A short survey on Pyragas time-delay feedback stabilization and odd number limitation

Kuznetsov, Nikolay V.; Leonov, Gennadij A.; Shumafov, M. M.

doi:10.1016/j.ifacol.2015.09.271

Cited by 22 publications

(7 citation statements)

References 28 publications

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“…Consider the challenges of the finite-time Lyapunov dimension computation along the trajectories over large time intervals (see, e.g., [48][49][50][51]), which is connected with the existence of unstable periodic orbits (UPOs) embedded in chaotic attractor. The "skeleton" of a chaotic attractor comprises embedded unstable periodic orbits (UPOs) (see, e.g., [1,6,17]), and one of the effective methods among others for the computation of UPOs is the delayed feedback control (DFC) approach, suggested by K. Pyragas [85] (see also discussions in [15,45,55]). This approach allows Pyragas and his progeny to stabilize and study UPOs in various chaotic dynamical systems.…”

Section: 2mentioning

confidence: 99%

The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

et al. 2020

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On the example of the famous Lorenz system, the difficulties and opportunities of reliable numerical analysis of chaotic dynamical systems are discussed in this article. For the Lorenz system, the boundaries of global stability are estimated and the difficulties of numerically studying the birth of selfexcited and hidden attractors, caused by the loss of global stability, are discussed. The problem of reliable numerical computation of the finite-time Lyapunov dimension along the trajectories over large time intervals is discussed. Estimating the Lyapunov dimension of attractors via the Pyragas time-delayed feedback control technique and the Leonov method is demonstrated. Taking into account the problems of reliable numerical experiments in the context of the shadowing and hyperbolicity theories, experiments are carried out on small time intervals and for trajectories on a grid of initial points in the attractor's basin of attraction.

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Section: 2mentioning

confidence: 99%

The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

et al. 2020

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“…One effective method among others for the stabilization of UPOs is the delay feedback control (DFC) approach, suggested by Pyragas (Pyragas, 1992) (see discussions of its advantages, limitations and modifications in (Kuznetsov et al, 2015;Chen and Yu, 1999;Lehnert et al, 2011;Hooton and Amann, 2012)). This approach allows Pyragas and his progeny to stabilize and study UPOs in various chaotic dynamic systems.…”

Section: Stabilization Of Periodic Orbits In the Shapovalov Model Via...mentioning

confidence: 99%

Time-delay control for stabilization of the Shapovalov mid-size firm model

Alexeeva

Barnett

Kuznetsov

et al. 2020

Preprint

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Control and stabilization of irregular and unstable behavior of dynamic systems (including chaotic processes) are interdisciplinary problems of interest to a variety of scientific fields and applications. Using the control methods allows improvements in forecasting the dynamics of unstable economic processes and offers opportunities for governments, central banks, and other policy makers to modify the behaviour of the economic system to achieve its best performance. One effective method for control of chaos and computation of unstable periodic orbits (UPOs) is the unstable delay feedback control (UDFC) approach, suggested by K. Pyragas. This paper proposes the application of the Pyragas' method within framework of economic models. We consider this method through the example of the Shapovalov model, by describing the dynamics of a mid-size firm. The results demonstrate that suppressing chaos is capable in the Shapovalov model, using the UDFC method.

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“…In particular, for the autonomous case, if the linearization matrix has an odd number of positive eigenvalues then stabilization is impossible by means of DFC methods ([8], [15]). An interesting review on the evolution of the ONL problem and its derivations may be found in [2].…”

Section: Introductionmentioning

confidence: 99%

“…It makes use of a control signal obtained from the difference between the current state of the system and the state of the system delayed by the period of the UPO. The DFC method is also reformulated as a tool to stabilize equilibria embedded in chaotic attractors (see [9] and references within it). With this objective, it is implemented on known chaotic systems as Chen system [25] or Rossler system [2] and on technical applications like [10] or [26] among others.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, for the autonomous case, if the linearization matrix has an odd number of positive eigenvalues then stabilization is impossible by means of DFC methods ( [19], [7]). An interesting review on the evolution of the ONL problem and its derivations may be found in [9]. Another drawback of time delayed feedback is that the controlled system comes out a delayed differential equation, the state space of which is infinite dimensional and hence it is quite difficult to state analytical results and to get effective stabilization criteria.…”

Section: Introductionmentioning

confidence: 99%

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Oscillating delayed feedback control schemes for stabilizing equilibrium points

Pastor

González

2019

Heliyon

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Limitations of the delayed feedback control and of its extended versions have been fully treated in the literature. The oscillating delayed feedback control appears as a promising scheme to overcome this problem. Two methods based on it are dealt with in this work. It is rigorously proven that for a nonlinear scalar system, stabilization in one of its (unstable) equilibrium points is achieved if any of these methods is applied. An ad-hoc map is associated to the (continuous) controlled system and the results are derived using discrete-time system stabilization tools. Moreover, the stability parameters region is fully described and issues like control performance, rate of convergence or robustness aspects are carefully analyzed.

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A short survey on Pyragas time-delay feedback stabilization and odd number limitation

Cited by 22 publications

References 28 publications

The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension

Time-delay control for stabilization of the Shapovalov mid-size firm model

Oscillating delayed feedback control schemes for stabilizing equilibrium points

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