Dynamics and Control of a Chaotic Electromagnetic System
In this paper, different nonlinear dynamics analysis techniques are employed to unveil the rich nonlinear phenomena of the electromagnetic system. In particular, bifurcation diagrams, time responses, phase portraits, Poincare maps, power spectrum analysis, and the construction of basins of attraction are all powerful and effective tools for nonlinear dynamics problems. We also employ the method of Lyapunov exponents to show the occurrence of chaotic motion and to verify those numerical simulation results. Finally, two cases of a chaotic electromagnetic system being effectively controlled by a reference signal or being synchronized to another nonlinear electromagnetic system are presented.
[1] S. C. Chang, T. C. Tung, "Identification of a non-linear electromagnetic
system: an experimental study," Journal of Sound and Vibration, vol. 214, pp. 853-872, 1998.
[2] A. Wolf, J. B. Swift, H. Swinney and J. A. Vastano, "Determining
Lyapunov exponents from a time series," Physica D, vol. 16, pp. 285-317, 1985.
[3] E. Ott, C. Grebogi and J. A. Yorke, "Controlling chaos," Physical Review Letters, vol. 64, pp. 1196-1199, 1990.
[4] W. L. Ditto, S. N. Rauseo and M. L. Spano, "Experimental control of
chaos," Physical Review Letters, vol. 65, pp. 3211-3214, 1990.
[5] E. R. Hunt, "Stabilizing high-period orbits in a chaotic system: The diode
resonator," Physical Review Letters, vol. 67, pp. 1953-1955, 1991.
[6] Y. C. Lai, M. Ding and C. Grebogi, "Controlling Hamiltonian chaos,"
Physical Review E, vol. 67, pp. 86-92, 1993.
[7] C. Cai, Z. Xu, W. Xu and B. Feng, "Notch filter feedback control in a
class of chaotic systems," Automatica, vol. 38, pp. 695-701, 2002.
[8] C. Cai, Z. Xu and W. Xu, "Converting chaos into periodic motion by
feedback control," Automatica, vol. 38, pp. 1927-1933, 2002.
[9] C. C. Fun, P. C. Tung, "Experimental and analytical study of dither signals in a class of chaotic system," Physics Letters A, vol. 229, pp. 228-234, 1997.
[10] L. M. Pecora and T. L. Carroll, "Synchronization in Chaotic Systems,"
Physical Review Letters, vol. 64, pp. 821-823, 1990.
[11] J. K. John and R. E. Amritkar, "Synchronization by feedback and
adaptive control," International Journal of Bifurcation and Chaos, vol. 4,
pp. 1687-1695, 1994.
[12] S. Li and Y. P. Tian, "Finite time synchronization of chaotic systems,"
Chaos, Solitons & Fractals, vol. 15, pp. 303-310, 2003.
[13] E. W. Bai and K. E. Lonngren, "Synchronization and Chaos of Chaotic
Systems," Chaos, Solitons & Fractals, vol. 10, pp. 1571-1575, 1999.
[14] T. L. Carroll, L. M. Pecora, "Synchronizing Chaotic Circuits," IEEE
Trans. Circ Syst. I, vol. 38, pp. 453-456, 1991.
[15] T. L. Liao and S. H. Tsai, "Adaptive Synchronization of Chaotic Suystem
and its Application to Secure Communications," Chaos, Solitons &
Fractals, vol. 11, pp. 1387-1396, 2000.
[16] IMSL, Inc, User-s manual - IMSL MATH/LIBRARY, 1989, pp. 633.
[17] J. L. Kaplan, J. A. Yorke, Chaotic behavior of multidimensional
difference equations, Lecture Notes in Mathematics. New York: Springer-Verlag, 1979, pp.228-237.
[18] W. Szemplinska-Stupnicka, G. Iooss and F. C. Moon, Chaotic Motions in
Nonlinear Dynamical Systems. New York: Springer-Verlag, 1988.
[19] C. Y. Tseng, P. C. Tung, "Stability, bifurcation, and chaos of a structure
with a non-linear actuator," Japanese Journal of Applied Physics, vol. 34, pp. 3766-3774, 1995.
[1] S. C. Chang, T. C. Tung, "Identification of a non-linear electromagnetic
system: an experimental study," Journal of Sound and Vibration, vol. 214, pp. 853-872, 1998.
