Chaos generalized synchronization of an inertial tachometer with new Mathieu-Van der Pol systems as functional system by GYC partial region stability theory. (English) Zbl 1250.34044

The authors implement a form of generalised synchronisation of a pair of identical chaotic ODE systems. Generalised synchronisation means that there is a functional relationship between the variables of one system and the variables of the other system. The synchronisation is implemented by driving one system by a signal which depends on the state of the driving system, the state of the driven system, and the value of an “error” function which is determined by the form of the functional relationship that one is trying to obtain. The exposition of the theory is difficult to follow. The results are demonstrated on a pair of “inertial tachometer” systems, and a pair of four ODEs formed by coupling a Mathieu equation with a van der Pol equation.


34D06 Synchronization of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
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[1] Moon, F. C., Chaotic Vibrations: An Introduction for Applied for Scientists and Engineers (1987), Wiley: Wiley New York · Zbl 0745.58003
[2] Thompson, J. M.T.; Stewart, H. B., Nonlinear Dynamics and Chaos (1986), Wiley: Wiley Chichester · Zbl 0601.58001
[3] Brockett TW. On conditions leading to chaos in feedback system. In: Proceedings of IEEE 21st conference on decision and control; 1982, p. 932-936.; Brockett TW. On conditions leading to chaos in feedback system. In: Proceedings of IEEE 21st conference on decision and control; 1982, p. 932-936.
[4] Holems, P., Bifurcation and chaos is a simple feedback control system, Proceedings of IEEE 22nd conference on decision and control, 365-370 (1983)
[5] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic system, Phys. Rev. Lett., 64, 821-824 (1990) · Zbl 0938.37019
[6] Femat, R.; Perales, G. S., On the chaos synchronization phenomenon, Phys Lett, A262, 50-60 (1999) · Zbl 0936.37010
[7] Krawiecki, A.; Sukiennicki, A., Generalizations of the concept of marginal synchronization of chaos, Chaos, Solitons Fract, 11, 9, 1445-1458 (2000) · Zbl 0982.37022
[8] Wang, C.; Ge, S. S., Adaptive synchronization of uncertain chaotic systems via backstepping design, Chaos, Solitons Fract, 12, 1199-1206 (2001) · Zbl 1015.37052
[9] Femat, R.; Ramirez, J. A.; Anaya, G. F., Adaptive synchronization of high-order chaotic systems: a feedback with low-order parameterization, Physica D, 139, 231-246 (2000) · Zbl 0954.34037
[10] Morgul, O.; Feki, M., A chaotic masking scheme by using synchronized chaotic systems, Phys Lett, A251, 169-176 (1999)
[11] Chen, S.; Lu, J., Synchronization of uncertain unified chaotic system via adaptive control, Chaos, Solitons Fract, 14, 4, 643-647 (2002) · Zbl 1005.93020
[12] Park Ju, H., Adaptive synchronization of hyperchaotic Chen system with uncertain parameters, Chaos, Solitons Fract, 26, 959-964 (2005) · Zbl 1093.93537
[13] Park Ju, H., Adaptive synchronization of Rossler system with uncertain parameters, Chaos, Solitons Fract, 25, 333-338 (2005) · Zbl 1125.93470
[14] Elabbasy, E. M.; Agiza, H. N.; El-Desoky, M. M., Adaptive synchronization of a hyperchaotic system with uncertain parameter, Chaos, Solitons Fract, 30, 1133-1142 (2006) · Zbl 1142.37325
[15] Ge, Z.-M.; Chen, C.-C., Phase synchronization of coupled chaotic multiple time scales systems, Chaos, Solitons Fract, 20, 639-647 (2004) · Zbl 1069.34056
[16] Ge, Z.-M.; Leu, W.-Y., Chaos synchronization and parameter identification for identical system, Chaos, Solitons Fract, 21, 1231-1247 (2004) · Zbl 1060.93523
[17] Ge, Z.-M.; Leu, W.-Y., Anti-control of chaos of two-degrees-of- freedom louderspeaker system and chaos synchronization of different order systems, Chaos, Solitons Fract, 20, 503-521 (2004) · Zbl 1048.37077
[18] Ge, Z.-M.; Chen, Y.-S., Synchronization of unidirectional coupled chaotic systems via partial stability, Chaos, Solitons Fract, 21, 101-111 (2004) · Zbl 1048.37027
[19] Ge, Z.-M.; Yu, J.-K., Pragmatical asymptotical stability theorem on partial region and for partial variable with applications to gyroscopic systems, Chinses J Mech, 16, 4, 179-187 (2000)
[20] Ge, Z.-M.; Chang, C.-M., Chaos synchronization and parameters identification of single time scale brushless DC motors, Chaos, Solitons Fract, 20, 883-903 (2004) · Zbl 1071.34048
[21] Ge, Zheng-Ming; Chen, Yen-Sheng, Synchronization of unidirectional coupled chaotic systems via partial stability, Chaos, Solitons Fract, 21, 101 (2004) · Zbl 1048.37027
[22] Ge, Z.-M.; Yao, C.-W.; Chen, H.-K., Stability on partial region in dynamics, J Chinese Soc Mech Eng, 15, 2, 140-151 (1994)
[23] Ge, Z.-M.; Chen, H.-K., Three asymptotical stability theorems on partial region with applications, Japanse J Appl Phys, 37, 2762-2773 (1998)
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