
Chaotic behavior is a very interesting nonlinear phenomena, but in many situations, it is desirable to be avoided, for example, when it restricts the operating range of electronic or mechanics devices
It is well known that the advantage of any delayed feedback control (DFC) method over simple proportional feedback (SPF) methods is just that the full knowledge of the unstable periodic orbit (UPO) to be stabilized is not needed, instead, limitations on convergence issues arise
Both SPF and DFC methods have been revisited, and, based on them, new modifications yielding to a “switching” SPF- and DFC-type strategies have been proposed
Summary
Chaotic behavior is a very interesting nonlinear phenomena, but in many situations, it is desirable to be avoided, for example, when it restricts the operating range of electronic or mechanics devices. Its usefulness arises when controlling chaos in highly dissipative systems; as there are cases in which it loses validity because Poincaré sections changes in each iteration, a recursive proportional feedback (RPF) has been proposed [36] Both SPF and RPF methods require the exact knowledge of the UPO to be stabilized and the linearized dynamics about it. Different from the original OGY method, the SPF method does not require exact knowledge on the linearization data of the UPO to stabilize Based on these ideas, modifications to the original Pyragas method [27] are introduced even in the extended version presented in [32], and alternative types of DFC and EDFC methods are proposed.
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