Singularly Perturbed Hopf Bifurcation

IEEE Transactions on Circuits and Systems

Volumn 32, Issue 12, 1985, Pages 1270-1280

  Abed, Eyad H ^a  

^a UNIVERSITY OF MARYLAND   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ELECTRIC NETWORKS, ACTIVE - MATHEMATICAL MODELS; ELECTRONIC CIRCUITS - OSCILLATIONS; SYSTEM STABILITY;

BIFURCATED PERIODIC SOLUTIONS DEGENERATION; SINGULARLY PERTURBED HOPF BIFURCATION; TWO-SCALE AND MULTIPLE-TIME-SCALE SYSTEMS;

MATHEMATICAL TECHNIQUES;

EID: 0022242033     PISSN: 00984094     EISSN: None     Source Type: Journal    
DOI: 10.1109/TCS.1985.1085669     Document Type: Article

References (49)

1. [1] N. Fenichel, “Geometric singular perturbation theory for ordinary differential differential equations, J. Differential Equations, vol. 31, pp. 53–98, 1979.
2. [2] E. Hopf, “Abzweigung einer periodischen Losung von einer stationaren Losung eines Differentialsystems,” Ber. Math. Phys. Kl. Sacks. Akad., vol. 94, 1–22, 1942. (Translation to English with commentary by L. N. Howard and N. Kopell in [5, pp. 163-205].)
3. [3] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory. New York: Springer-Verlag, 1982.
4. [4] B. D. Hassard, N. D. Kazarinoff, and Y.-H. Wan, Theory and Applications of Hopf Bifurcation. Cambridge, UK: Cambridge Univ. Press, 1981.
5. [5] J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications. New York: Springer-Verlag, 1976.
6. [6] A. B. Poore, “On the theory and application of the Hopf-Friedrichs bifurcation theory,” Arch. Rational Mech. Anal., vol. 60, pp. 371–393, 1976.
7. [7] D. Ruelle and F. Takens, “On the nature of turbulence,” Comm. Math. Phys., vol. 20, pp. 167–192, 1971; vol. 23, pp. 343–344, 1971.
8. [8] D. V. Anosov, “On limit cycles in systems of differential equations with a small parameter in the highest derivatives,” AMS Translations Ser. 2, vol. 33, pp. 233–275, 1963.
9. [9] H. K. Khalil and P. V. Kokotovic, “D-stability and multiparameter singular, perturbation,” SIAM J. Contr. Opt., vol. 17, pp. 56–65, 1979.
10. [10] J. K. Hale, Ordinary Differential Equations. 2nd edition. Malabar, FL: Krieger, 1980.
11. [11] N. Fenichel, “Oscillatory bifurcations in singular perturbation theory, I. Slow oscillations,” SIAM J. Math. Anal., vol. 14, pp. 861–867, 1983.
12. [12] E. H. Abed, “Singularly perturbed Hopf bifurcation and oscillations in power systems, Ph.D. dissertation, Univ. of California, Berkeley, Dec. 1982.
13. [13] A. I. Mees and L. O. Chua, “The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems,” IEEE Trans. Circuits Syst., vol. CAS-26, 235–254, 1979.
14. [14] E. Abed and P. Varaiya, “Oscillations in power systems via Hopf bifurcation,” in Proc. 20th IEEE Conf. Decision and Control, San Diego, CA, pp. 926–929, 1981.
15. [15] E. Abed and P. Varaiya, “Nonlinear oscillations in power systems,” Int. J. Elect. Power Energy Syst., vol. 6, pp. 37–43, 1984.
16. [16] E. Abed, N. Tsolas, and P. Varaiya, “Study of nonlinear oscillations due to exciter control using Hopf bifurcation,” in Proc. 1983 IEEE Int. Symp. Circuits Syst., Newport Beach, CA, pp. 1410–1413, 1983.
17. [17] E. Abed, N. Tsolas, and P. Varaiya, “Study of nonlinear oscillations due to exciter control,” in Proc. 1983 Amer. Control Conf., San Francisco, CA, pp. 674–678, 1983.
18. [18] J. C. Alexander, “Oscillatory solutions of a model nonlinear swing equation,” preprint, Dept. of Math., Univ. of Maryland, College Park, 1985.
19. [19] P. V. Kokotovic, R. E. O’Malley, Jr., and P. Sannuti, “Singular perturbations and order reduction in control theory—An overview,” Automatica, vol. 12, pp. 123–132, 1976.
20. [20] V. R. Saksena, J. O’Reilly, and P. V. Kokotovic, “Singular perturbation and time-scale methods in control theory: Survey, 1976-1983,” Automatica vol. 20, 273–294, 1984.
21. [21] P. V. Kokotovic, “Applications of singular perturbation techniques to control problems, SIAM Rev., vol. 26, 501–550, 1984.
22. [22] S. J. Merrill, “A model of the stimulation of B-cells by replicating antigen, I.,” Math. Biosci., vol. 41, pp. 125–141, 1978; II. vol. 41, pp. 143–155, 1978.
