Investigations of stochastic layers in nonlinear dynamics

Journal of Vibration and Acoustics, Transactions of the ASME

Volumn 122, Issue 1, 2000, Pages 36-41

  Luo, Albert C J ^a   Han, Ray P S ^b  

^a SOUTHERN ILLINOIS UNIVERSITY EDWARDSVILLE   (United States)

^b UNIVERSITY OF IOWA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

STOCHASTIC LAYERS;

PENDULUMS; STOCHASTIC CONTROL SYSTEMS;

NONLINEAR CONTROL SYSTEMS;

EID: 0033721122     PISSN: 10489002     EISSN: 15288927     Source Type: Journal    
DOI: 10.1115/1.568439     Document Type: Article

References (34)

1. Abdullaev, S. S., and Zaslavsky, G. M., 1995, “Self-similarity of Stochastic Magnetic Field Lines near the X-Point,” Phys. Plasmas, Vol. 2, pp. 4533-4541.
2. Abdullaev, S. S., and Zaslavsky, G. M., 1996, “Application of the Separatrix Map to Study Perturbed Magnetic Field Lines near the Separatrix,” Phys. Plasmas, Vol. 3, pp. 516-528.
3. Ahn, T., Kim, G., and Kim, S., 1996, “Analysis of the Separatrix Map in Hamiltonian Systems,” Physica D, Vol. 89, pp. 315-328.
4. Chirikov, B. V., 1979, “A Universal Instability of Many-Dimensional Oscillator System,” Phys. Rep., Vol. 52, pp. 263-379.
5. Escade, D. F., 1985, “Stochasticity in Classic Hamiltonian Systems: Universal Aspects,” Phys. Rep., Vol. 121, pp. 165-261.
6. Escande, D. F., and Doveil, F., 1981, “Renormalization Method for the Onset of Stochasticity in a Hamiltonian System,” Phys. Lett., Vol. 83A, pp. 307-310.
7. Gelfreich, V. G., Lazutkin, V. F., and Tabanov, M. B., 1991, “Exponentially Small Splitting in Hamiltonian Systems,” Chaos, Vol. 1, pp. 137-142.
8. Gelfreich, V. G., Lazutkin, V. F., and Svanidze, N. V., 1994, “A Refined Formula for the Separatrix Splitting for the Standard Map,” Physica D, Vol. 71, pp. 82-101.
9. Han, Ray P. S., and Luo, Albert, C. J., 1998, “Resonant Layers in Nonlinear Dynamics,” ASME J. Appl. Mech., Vol. 65, pp. 727-736.
10. Holmes, P. J., Marsden, J. E., and Scheurle, J., 1988, “Exponentially Small Splittings of Separatrices with Applications to KAM Theory and Degenerate Bifurcations,” Contemp. Math., Vol. 81, pp. 213-244.
11. Iomin, A., and Fishman, S., 1996, “Semiclassical Quantization of a Separatrix Map,” Phys. Rev., Vol. 54, pp. R1-R5.
12. Lazukin, V. F., Schachmannski, I. G., and Tabanov, M. B., 1989, “Splitting of Separatrices for Standard and Semistandard Mappings,” Physica D, Vol. 40, pp. 235-248.
13. Lichtenberg, A. J., and Lieberman, M. A., 1992, Regular and Chaotic Dynamics, 2nd ed., Springer-Verlag, New York.
14. Lin, W. A., and Reichl, L. E., 1986, “External Field Induced Chaos in an Infinite Square Well Potential,” Physica D, Vol. 19, pp. 145-152.
15. Luo, A. C. J., 1995, Analytical Modeling of Bifurcations, Chaos, and Multifractals in Nonlinear Dynamics, Ph.D. dissertation, University of Manitoba, Winnipeg, Manitoba, Canada.
16. Luo, A. C. J., 1999, “Resonant-Overlap Phenomena in Stochastic Layers of 1.5 Degrees of Freedom Nonlinear Systems,” Journal of Sound and Vibration, submitted.
17. Luo, A. C. J., Gu, K., and Han, R. P. S., 1999, “Resonant Separatrix Webs in the Stochastic Layers of the Twin-Well Duffing Oscillator,” Nonlinear Dynamics, Vol. 19, pp. 37-48.
18. Luo, A. C. J., and Han, R. P. S., 1997, “Stochastic and Resonant Layers in a Periodically Driven Pendulum,” International Mechanical Engineering Congress and Explosion, Dallas, Texas, USA. DE-Vol. 95/AMD-Vol. 223, pp. 207-215.
19. Luo, A. C. J., and Han, R. P. S., 1999a, “Analytical Predictions of Chaos in a Nonlinear Rod,” Journal of Sound and Vibration, in press.
20. Luo, A. C. J., and Han, R. P. S., 1999b, “The Resonant Mechanism of Stochastic Layers in Nonlinear Dynamic Systems with 1.5 Degrees of Freedom,” International Journal of Nonlinear Mechanics, submitted.
21. Luo, A. C. J., Han, R. P. S., and Xiang, Y. M., 1995, “Chaotic Analysis of Subharmonic Resonant Waves in Undamped and Damped Strings,” Journal ofHy-drodynamics, Vol. 7B, pp. 92-104.
22. Melnikov, V. K., 1962, “On the Behavior of Trajectories of System near to Autonomous Hamiltonian Systems,” Sov. Math. Dokl., Vol. 3, pp. 109-112.
23. Melnikov, V. K., 1963, “On the Stability of the Center for Time Periodic Perturbations,” Trans. Moscow Math. Soc., Vol. 12, pp. 1-57.
24. Poincare, H., 1899, LesMethods Nouvelles de la Mecanique Celests, 3 Vols. Gauthier-Zvillars: Paris.
25. Reichl, L. E., 1992, The Transition to Chaos in Conservative Classic System: Quantum Manifestations, Spring-Verlag, New York.
26. Reichl, L. E., and Zheng, W. M., 1984a, “Field-Induced Barrier Penetration in the Quadratic Potential,” Phys. Rev., Vol. 29A, pp. 2186-2193.
27. Reichl, L. E., and Zheng, W. M., 1984b, “Perturbed Double-Well System: The Pendulum Approximation and Low-Frequency Effect,” Phys. Rev., Vol. 30A, pp. 1068-1077.
28. Rom-Kedar, V., 1990, “Transport Rates of a Class of Two-Dimensional Maps and Flow,” Physica D, Vol. 43, pp. 229-268.
29. Rom-Kedar, V., 1994, “Homoclinic Tangles-Classification and Applications,” Nonlinearity, Vol. 7, pp. 441-473.
30. Rom-Kedar, V., 1995, “Secondary Homoclinic Bifurcation Theorems,” Chaos, Vol. 5, pp. 385-401.
31. Treschev, D. V., 1995, “An Averaging Method for Hamiltonian Systems, Exponentially Close to Integrable Ones,” Chaos, Vol. 6, pp. 6-14.
32. Treschev, D. V., 1998, “Width of Stochastic Layers in Near-Integrable TwoDimensional Symplectic Maps,” Physica D, Vol. 116, pp. 21-43.
33. Zaslavsky, G. M., and Filonenko, N. N., 1968, “Stochastic Instability of Trapped Particles and Conditions of Application of the Quasi-Linear Approximation,” Sov. Phys. JETP, Vol. 27, pp. 851-857.
34. Zaslavsky, G. M., and Abdullaev, S. S., 1995, “Scaling Properties and Anomalous Transport of Particles inside the Stochastic Layer,” Phys. Rev., Vol. 51, pp. 3901-3910.