Controlling Hamiltonian chaos

Ying-Cheng Lai; Mingzhou Ding; Celso Grebogi

doi:10.1103/PhysRevE.47.86

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Controlling Hamiltonian chaos

Ying-Cheng Lai

Mingzhou Ding

Celso Grebogi

Laboratory for Plasma Research, University of Maryland, College Park, Maryland 20742
Department of Biomedical Engineering, The Johns Hopkins University School of Medicine, Baltimore, Maryland 21205

Center for Complex Systems, Florida Atlantic University, Boca Raton, Florida 33431
Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431

Laboratory for Plasma Research, University of Maryland, College Park, Maryland 20742
Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Department of Mathematics, University of Maryland, College Park, Maryland 20742

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Phys. Rev. E 47, 86 – Published 1 January, 1993

DOI: https://doi.org/10.1103/PhysRevE.47.86

Abstract

The method for stabilizing an unstable periodic orbit in chaotic dynamical systems originally formulated by Ott, Grebogi, and Yorke (OGY) is not directly applicable to chaotic Hamiltonian systems. The reason is that an unstable periodic orbit in such systems often exhibits complex-conjugate eigenvalues at one or more of its orbit points. In this paper we extend the OGY stabilization method to control Hamiltonian chaos by incorporating the notion of stable and unstable directions at each periodic point. We also present an algorithm to calculate the stable and unstable directions. Other issues specific to the control of Hamiltonian chaos are also discussed.

References (25)

E. Ott, C. Grebogi and J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990); in Chaos: Soviet-American Perspectives on Nonlinear Science, edited by D. K. Campbell (American Institute of Physics, New York, 1990).
U. Dressler and G. Nitsche, Phys. Rev. Lett. 68, 1 (1992).
D. Auerbach, C. Grebogi, E. Ott, and J. A. Yorke (unpublished).
F. J. Romeiras, C. Grebogi, E. Ott and W. P. Dayawansa, Physica D 58, 165 (1992).
T. Tel, J. Phys. A 24, L1359 (1991).
W. L. Ditto, S. N. Rauseo and M. L. Spano, Phys. Rev. Lett. 65, 3211 (1990).
J. Singer, Y.-Z. Wang and H. H. Bau, Phys. Rev. Lett. 66, 1123 (1991).
A. Azevedo and S. M. Rezende, Phys. Rev. Lett. 66, 1342 (1991).
B. Peng, V. Petrov and K. Showalter, J. Phys. Chem. 95, 4957 (1991).
E. R. Hunt, Phys. Rev. Lett. 67, 1953 (1992).
R. Roy, T. W. Murphy, Jr., T. D. Maier and Z. Gills, Phys. Rev. Lett. 68, 1259 (1992); C. Reyl, L. Flepp, R. Badii, and E. Brun (unpublished).
A. Garfinkel, W. L. Ditto, M. L. Spano and J. Weiss, Science 257, 1230 (1992).
B. V. Chirikov, Phys. Rep. 52, 263 (1979).
Note that a trajectory originally in the neighborhood of a period-m orbit leaves the orbit exponentially like exp ( $λ_{u}$ m), where exp ( $λ_{u}$ ) > 1 is the unstable Lyapunov number of the orbit. Under the influence of noise, if control is applied at every mth time step, the trajectory will deviate from the orbit rapidly before the next parameter control can be applied and one may lose control. Hence, it is desirable that the control be applied at each time step.
Y. C. Lai, C. Grebogi, J. A. Yorke, and I. Kan, Nonlinearity (to be published).
C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. Lett. 57, 1284 (1986); P. Romeiras, C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. A 36, 5365 (1987).
The algebraic probability distribution of tau in Hamiltonian systems can be seen as follows. The phase space of typical Hamiltonian systems consists of KAM islands and chaotic regions. The KAM islands appear to exist in all scales and typically form an island-around-island phase-space structure. Hence, a particle initialized in the chaotic region may spend an arbitrarily long time in the island-around-island structure. A physical consequence of this is that particles exit algebraically [18–20] from some predefined (large) phase-space region containing the island-around-island structure. To be specific, imagine that there is some large region surrounded by an outermost KAM island curve and that outside this island there is a connected chaotic region. We then draw a large circle which contains the large island at its center and which also encloses a substantial part of the outer chaotic region. Now initialize a large number of randomly chosen initial conditions in the connected chaotic region outside the large island and evolve them under the dynamics. If a particle leaves the circle, we regard it as having exited the system. Let P (t) denote the probability of particles inside the circle at time t; then one typically observes [18] that P (t) app $t^{- α}$ .
C. F. F. Karney, Physica D 8, 360 (1983).
B. V. Chirikov and D. L. Shepelyansky, Physica D 13, 395 (1984).
J. D. Meiss and E. Ott, Phys. Rev. Lett. 55, 2741 (1985); Physica D 20, 387 (1986).
T. Shinbrot, E. Ott, C. Grebogi and J. A. Yorke, Phys. Rev. Lett. 65, 3215 (1990).
T. Shinbrot (private communication).
J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).
R. S. Mackay, J. D. Meiss and I. C. Pervival, Phys. Rev. Lett. 52, 697 (1984); and Physica D 13, 55 (1984).
After we completed the present work, we received a report by C. Reyl, L. Flepp, R. Badii, and E. Brun, who de- monstrated the control of NMR-laser chaos in high-dimensional embedding space. In their system, there also exist complex eigenvalues. Since the system is dissipative, those complex eigenvalues are not on the unit circle. They developed a modified OGY method in which parameter perturbations are chosen to minimize the norm || X ${vec}_{n + 1}$ - X ${vec}_{O (n + 1)}$ ( $p_{0}$ ) || contained in Eq. (4). Hence, their method does not make use of the geometric structure of hyperbolic period orbits, and consequently is quite different from the method we present in this paper.

