Melnikov method approach to control of homoclinic/heteroclinic chaos by weak harmonic excitations

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 364, Issue 1846, 2006, Pages 2335-2351

  Chacón, Ricardo ^a  

^a UNIVERSITY OF EXTREMADURA   (Spain)

Author keywords

Chaos control; Harmonic perturbation; Heteroclinic chaos; Homoclinic chaos; Melnikov method

Indexed keywords

EID: 33748302815     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2006.1828     Document Type: Review

References (65)

1. Alekseev, V. V. & Loskutov, A. Y. 1987 Control of a system with a strange attractor through periodic parametric action. Sov. Phys. Dokl. 32, 1346-1348.
2. Alonso, S., Sagués, F. & Mikhailov, A. S. 2003 Taming Winfree turbulence of scroll waves in excitable media. Science 299, 1722-1725. (doi:10.1126/science.1080207)
3. Arnold, V. I. 1964 Instability of dynamical systems with many degrees of freedom. Sov. Math. Dokl. 5, 581-585.
4. Arrowsmith, D. K. & Place, C. M. 1990 An introduction to dynamical systems. Cambridge, UK: Cambridge University Press.
5. Azevedo, A. & Rezende, S. M. 1991 Controlling chaos in spin-wave instabilities. Phys. Rev. Lett. 66, 1342-1345. (doi:10.1103/PhysRevLett.66.1342)
6. Balibrea, F., Chacón, R. & López, M. A. 1998 Inhibition of chaotic escape by an additional driven term. Int. J. Bif. Chaos 8, 1719-1723. (doi:10.1142/S0218127498001406)
7. Barone, A. & Paterno, G. 1982 Physics and applications of the Josephson effect. New York, NY: Wiley.
8. Braiman, Y. & Goldhirsch, I. 1991 Taming chaotic dynamics with weak periodic perturbations. Phys. Rev. Lett. 66, 2545-2548. (doi:10.1103/PhysRevLett.66.2545)
9. Cai, C., Xu, Z. & Xu, W. 2002 Melnikov s analysis of time-delayed feedback control in chaotic dynamics. IEEE Trans. Circuits Syst. 749, 1724-1728. (doi:10.1109/TCSI.2002.805738)
10. Chacón, R. 1995a Suppression of chaos by selective resonant parametric perturbations. Phys. Rev. E 51, 761-764.
11. Chacón, R. 1995b Natural symmetries and regularization by means of weak parametric modulations in the forced pendulum. Phys. Rev. E 52, 2330-2337.
12. Chacón, R. 1998 Comparison between parametric excitation and additional forcing terms as chaos-suppressing perturbations. Phys. Lett. 249A, 431-436.
13. Chacón, R. 1999 General results on chaos suppression for biharmonically driven dissipative systems. Phys. Lett. 257A, 293-300.
14. Chacón, R. 2001a Maintenance and suppression of chaos by weak harmonic perturbations: a unified view. Phys. Rev. Lett. 86, 1737-1740.
15. Chacón, R. 2001b Role of ultrasubharmonic resonances in taming chaos by weak harmonic perturbations. Europhys. Lett. 54, 148-153.
16. Chacón, R. 2002 Relative effectiveness of weak periodic excitations in suppressing homoclinic/heteroclinic chaos. Eur. Phys. J. B 30, 207-210.
17. Chacón, R. 2004 Dissipative dynamics of a charged particle in the field of three plane waves: chaos and control. arXiv:nlin.CD/0406066.
18. Chacón, R. & Cirac, J. I. 1995 Chaotic and regular behavior of a trapped ion interacting with a laser field. Phys. Rev. A 51, 4900-4906.
19. Chacón, R. & Díaz Bejarano, J. 1993 Routes to suppressing chaos by weak periodic perturbations. Phys. Rev. Lett. 71, 3103-3106.
20. Chacón, R. & Martínez, J. A. 2002 Inhibition of chaotic escape from a potential well by incommensurate escape-suppressing excitations. Phys. Rev. E 65, 036 213/1-7.