23. [23] J. Carr, Applications of Centre Manifold Theory. New York: Springer-Verlag, 1981.
24. [24] D. Hart, “The generator of a C’ flow,” in Dynamical Systems II, A. R. Bednarek and L. Cesari, Eds. New York: Academic, 1982, pp. 547–553.
25. [25] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. New York: Springer-Verlag, 1983.
26. [26] D. D. Siljak, Large Scale Dynamic Systems: Stability and Structure. New York: North-Holland, 1978.
27. [27] L. N. Howard, “Nonlinear oscillations,” in Nonlinear Oscillations in Biology, F. C. Hoppensteadt, Ed., Amer. Math. Soc., Providence, RI, 1979, pp. 1–63.
28. [28] M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems and Linear Algebra. New York: Academic, 1974.
29. [29] E. H. Abed, “Multiparameter singular perturbation problems: Iterative expansions and asymptotic stability,” Systems and Control Letters, vol. 5, pp. 279–282, 1985.
30. [30] E. H. Abed, “A new parameter estimate in singular perturbations,” Systems and Control Letters, to appear.
31. [31] E. H. Abed, “Stability of multiparameter singular perturbation problems with parameter bounds, I. Time-varying systems with cone-restricted restricted perturbations,” Systems Res. Rep. 85–5, Elect. Eng. Dep., Univ. of Maryland, College Park, 1985.
32. [32] E. H. Abed, “Stability of multiparameter singular perturbation problems with parameter bounds, II. Time-invariant systems with arbitrary perturbations,” Systems Res. Rep. 85–6, Elect. Eng. Dep., Univ. of Maryland, College Park, 1985.
33. [33] N. Kopell, “A geometric approach to boundary layer problems exhibiting resonance,” SIAM J. Appl. Math., vol. 37, pp. 436–458.
34. [34] R. E. Meyer and S. V. Partner, Eds., Singular Perturbations and Asymptotics. New York: Academic, 1980.
35. [35] J. H. Chow, Ed., Time-Scale Modeling of Dynamic Networks with Applications to Power Systems. Berlin, Germany: Springer-Verlag, 1982.
36. [36] D. J. Allwright, “Harmonic balance and the Hopf bifurcation theorem,” Math. Proc. Cambridge Philosophical Soc., vol. 82, pp. 453–467, 1977.
37. [37] S. L. Campbell, Singular Systems of Differential Equations. London, UK: Pitman, 1980.
38. [38] S. L. Campbell, Singular Systems of Differential Equations II. London, UK: Pitman, 1982.
39. [39] K. W. Chang and F. A. Howes, Nonlinear Singular Perturbation Phenomena: Theory and Application. New York: Springer-Verlag, 1984
40. [40] F. Hoppensteadt, “Properties of solutions of ordinary differential equations with small parameters,” Comm. Pure Appl. Math., vol. 24, p 807-840, 1971.
41. [41] M. Coderch, A. S. Willsky, S. S. Sastry, and D. A. Castanon, “Hierarchical aggregation of linear systems with multiple time scales,” IEEE Trans. Automat. Contr., vol. AC-28, pp. 1017–1030, 1983.
42. [42] R. E. O’Malley, Jr., “Boundary value problems for linear systems of ordinary differential equations involving many small parameters,” J. Math. Mech., vol. 18, pp. 835–855, 1969.
43. [43] H. K. Khalil and P. V. Kokotovic, “Control strategies for decision makers using different models of the same system,” IEEE Trans. Automat. Contr., vol. AC-23, pp. 289–298, 1978.
44. [44] H. K. Khalil, “Stabilization of multiparameter singularly perturbed systems,” IEEE Trans. Automat. Contr., vol. AC-24, pp. 790–791, 1979.
45. [45] H. K. Khalil and P. V. Kokotovic, “Decentralized stabilization of systems with slow and fast modes,” Large Scale Systems, vol. 1, pp. 141–148, 1980.
46. [46] H. K. Khalil, “Asymptotic stability of nonlinear multiparameter singularly perturbed systems,” Automatica, vol. 17, pp. 797–804, 1981.
47. [47] E. H. Abed, “Persistence of the Hopf bifurcation under singular• perturbations,” in Proc. 1984 IEEE Int. Symp. Circuits Syst., Montreal, Canada, pp. 1339–1342, 1984.
48. [48] E. H. Abed, “Strong D-stability,” preprint, Electrical Engineering Dep., Univ. of Maryland, College Park, 1985.
49. [49] A. N. Tikhonov, “Systems of differential equations containing small parameters in the derivatives,” Mat. Sb., (in Russian), vol. 31, no. 73, pp. 575-586 1952.