Analysis of static and dynamic bifurcations from a feedback systems perspective

Dynamical Systems

Volumn 12, Issue 4, 1997, Pages 293-317

  Moiola, Jorge L ^a   Colantonio, María C ^a   Doñate, Pedro D ^a  

^a UNIVERSIDAD NACIONAL DEL SUR   (Argentina)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS; BIFURCATION (MATHEMATICS); FEEDBACK; FREQUENCY DOMAIN ANALYSIS; NYQUIST DIAGRAMS;

GENERALIZED NYQUIST STABILITY CRITERION; HOPF BIFURCATIONS;

DYNAMICS;

EID: 0031369597     PISSN: 14689367     EISSN: None     Source Type: Journal    
DOI: 10.1080/02681119708806249     Document Type: Article

References (46)

1. Abed, E. H. (1995) Bifurcation-theoretic issues in the control of voltage collapse. Proceedings of an IMA Workshop on System and Control Theory for Power Systems (eds J. H. Chow, P. V. Kokotovic and R. J. Thomas; Springer, New York), pp. 1-21.
2. Abed, E. H. and Fu, J.-H. (1986) Local feedback stabilization and bifurcation control, I. Hopf bifurcation. Systems & Control Letters 7, 11-17.
3. Abed, E. H. and Wang, H. O. (1995) Feedback control of bifurcation and chaos in dynamical systems. Nonlinear Dynamics and Stochastic Mechanics (eds Sri Namachchivaya and W. Kliemann; CRC Press, Boca Raton), pp. 153-173.
4. Abed, E. H., Wang, H. O., Alexander, J. C., Hamdan, A. M. A. and Lee, H.-C. (1993) Dynamic bifurcations in a power system model exhibiting voltage collapse. International Journal of Bifurcation and Chaos 3, 1169-1176.
5. Ajjarapu, V. and Lee, B. (1992) Bifurcation theory and its application to nonlinear dynamical phenomena in an electrical power system. IEEE Transactions on Power Systems 7, 424-431.
6. Allwright, D. J. (1977) Harmonic balance and the Hopf bifurcation theorem. Mathematical Proceedings of Cambridge Philosophical Society 82, 453-467.
7. Atherton, D. P. (1975) Nonlinear Control Engineering (Van Nostrand Reinhold, London).
8. Bressloff, P. C. (1994) Dynamics of compartmental model recurrent neural networks. Physical Review E 50, 2308-2319.
9. Chen, G. and Dong, X. (1993) From chaos to order - perspectives and methodologies in controlling nonlinear dynamical systems. International Journal of Bifurcation and Chaos 3, 1363-1409.
10. Chen, G. and Moiola, J. L. (1994) An overview on bifurcation, chaos and nonlinear dynamics in control systems. Journal of the Franklin Institute 331B, 819-858.
11. Chiang, H.-D., Liu, C. W., Varaiya, P. P., Wu, F. F. and Lauby, M. G. (1993) Chaos in a simple power system. IEEE Transactions on Power Systems 8, 1407-1417.
12. Chiang, H. D., Conneen, T. P. and Flueck, A. J. (1994) Bifurcations and chaos in electric power systems: numerical studies. Journal of the Franklin Institute 331B, 1001-1036.
13. Colantonio, M. C., Moiola, J. L., Desages, A. C. and Romagnoli, J. A. (1989) Hopf bifurcation in a CSTR using characteristic loci. Chemical Engineering Science 44, 595-602.
14. Colantonio, M. C., Padin, M., Desages, A. C. and Romagnoli, J. A. (1991) Application of higher-order bifurcation formulas. IEEE Transactions on Automatic Control 36, 746-749.
15. Dobson, I. and Chiang, H.-D. (1989) Towards a theory of voltage collapse in electric power systems. Systems Gf Control Letters 13, 253-262.
16. Doedel, E. J. (1986) AUTO: software for continuation and bifurcation problems in ordinary differential equations. Monograph, Applied Mathematics, Caltech.
17. Freire, E., Rodríguez-Luis, A. J., Gamero, E. and Ponce, E. (1993) A case study for homoclinic chaos in an autonomous electronic circuit. A trip from Takens-Bogdanov to Hopf-Šil'nikov. Physica D 62, 230-253.
18. Genesio, R. and Tesi, A. (1991) Chaos prediction in nonlinear feedback systems. IEE Proceedings Part D 138, 313-320.
19. Genesio, R. and Tesi, A. (1992) Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28, 531-548.
20. Genesio, R., Tesi, A., Wang, H. O. and Abed, E. H. (1993) Control of period doubling bifurcations using harmonic balance. Proceedings of the 32nd Conference on Decision and Control, San Antonio, Texas, USA, pp. 492-497.
21. Golubitsky, M. and Langford, W. F. (1981) Classification and unfoldings of degenerate Hopf bifurcations. Journal of Differential Equations 41, 375-415.