[2] A. Wolf, J. B. Swift, H. Swinney and J. A. Vastano, "Determining
Lyapunov exponents from a time series," Physica D, vol. 16, pp. 285-317, 1985.
[3] E. Ott, C. Grebogi and J. A. Yorke, "Controlling chaos," Physical Review Letters, vol. 64, pp. 1196-1199, 1990.
[4] W. L. Ditto, S. N. Rauseo and M. L. Spano, "Experimental control of
chaos," Physical Review Letters, vol. 65, pp. 3211-3214, 1990.
[5] E. R. Hunt, "Stabilizing high-period orbits in a chaotic system: The diode
resonator," Physical Review Letters, vol. 67, pp. 1953-1955, 1991.
[6] Y. C. Lai, M. Ding and C. Grebogi, "Controlling Hamiltonian chaos,"
Physical Review E, vol. 67, pp. 86-92, 1993.
[7] C. Cai, Z. Xu, W. Xu and B. Feng, "Notch filter feedback control in a
class of chaotic systems," Automatica, vol. 38, pp. 695-701, 2002.
[8] C. Cai, Z. Xu and W. Xu, "Converting chaos into periodic motion by
feedback control," Automatica, vol. 38, pp. 1927-1933, 2002.
[9] C. C. Fun, P. C. Tung, "Experimental and analytical study of dither signals in a class of chaotic system," Physics Letters A, vol. 229, pp. 228-234, 1997.
[10] L. M. Pecora and T. L. Carroll, "Synchronization in Chaotic Systems,"
Physical Review Letters, vol. 64, pp. 821-823, 1990.
[11] J. K. John and R. E. Amritkar, "Synchronization by feedback and
adaptive control," International Journal of Bifurcation and Chaos, vol. 4,
pp. 1687-1695, 1994.
[12] S. Li and Y. P. Tian, "Finite time synchronization of chaotic systems,"
Chaos, Solitons & Fractals, vol. 15, pp. 303-310, 2003.
[13] E. W. Bai and K. E. Lonngren, "Synchronization and Chaos of Chaotic
Systems," Chaos, Solitons & Fractals, vol. 10, pp. 1571-1575, 1999.
[14] T. L. Carroll, L. M. Pecora, "Synchronizing Chaotic Circuits," IEEE
Trans. Circ Syst. I, vol. 38, pp. 453-456, 1991.
[15] T. L. Liao and S. H. Tsai, "Adaptive Synchronization of Chaotic Suystem
and its Application to Secure Communications," Chaos, Solitons &
Fractals, vol. 11, pp. 1387-1396, 2000.
[16] IMSL, Inc, User-s manual - IMSL MATH/LIBRARY, 1989, pp. 633.
[17] J. L. Kaplan, J. A. Yorke, Chaotic behavior of multidimensional
difference equations, Lecture Notes in Mathematics. New York: Springer-Verlag, 1979, pp.228-237.
[18] W. Szemplinska-Stupnicka, G. Iooss and F. C. Moon, Chaotic Motions in
Nonlinear Dynamical Systems. New York: Springer-Verlag, 1988.
[19] C. Y. Tseng, P. C. Tung, "Stability, bifurcation, and chaos of a structure
with a non-linear actuator," Japanese Journal of Applied Physics, vol. 34, pp. 3766-3774, 1995.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:64470", author = "Shun-Chang Chang", title = "Dynamics and Control of a Chaotic Electromagnetic System", abstract = "In this paper, different nonlinear dynamics analysis techniques are employed to unveil the rich nonlinear phenomena of the electromagnetic system. In particular, bifurcation diagrams, time responses, phase portraits, Poincare maps, power spectrum analysis, and the construction of basins of attraction are all powerful and effective tools for nonlinear dynamics problems. We also employ the method of Lyapunov exponents to show the occurrence of chaotic motion and to verify those numerical simulation results. Finally, two cases of a chaotic electromagnetic system being effectively controlled by a reference signal or being synchronized to another nonlinear electromagnetic system are presented.
", keywords = "bifurcation, Poincare map, Lyapunov exponent, chaotic motion.", volume = "6", number = "5", pages = "619-6", }
