Chaos intermittency is composed of a laminar regime, which exhibits almost periodic motion, and a burst regime, which exhibits chaotic motion; it is known that in chaos intermittency, switching between these regimes occurs irregularly. In the laminar regime of chaos intermittency, the periodic solution before the saddle node bifurcation is closely related to its generation, and its behavior becomes periodic in a short time; the laminar is not, however, a periodic solution, and there are no unstable periodic solutions nearby. Most chaos control methods cannot be applied to the problem of stabilizing a laminar response to a periodic solution since they refer to information about unstable periodic orbits. In this paper, we demonstrate a control method that can be applied to the control target with laminar phase of a dynamical system exhibiting chaos intermittency. This method records the time series of a periodic solution prior to the saddle node bifurcation as a pseudo-periodic orbit and feeds it back to the control target. We report that when this control method is applied to a circuit model, laminar motion can be stabilized to a periodic solution via control inputs of very small magnitude, for which robust control can be obtained.

Keywords:

References

Alexeeva, T., Diep, Q. B., Kuznetsov, N. & Zelinka, I. [2023] “ Forecasting and stabilizing chaotic regimes in two macroeconomic models via artificial intelligence technologies and control methods,” Chaos Solit. Fract. 170, 113377. Crossref, Web of Science, Google Scholar
Boccaletti, S., Grebogi, C., Lai, Y. -C., Mancini, H. & Maza, D. [2000] “ The control of chaos: Theory and applications,” Phys. Rep. 329, 103–197. Crossref, Web of Science, Google Scholar
Ding, J. & Lei, Y. [2023] “ Control of chaos with time-delayed feedback based on deep reinforcement learning,” Physica D 451, 133767. Crossref, Web of Science, Google Scholar
Dousseh, Y. P., Monwanou, A. V., Koukpémédji, A. A., Miwadinou, C. H. & Chabi Orou, J. B. [2023] “ Dynamics analysis, adaptive control, synchronization and anti-synchronization of a novel modified chaotic financial system,” Int. J. Dyn. Control 11, 862–876. Crossref, Google Scholar
Elaskar, S., del Río, E. & Elaskar, S. [2022a] “ Intermittency reinjection in the logistic map,” Symmetry 14, 481. Crossref, Web of Science, Google Scholar
Elaskar, S., del Rio, E. & Schulz, W. [2022b] “ Analysis of the type V intermittency using the Perron–Frobenius operator,” Symmetry 14, 2519. Crossref, Web of Science, Google Scholar
Goulefack, L. M., Chamgoue, A. C., Anteneodo, C. & Yamapi, R. [2022] “ Stability analysis of the Hindmarsh–Rose neuron under electromagnetic induction,” Nonlin. Dyn. 108, 2627–2642. Crossref, Web of Science, Google Scholar
Hou, L. & Cao, S. [2019] “ Nonlinear dynamic analysis on planetary gears-rotor system in geared turbofan engines,” Int. J. Bifurcation and Chaos 29, 1950076-1–20. Link, Web of Science, Google Scholar
Miino, Y., Ito, D., Ueta, T. & Kawakami, H. [2021] “ Locating and stabilizing unstable periodic orbits embedded in the horseshoe map,” Int. J. Bifurcation and Chaos 31, 2150110-1–12. Link, Web of Science, Google Scholar
Nayfeh, A. H. & Balachandran, B. [2008] Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods (John Wiley & Sons). Google Scholar
Nieto, A. R., Lilienkamp, T., Seoane, J. M., Sanjuán, M. A. & Parlitz, U. [2022] “ Control of escapes in two-degree-of-freedom open Hamiltonian systems,” Chaos 32, 063118. Crossref, Web of Science, Google Scholar
Nishiuchi, Y., Ueta, T. & Kawakami, H. [2006] “ Stable torus and its bifurcation phenomena in a simple three-dimensional autonomous circuit,” Chaos Solit. Fract. 27, 941–951. Crossref, Web of Science, Google Scholar
Ott, E., Grebogi, C. & Yorke, J. A. [1990] “ Controlling chaos,” Phys. Rev. Lett. 64, 1196. Crossref, Web of Science, Google Scholar
Peng, B., Petrov, V. & Showalter, K. [1991] “ Controlling chemical chaos,” J. Phys. Chem. 95, 4957–4959. Crossref, Web of Science, Google Scholar
Petrov, V., Gáspár, V., Masere, J. & Showalter, K. [1993] “ Controlling chaos in the Belousov–Zhabotinsky reaction,” Nature 361, 240–243. Crossref, Web of Science, Google Scholar
Pomeau, Y. & Manneville, P. [1980] “ Intermittent transition to turbulence in dissipative dynamical systems,” Commun. Math. Phys. 74, 189–197. Crossref, Web of Science, Google Scholar
Pyragas, K. [1992] “ Continuous control of chaos by self-controlling feedback,” Phys. Lett. A 170, 421–428. Crossref, Web of Science, Google Scholar
Pyragas, K. [2012] “ A twenty-year review of time-delay feedback control and recent developments,” Int. Symp. Nonlin. Th. Appl. (Citeseer), pp. 683–686. Google Scholar
Pyragas, V. & Pyragas, K. [2016] “ Act-and-wait time-delayed feedback control of nonautonomous systems,” Phys. Rev. E 94, 012201. Crossref, Web of Science, Google Scholar
Pyragas, V. & Pyragas, K. [2019] “ State-dependent act-and-wait time-delayed feedback control algorithm,” Commun. Nonlin. Sci. Numer. Simul. 73, 338–350. Crossref, Web of Science, Google Scholar
Strogatz, S. H. [2018] Nonlinear Dynamics and Chaos with Student Solutions Manual: With Applications to Physics, Biology, Chemistry, and Engineering (CRC Press). Crossref, Google Scholar
Trikha, P., Mahmoud, E. E., Jahanzaib, L. S., Matoog, R. & Abdel-Aty, M. [2021] “ Fractional order biological snap oscillator: Analysis and control,” Chaos Solit. Fract. 145, 110763. Crossref, Web of Science, Google Scholar
Ueta, T., Ito, D. & Aihara, K. [2015] “ Can a pseudo periodic orbit avoid a catastrophic transition?,” Int. J. Bifurcation and Chaos 25, 1550185-1–10. Link, Web of Science, Google Scholar

Vol. 34, No. 02

Metrics

History

Received 21 November 2023

Accepted 1 December 2023

Keywords

PDF download

Stabilization of Laminars in Chaos Intermittency

Abstract

References

Recommended