21. Chacón R., Balibrea, F. & López, M. A. 1996 Inhibition of chaotic escape from a potential well using small parametric modulations. J. Math. Phys. 37, 5518-5523. (doi:10.1063/1.531720)
22. Chacón, R., Balibrea, F. & López, M. A. 1997 Role of parametric resonance in the inhibition of chaotic escape from a potential well. Phys. Lett. 235A, 153-158.
23. Chacón, R., Balibrea, F. & López, M. A. 2001a Role of nonlinear dissipation in the suppression of chaotic escape from a potential well. Phys. Lett. 279A, 38-46.
24. Chacón, R., Palmero, F. & Balibrea, F. 2001b Taming chaos in a driven Josephson junction. Int. J. Bif. Chaos 11, 1897-1909. (doi:10.1142/S0218127401003073)
25. Chacón, R., Sánchez-Bajo, F. & Martínez, J. A. 2002 Robustness in the suppression of bidirectional chaotic escape from a potential well by weak parametric excitations. Phys. Lett. 303A, 190-196.
26. Chacón, R., Preciado, V. & Tereshko, V. 2003 Suppressing temporal and spatio-temporal chaos by multiple weak resonant excitations: application to arrays of chaotic coupled oscillators. Europhy. Lett. 63, 667-673.
27. Chen, G. & Dong, X. 1998 From chaos to order: perspectives, methodologies and applications. Singapore: World Scientific.
28. 2 laser. Phys. Rev. E 54, 4576-4579. (doi:10.1103/PhysRevE.54.4576)
29. Chow, S. N., Hale, J. K. & Mallet-Paret, J. 1980 An example of bifurcation to homoclinic orbits. J. Diff. Eqns. 37, 351-373. (doi:10.1016/0022-0396(80)90104-7)
30. Cicogna, G. & Fronzoni, L. 1990 Effects of parametric perturbations on the onset of chaos in the Josephson-junction model: theory and analog experiments. Phys. Rev. A 42, 1901-1906. (doi:10.1103/PhysRevA.42.1901)
31. Cicogna, G. & Fronzoni, L. 1993 Modifying the onset of homoclinic chaos: application to a bistable potential. Phys. Rev. E 47, 4585-4588. (doi:10.1103/PhysRevE.47.4585)
32. Contopoulos, G., Kandrup, H. & Kaufmann, D. 1993 Fractal properties of escape from a two-dimensional potential. Physica 64D, 310-323.
33. Cuadros, F. & Chacón, R. 1993 Comment on suppression of chaos by resonant parametric perturbations. Phys. Rev. E 47, 4628-4629. (doi:10.1103/PhysRevE.47.4628)
34. Ding, W. X., She, H. Q., Huang, W. & Yu, C. X. 1994 Controlling chaos in a discharge plasma. Phys. Rev. Lett. 72, 96-99. (doi:10.1103/PhysRevLett.72.96)
35. Ditto, W. L., Rauseo, S. N. & Spano, M. L. 1990 Experimental control of chaos. Phys. Rev. Lett. 65, 3211-3214. (doi:10.1103/PhysRevLett.65.3211)
36. Greenspan, B. D. 1981 Bifurcations in periodically forced oscillations: subharmonic and homoclinic orbits. Ph.D. thesis, Cornell University.
37. n. SIAM J. Math. Anal. 16, 907-931. (doi:10.1137/0516069)
38. Guckenheimer, J. & Holmes, P. 1983 Nonlinear oscillations, dynamical systems and bifurcations of vector fields. New York, NY: Springer.
39. Holmes, P. J. 1979 A nonlinear oscillator with a strange attractor. Phil. Trans. R. Soc. A 292, 419-448.
40. Holmes, P. J. & Marsden, J. E. 1982 Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom. Comm. Math. Phys. 82, 523-544. (doi:10.1007/BF01961239)
41. Hübler, A. W. & Lüscher, E. 1989 Resonant stimulation and control of nonlinear oscillators. Naturwissenschaft 76, 67-69.
42. Hunt, E. R. 1991 Stabilizing high-period orbits in a chaotic system: the diode resonator. Phys. Rev. Lett. 68, 1953-1955. (doi:10.1103/PhysRevLett.67.1953)
43. Kivshar, Y. S., Rödelsperger, F. & Benner, H. 1994 Suppression of chaos by nonresonant parametric perturbations. Phys. Rev. E 49, 319-324. (doi:10.1103/PhysRevE.49.319)
