Constructing normal forms from experimental observations and time series analysis

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 7, Issue 1, 1997, Pages 107-127

  Karantonis, Antonis ^a   Pagitsas, Michael ^a  

^a ARISTOTLE UNIVERSITY OF THESSALONIKI   (Greece)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030635262     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S021812749700008X     Document Type: Article

References (42)

1. Arnold, V. I. [1971] "On matrices depending on parameters," Russian Math. Surveys 26, 29-43.
2. Arnold, V. I. [1972] "Lectures on bifurcations in versal families," Russian Math. Surveys 27, 54-121.
3. Arnold, V. I. [1973] Ordinary Differential Equations (MIT Press), pp. 119-121.
4. Arnold, V. I. [1988] Geometrical Methods in the Theory of Ordinary Differential Equations (Springer-Verlag, New York), pp. 222-246.
5. Ashkenazi, M. & Chow, S.-N. [1988] "Normal forms near critical points for differential equations and maps," IEEE Trans. Circ. Sys. 35, 850-862.
6. Carr, J. [1981] Applications of the Centre Manifold Theory (Springer-Verlag, New York), pp. 1-13.
7. Chua, L. O. & Kokubu, H. [1988] "Normal forms for nonlinear vector fields - Part I: Theory and algorithm," IEEE Trans. Circ. Sys. 35, 863-880.
8. Chua, L. O. & Kokubu, H. [1989] "Normal forms for nonlinear vector fields - Part II: Applications," IEEE Trans. Circ. Sys. 36, 51-70.
9. Crawford, J. D. [1991] "Introduction to bifurcation theory," Rev. of Mod. Phys. 63, 991-1037.
10. Diem, C. B. & Hudson, J. L. [1987] "Chaos during the electrodissolution of iron," AIChE J. 33, 218-224.
11. Elphick, C., Tirapegui, E., Brachet, M. E., Coullet, P. & Iooss, G. [1987] "A simple global characterization for normal forms of singular vector fields," Physica D29, 95-127.
12. Franck, U. F. & FitzHugh, R. [1961] "Periodische Elektrodenprozesse und ihre Beschreibung durch ein mathematisches Modell," Z. Elektrochem. 65, 156-168.
13. Gantmacher, F. R. [1977] The Theory of Matrices (Chelsea, New York), pp. 215-227.
14. Guckenheimer, J. [1981] "On a codimension two bifurcation," in Dynamical Systems and Turbulence eds. Rands, D. A. & Young, L. S., Lecture Notes in Mathematics vol. 898 (Springer-Verlag, New York), pp. 99-153.
15. Guckenheimer, J. [1984] "Multiple bifurcation problems of codimension two," SIAM J. Math. Anal. 15, 1-49.
16. Guckenheimer, J. & Holmes, Ph. [1983] Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer-Verlag, New York).
17. Holmes, Ph. [1980a] "Unfolding a degenerate nonlinear oscillator: A codimension two bifurcation," in Nonlinear Dynamics, ed. Heleman, R. H. G. (Annals New York Academy of Sciences), pp. 473-488.
18. Holmes, Ph. [1980b] "A strange family of three-dimensional vector fields near a degenerate singularity," J. Diff. Eq. 37, 382-409.
19. Holmes, Ph. [1981] "Center manifolds, normal forms and bifurcations of vector fields with applications to coupling between periodic and steady motions," Physica D2, 449-481.
20. Hudson, J. L. & Tsotsis, T. T. [1994] "Electrochemical reaction dynamics: A review," Chem. Eng. Sci. 49, 1493-1572.
21. Hynne, F., Sorensen, Graae, P. & Nielsen, K. [1990] "Quenching of chemical oscillations: General theory," J. Chem. Phys. 92, 1747-1757.
22. Iooss, G. & Adelmeyer, M., [1992] Topics in Bifurcation Theory and Applications (World Scientific Singapore), pp. 12-21.
23. Karantonis, A. & Pagitsas, M. [1996] "A comparative study for the calculation of the Lyapunov spectrum from non-linear experimental signals," Phys. Rev. E 53, 5428-5444.
24. Kleinke, M. U. [1995] "Chaotic behavior of current oscillations during iron electrodissolution in sulfuric acid," J. Phys. Chem. 99, 17403-17409.
25. Koper, M. T. M. [1996] "Oscillations and complex dynamical bifurcations in electrochemical systems," Adv. Chem. Phys. 92, 161-298.
26. Langford, W. F. [1979] "Periodic and steady-state mode interactions lead to tori," SIAM J. Appl. Math. 37, 22-48.
27. Langford, W. F. [1985] "A Review of interactions of Hopf steady-state bifurcations," in Nonlinear Dynamics and Turbulence eds. Barenblatt, G. Iooss, G. & Joseph, D. D. (Printman, London), pp. 215-237.
28. Langford, W. F. & Iooss, G. [1980] "Interactions of Hopf and pitchfork bifurcations," in Bifurcation Problems and Their Numerical Solution, (Birkhowsen, Basel), pp. 103-134.
29. Marsden, J. E. & McCracken, H. B. [1976] The Hopf Bifurcation and its Applications (Springer-Verlag, New York), pp. 27-50, 219-224.
30. Nielsen, K., Hynne, F. & Sorensen, Graae, P. [1991] "Hopf bifurcation in chemical kinetics," J. Chem. Phys. 94, 1020-1029.
31. Russell, P. & Newman, J. [1983] "Experimental determination of the passive-active transition for iron in 1M sulfuric acid," J. Electrochem. Soc. 130, 547-553.
32. Russell, P. & Newman, J. [1986a] "Anodic dissolution of iron in acidic sulfate electrolytes," J. Electrochem. Soc. 133, 59-69.
33. Russell, P. & Newman, J. [1986b] "Current oscillations observed within the limiting current plateau for iron in sulfuric acid," J. Electrochem. Soc. 133, 2094-2097.
34. Sano, M. & Sawada, Y. [1986] "Measurement of the Lyapunov spectrum from a chaotic time series," Phys. Rev. Lett. 55, 1082-1085.
35. Sazou D., Karantonis A. & Pagitsas, M. [1993] "Generalized Hopf, saddle node infinite period bifurcations and excitability during the electrodissolution of iron in a sulfuric acid solution," Int. J. Bifurcation and Chaos 3, 981-997.
36. Scully, J. C. [1975] Fundamentals of Corrosion (Pergamon), pp. 72-74.
37. Sorensen, Graae, P., Hynne, F. & Nielsen, K. [1990] "Characteristic modes of oscillatory chemical reactions," J. Chem. Phys. 92, 4778-4785.
38. Takens, F. [1974] "Singularities of vector fields," IHES 43, 47-100.
39. Takens, F. [1981] "Detecting strange attractors in fluid turbulence," in Dynamical Systems and Turbulence, eds. Rand, D. & Young, L. S., Lecture Notes in Mathematics, vol. 898 (Springer, Berlin) pp. 366-381.
40. Wang, Y. & Hudson, J. L. [1991] "Effect of electrode surface area on chaotic attractor dimension" AIChE J. 37, 1833-1843.
41. Wiggins, S. [1990] Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer-Verlag, New York), pp. 331-357, 596-603.
42. Zauderer, E. [1989] Partial Differential Equations (Wiley), pp. 48-51.