22. Golubitsky, M. and Schaeffer, D. G. (1985) Singularities and Groups in Bifurcation Theory 1 (Springer, New York).
23. Hale, J. K. and Koçak, H. (1991) Dynamics and Bifurcations (Springer, New York).
24. Healey, J. J., Broomhead, D. S., Cliffe, K. A., Jones, R. and Mullin, T. (1991) The origins of chaos in a modified Van der Pol oscillator. Physica D 48, 322-339.
25. Kwatny, H. G. and Piper, G. E. (1990) Frequency domain analysis of Hopf bifurcations in electric power networks. IEEE Transactions on Circuits and Systems 37, 1317-1321.
26. Kwatny, H. G., Pasrija, A. K. and Bahar, L. Y. (1986) Static bifurcations in electric power networks: loss of steady-state stability and voltage collapse. IEEE Transactions on Circuits and Systems 33, 981-991.
27. Lee, B. and Ajjarapu, V. (1993) Period-doubling route to chaos in an electrical power system. IEE Proceedings Part C 140, 490-496.
28. Leonov, G. A., Burkin, I. M. and Shepeljavyi, A. I. (1996) Frequency Methods in Oscillation Theory (Mathematics and Its Applications, Vol. 357; Kluwer, Dordrecht).
29. Llibre, J. and Ponce, E. (1996) Global first harmonic bifurcation diagram for odd piecewise linear control systems. Dynamics and Stability of Systems 11, 49-88.
30. Longtin, A. and Milton, J. G. (1989) Insight into the transfer function, gain, and oscillation onset for the pupil light reflex using nonlinear delay-differential equations. Biological Cybernetics 61, 51-58.
31. MacFarlane, A. G. J. and Postlethwaite, I. (1977) The generalized Nyquist stability criterion and multivariable root loci. International Journal of Control 25, 81-127.
32. Mees, A. I. (1981) Dynamics of Feedback Systems (John Wiley, Chichester).
33. Mees, A. I. and Allwright, D. J. (1979) Using characteristic loci in the Hopf bifurcation. IEE Proceedings Part D 126, 628-632.
34. Mees, A. I. and Chua, L. O. (1979) The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems. IEEE Transactions on Circuits and Systems 26, 235-254.
35. Moiola, J. L. (1993) On the computation of local bifurcation diagrams near degenerate Hopf bifurcations of certain types. International Journal of Bifurcation and Chaos 3, 1103-1122.
36. Moiola, J. L. and Chen, G. (1993) Frequency domain approach to computation and analysis of bifurcations and limit cycles: a tutorial. International Journal of Bifurcation and Chaos 3, 843-867.
37. Moiola, J. L. and Chen, G. (1996) Hopf Bifurcation Analysis - A Frequency Domain Approach (World Scientific, Singapore).
38. Moiola, J. L., Castro, L., Cendra, H. and Desages, A. C. (1990) Degenerate bifurcations and catastrophe sets via frequency analysis. Dynamics and Stability of Systems 5, 163-181.
39. Nayfeh, A. H., Harb, A. M. and Chin, C. M. (1995) Bifurcations in a power system model. Proceedings of the IEEE International Symposium on Circuits and Systems ISCAS'95, Seattle, WA, USA, pp. 283-286.
40. Piccardi, C. (1995) Bifurcation analysis via harmonic balance in periodic systems with feedback structure. International Journal of Control 62, 1507-1515.
41. Piper, G. E. and Kwatny, H. G. (1992) Complicated dynamics in a spacecraft attitude control systems. Journal of Guidance, Control and Dynamics 15, 825-831.
42. Planeaux, J. B. (1993) Bifurcation phenomena in CSTR dynamics. PhD thesis, University of Minnesota.
43. Razón, L. F. and Schmitz, R. A. (1987) Multiplicities and instabilities in chemically reacting systems - a review. Chemical Engineering Science 42, 1005-1047.
44. Tan, C. W., Varghese, M., Varaiya, P. and Wu, F. F. (1995) Bifurcation, chaos and voltage collapse in power systems. Proceedings of the IEEE 83, 1484-1496.
45. Tesi, A., Abed, E. H., Genesio, R. and Wang, H. O. (1996) Harmonic balance analysis of period-doubling bifurcations with implications for control of nonlinear dynamics. Automatica 32, 1255-1271.
46. Wang, H. O., Abed, E. H. and Hamdan, A. M. A. (1994) Bifurcations, chaos, and crises in voltage collapse of a model power system. IEEE Transactions on Circuits and Systems - Part I: Fundamental Theory and Applications 41, 294-302.