44. Lenci, S. & Rega, G. 2003 Optimal control of homoclinic bifurcation: theoretical treatment and practical reduction of safe basin erosion in the Helmholtz oscillator. J. Vib. Control 9, 281-315.
45. Lerman, L. M. & Umanski, Ia. L. 1984 On the existence of separatrix loops in four dimensional systems similar to integrable Hamiltonian systems. PMM USSR 47, 335-340.
46. Lichtenberg, A. J. & Lieberman, M. A. 1983 Regular and stochastic motion. New York, NY: Springer.
47. Lima, R. & Pettini, M. 1990 Suppression of chaos by resonant parametric perturbations. Phys. Rev. A 41, 726-733. (doi:10.1103/PhysRevA.41.726)
48. Lima, R. &: Pettini, M. 1993 Reply to 'comment on suppression of chaos by resonant parametric perturbations'. Phys. Rev. E 47, 4630-4631. (doi:10.1103/PhysRevE.47.4630)
49. Martínez, P. J. & Chacón, R. 2004 Taming chaotic solitons in Frenkel-Kontorova chains by weak periodic excitations. Phys. Rev. Lett. 93, 237006/1-4. (doi:10.1103/PhysRevLett.93.237006)
50. Melnikov, V. K. 1963 On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc. 12, 1-57.
51. Meucci, R., Gadomski, W., Ciofini, M. & Arecchi, F. T. 1994 Experimental control of chaos by means of weak parametric perturbations. Phys. Rev. E 49, R2528-R2531. (doi:10.1103/PhysRevE.49.R2528)
52. Petrov, V., Gaspar, V., Masere, J. & Showalter, K. 1993 Controlling chaos in the Belousov-Zhabotinsky reaction. Nature 361, 240-243. (doi:10.1038/361240a0)
53. Rajasekar, S. 1993 Controlling of chaos by weak periodic perturbations in Duffing-van der Pol oscillator. Pramana J. Phys. 41, 295-309.
54. Roy, R., Murphy, T. W., Maier, T. D., Gills, Z. & Hunt, E. R. 1992 Dynamical control of a chaotic laser: experimental stabilization of a globally coupled system. Phys. Rev. Lett. 68, 1259-1262. (doi:10.1103/PhysRevLett.68.1259)
55. Salam, F. M. A. 1987 The Melnikov technique for highly dissipative systems. SIAM J. Appl. Math. 47, 232-243. (doi:10.1137/0147015)
56. Salerno, M. 1991 Suppression of phase-locking chaos in long Josephson junctions by biharmonic microwave fields. Phys. Rev. 544, 2720-2726. (doi:10.1103/PhysRevB.44.2720)
57. Salerno, M. & Samuelsen, M. R. 1994 Stabilization of chaotic phase locked dynamics in long Josephson junctions. Phys. Lett. 190A, 177-181.
58. Schecter, S. 1987 Melnikov's method at a saddle-node and the dynamics of the forced Josephson junction. SIAM J. Math. Anal. 18, 1699-1715. (doi:10.1137/0518122)
59. Schiff, S. J., Jerger, K., Duong, D. H., Chang, T., Spano, M. L. & Ditto, W. L. 1994 Controlling chaos in the brain. Nature 370, 615-620. (doi:10.1038/370615a0)
60. Schöll, E. 2001 Nonlinear spatio-temporal dynamics and chaos in semiconductors. Cambridge, UK: Cambridge University Press.
61. Thompson, J. M. T. 1989 Chaotic phenomena triggering the escape from a potential well. Proc. R. Soc. A 421, 195-225.
62. Uchida, A., Sato, T., Ogawa, T. & Kannari, F. 1998 Nonfeedback control of chaos in a microchip solid-state laser by internal frequency resonance. Phys. Rev. E 58, 7249-7255. (doi:10.1103/PhysRevE.58.7249)
63. Wiggings, S. 1987 Chaos in the quasi-periodically forced Duffing oscillator. Phys. Lett. 124A, 138-142.
64. Wiggings, S. 1988 Global bifurcations and chaos. New York, NY: Springer.
65. Wiggings, S. 1990 Introduction to applied nonlinear dynamical systems and chaos. New York, NY: Springer.