[c1] E. Ott, C. Grebogi and J. A. Yorke, Phys. Rev. Lett. 64, 1196 (1990); in Chaos: Soviet-American Perspectives on Nonlinear Science, edited by D. K. Campbell (American Institute of Physics, New York, 1990).

[c2] U. Dressler and G. Nitsche, Phys. Rev. Lett. 68, 1 (1992).

[c3] D. Auerbach, C. Grebogi, E. Ott, and J. A. Yorke (unpublished).

[c4] F. J. Romeiras, C. Grebogi, E. Ott and W. P. Dayawansa, Physica D 58, 165 (1992).

[c5] T. Tel, J. Phys. A 24, L1359 (1991).

[c6] W. L. Ditto, S. N. Rauseo and M. L. Spano, Phys. Rev. Lett. 65, 3211 (1990).

[c7] J. Singer, Y.-Z. Wang and H. H. Bau, Phys. Rev. Lett. 66, 1123 (1991).

[c8] A. Azevedo and S. M. Rezende, Phys. Rev. Lett. 66, 1342 (1991).

[c9] B. Peng, V. Petrov and K. Showalter, J. Phys. Chem. 95, 4957 (1991).

[c10] E. R. Hunt, Phys. Rev. Lett. 67, 1953 (1992).

[c11] R. Roy, T. W. Murphy, Jr., T. D. Maier and Z. Gills, Phys. Rev. Lett. 68, 1259 (1992); C. Reyl, L. Flepp, R. Badii, and E. Brun (unpublished).

[c12] A. Garfinkel, W. L. Ditto, M. L. Spano and J. Weiss, Science 257, 1230 (1992).

[c13] B. V. Chirikov, Phys. Rep. 52, 263 (1979).

[c14] Note that a trajectory originally in the neighborhood of a period-m orbit leaves the orbit exponentially like exp ( $λ_{u}$ m), where exp ( $λ_{u}$ ) > 1 is the unstable Lyapunov number of the orbit. Under the influence of noise, if control is applied at every mth time step, the trajectory will deviate from the orbit rapidly before the next parameter control can be applied and one may lose control. Hence, it is desirable that the control be applied at each time step.

[c15] Y. C. Lai, C. Grebogi, J. A. Yorke, and I. Kan, Nonlinearity (to be published).

[c16] C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. Lett. 57, 1284 (1986); P. Romeiras, C. Grebogi, E. Ott and J. A. Yorke, Phys. Rev. A 36, 5365 (1987).

[c17] The algebraic probability distribution of tau in Hamiltonian systems can be seen as follows. The phase space of typical Hamiltonian systems consists of KAM islands and chaotic regions. The KAM islands appear to exist in all scales and typically form an island-around-island phase-space structure. Hence, a particle initialized in the chaotic region may spend an arbitrarily long time in the island-around-island structure. A physical consequence of this is that particles exit algebraically [18–20] from some predefined (large) phase-space region containing the island-around-island structure. To be specific, imagine that there is some large region surrounded by an outermost KAM island curve and that outside this island there is a connected chaotic region. We then draw a large circle which contains the large island at its center and which also encloses a substantial part of the outer chaotic region. Now initialize a large number of randomly chosen initial conditions in the connected chaotic region outside the large island and evolve them under the dynamics. If a particle leaves the circle, we regard it as having exited the system. Let P (t) denote the probability of particles inside the circle at time t; then one typically observes [18] that P (t) app $t^{- α}$ .

[c18] C. F. F. Karney, Physica D 8, 360 (1983).

[c19] B. V. Chirikov and D. L. Shepelyansky, Physica D 13, 395 (1984).

[c20] J. D. Meiss and E. Ott, Phys. Rev. Lett. 55, 2741 (1985); Physica D 20, 387 (1986).

[c21] T. Shinbrot, E. Ott, C. Grebogi and J. A. Yorke, Phys. Rev. Lett. 65, 3215 (1990).

[c22] T. Shinbrot (private communication).

[c23] J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617 (1985).

[c24] R. S. Mackay, J. D. Meiss and I. C. Pervival, Phys. Rev. Lett. 52, 697 (1984); and Physica D 13, 55 (1984).

[c25] After we completed the present work, we received a report by C. Reyl, L. Flepp, R. Badii, and E. Brun, who de- monstrated the control of NMR-laser chaos in high-dimensional embedding space. In their system, there also exist complex eigenvalues. Since the system is dissipative, those complex eigenvalues are not on the unit circle. They developed a modified OGY method in which parameter perturbations are chosen to minimize the norm || X ${vec}_{n + 1}$ - X ${vec}_{O (n + 1)}$ ( $p_{0}$ ) || contained in Eq. (4). Hence, their method does not make use of the geometric structure of hyperbolic period orbits, and consequently is quite different from the method we present in this paper.

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