Regularization of the Trajectories of Dynamical Systems by Adjusting Parameters
A gradient learning method to regulate the trajectories
of some nonlinear chaotic systems is proposed. The method is
motivated by the gradient descent learning algorithms for neural
networks. It is based on two systems: dynamic optimization system
and system for finding sensitivities. Numerical results of several
examples are presented, which convincingly illustrate the efficiency
of the method.
[1] E. Ott, C. Grebogy, J. A. Yorke, "Controlling chaos," Phys. Rev. Lett.,
vol. 64, pp. 1196- 1199, 1990.
[2] J. Singer, Y. Z. Wang, H. H. Bau, ÔÇ×Controlling chaotic system," Phys.
Rev. Lett., vol. 66, pp. 1123- 1125, 1991.
[3] K. Pyragas, "Continuous control of chaos by self-controlling feedback,"
Phys. Lett., A, vol. 170, pp. 421-428, 1992.
[4] C. C. Hwang, R. F. Fung, J. Y. Hsieh, W. J. Li, ÔÇ×A nonlinear feedback
control of the Lorenz equation," Int. J. Eng. Sci., vol. 37, pp 1893-1900,
1999.
[5] Y. Braiman, I. Goldhirsh, "Taming chaotic dynamic with weak periodic
perturbations," Phys. Rev. Lett., vol. 66, pp. 2545- 2548, 1991.
[6] T. Kapitaniak, "The loss of chaos in a quasiperiodically-forced nonlinear
oscillator," Int. J. Bifurcation and Chaos, vol. 1, pp357-362, 1991.
[7] Ü. Lepik, H. Hein, "On response of nonlinear oscillators with random
frequency of excitation," J. Sound and Vibration, vol. 288, pp. 275-292,
2005.
[8] H. Hein, Ü. Lepik, "Response of nonlinear oscillators with random
frequency of excitation," revisited., Journal of Sound and Vibration, vol.
301, pp. 1040-1049, 2007.
[9] P. Baldi, "Gradient descent learning algorithm overview: A general
dynamical perspective," IEEE Transactions on neural networks, vol. 6,
no. 1, January 1995.
[10] I. P. Marino, J. Miguez, "An approximate gradient-descent method for
joint parameter estimation and synchronization of coupled chaotic
systems," Phys. Lett., A, vol. 351, pp. 262-267, 2006.
[11] A. Bryson, H. Yu-Chi, Applied optimal control, New York: Wiley,
1975.
[12] C. Sparrow, The Lorenz equations: Bifurcations, chaos and strange
attractors, Berlin: Springer, 1982.
[13] Y. Yao, "Dynamic tunneling algorithm for global optimization," IEEE
Transactions on systems, Man and Cybernetics, vol. 19, no. 5, pp. 1222-
1230, October, 1989.
[14] P. RoyChowdhury, Y. P. Singh and R. A. Chansarkar, "Dynamic
tunneling technique for efficient training of multilayer perceptrons,"
IEEE Transactions on neural networks, vol. 10, no. 1, pp.48-55,
January, 1999.
[1] E. Ott, C. Grebogy, J. A. Yorke, "Controlling chaos," Phys. Rev. Lett.,
vol. 64, pp. 1196- 1199, 1990.
[2] J. Singer, Y. Z. Wang, H. H. Bau, ÔÇ×Controlling chaotic system," Phys.
Rev. Lett., vol. 66, pp. 1123- 1125, 1991.
[3] K. Pyragas, "Continuous control of chaos by self-controlling feedback,"
Phys. Lett., A, vol. 170, pp. 421-428, 1992.
[4] C. C. Hwang, R. F. Fung, J. Y. Hsieh, W. J. Li, ÔÇ×A nonlinear feedback
control of the Lorenz equation," Int. J. Eng. Sci., vol. 37, pp 1893-1900,
1999.
[5] Y. Braiman, I. Goldhirsh, "Taming chaotic dynamic with weak periodic
perturbations," Phys. Rev. Lett., vol. 66, pp. 2545- 2548, 1991.
[6] T. Kapitaniak, "The loss of chaos in a quasiperiodically-forced nonlinear
oscillator," Int. J. Bifurcation and Chaos, vol. 1, pp357-362, 1991.
[7] Ü. Lepik, H. Hein, "On response of nonlinear oscillators with random
frequency of excitation," J. Sound and Vibration, vol. 288, pp. 275-292,
2005.
[8] H. Hein, Ü. Lepik, "Response of nonlinear oscillators with random
frequency of excitation," revisited., Journal of Sound and Vibration, vol.
301, pp. 1040-1049, 2007.
[9] P. Baldi, "Gradient descent learning algorithm overview: A general
dynamical perspective," IEEE Transactions on neural networks, vol. 6,
no. 1, January 1995.
[10] I. P. Marino, J. Miguez, "An approximate gradient-descent method for
joint parameter estimation and synchronization of coupled chaotic
systems," Phys. Lett., A, vol. 351, pp. 262-267, 2006.
[11] A. Bryson, H. Yu-Chi, Applied optimal control, New York: Wiley,
1975.
[12] C. Sparrow, The Lorenz equations: Bifurcations, chaos and strange
attractors, Berlin: Springer, 1982.
[13] Y. Yao, "Dynamic tunneling algorithm for global optimization," IEEE
Transactions on systems, Man and Cybernetics, vol. 19, no. 5, pp. 1222-
1230, October, 1989.
[14] P. RoyChowdhury, Y. P. Singh and R. A. Chansarkar, "Dynamic
tunneling technique for efficient training of multilayer perceptrons,"
IEEE Transactions on neural networks, vol. 10, no. 1, pp.48-55,
January, 1999.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:64971", author = "Helle Hein and Ülo Lepik", title = "Regularization of the Trajectories of Dynamical Systems by Adjusting Parameters", abstract = "A gradient learning method to regulate the trajectories
of some nonlinear chaotic systems is proposed. The method is
motivated by the gradient descent learning algorithms for neural
networks. It is based on two systems: dynamic optimization system
and system for finding sensitivities. Numerical results of several
examples are presented, which convincingly illustrate the efficiency
of the method.", keywords = "Chaos, Dynamical Systems, Learning, Neural
Networks", volume = "2", number = "7", pages = "514-4", }
