# zbMATH — the first resource for mathematics

Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems. (English) Zbl 1073.37512
Summary: It is a typical route to generate chaos via period-doubling bifurcations in some nonlinear systems. We propose a new hybrid control strategy in which state feedback and parameter perturbation are used to control the period-doubling bifurcations and to stabilize unstable periodic orbits embedded in the chaotic attractor of a discrete chaotic dynamical system. Simulation shows that the higher stable $$2^n$$-periodic orbit of the system can be controlled to lower stable $$2^m$$-periodic orbits, $$m<n$$, by this methods. Some other numerical simulations are also presented to verify the theoretical analysis.

##### MSC:
 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior 93C10 Nonlinear systems in control theory 93B52 Feedback control 37C27 Periodic orbits of vector fields and flows
Full Text:
##### References:
 [1] Ott, E.; Grebogi, C.; Yorke, J.A., Controlling chaos, Phys. rev. lett., 64, 1196-1199, (1990) · Zbl 0964.37501 [2] Chen, G., On some controllability condition for chaotic dynamics control, Chaos, solitons & fractals, 8, 9, 1461-1467, (1997) [3] Chen, G.; Yu, X., On time delayed feedback control of chaos, IEEE trans. circ. sys.-I, 46, 767-772, (1999) · Zbl 0951.93034 [4] Chen, G.; Dong, X., On feedback control of chaotic continuous-time systems, IEEE trans. circ. sys.-I, 40, 591-601, (1993) · Zbl 0800.93758 [5] Luo, X.S.; Fang, J.Q.; Wang, L.H., A new strategy of chaos control and a unified mechanism for several kinds of chaos control methods, Acta physica sinica, 8, 12, 895-901, (1999), [English edition] [6] Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys. lett. A, 170, 421-428, (1992) [7] Lima, R.; Pettini, M., Suppression of chaos by resonant parametric perturbations, Phys. rev. A, 41, 2, 726-733, (1990) [8] Braiman, Y., Taming chaotic dynamics with periodic perturbations, Phys. rev. lett., 66, 2545-2549, (1991) · Zbl 0968.37508 [9] Rajasekar, S.; Murreali, K.L., Control of chaos by nonfeedback methods in a simple electronic circuit system and the fitz huge-Nagumo equation, Chaos, solitons & fractals, 8, 9, 1554-1558, (1997) [10] Abed, F.H.; Wang, H.O.; Chen, R.C., Stabilization of period doubling bifurcations and implications for control of chaos, Physica D, 70, 154-164, (1994) · Zbl 0807.58033 [11] Yang, L.; Liu, Z.; Mao, J., Controlling hyperchaos, Phys. rev. lett., 84, 67-70, (2000) [12] Yang, L.; Liu, Z.; Chen, G., Chaotifying a continuous-time system via impulsive input, Int. J. bifurcation chaos, 12, 1411-1416, (2002)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.