Qualitative resonance of Shil'nikov-like strange attractors, Part I: Experimental evidence

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 14, Issue 3, 2004, Pages 873-891

  De Feo, Oscar ^a  

^a EPFL   (Switzerland)

Author keywords

Homoclinic chaos; Pattern recognition; Periodic control theory; Synchronization

Indexed keywords

BIFURCATION (MATHEMATICS); CHAOS THEORY; MATHEMATICAL MODELS; ORDINARY DIFFERENTIAL EQUATIONS; PATTERN RECOGNITION; PERTURBATION TECHNIQUES; PROBLEM SOLVING; SYNCHRONIZATION; THEOREM PROVING;

HOMOCLINIC BIFURCATION; HOMOCLINIC CHAOS; PERIODIC CONTROL THEORY; QUALITATIVE RESONANCE;

RESONANCE;

EID: 10444230715     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127404009570     Document Type: Article

References (29)

1. Aeyels, D. & Willems, J. [1995] "Pole assignment for linear periodic systems by memoryless output feedback," IEEE Trans. Automat. Contr. 40, 735-739.
2. Afraimovich, V., Verichev, N. & Rabinovich, M. [1986] "Stochastically synchronized oscillators in dissipative systems," Radiophys. Quant. Electron. 29, 795-803.
3. Bittanti, S. & Colaneri, P. [1999] "Periodic control," in Encyclopedya of Electrical and Electronics Engineering, ed. Webster, J. (John Wiley, NY), pp. 59-74.
4. Brogan, W. [1996] Modern Control Theory, 3rd edition (Prentice-Hall, NY).
5. Brunovski, P. [1970] "A classification of linear controllable systems," Kybernetica 3, 173-187.
6. Callier, F. & Desoer, C. [1991] Linear System Theory (Springer-Verlag, NY).
7. De Feo, O. & Rinaldi, S. [1997] "Yield and dynamics in tritrophic food chains," The American Naturalist 150, 328-345.
8. De Feo, O., Maggio, G. & Kennedy, M. [2000] "The Colpitts oscillator: Families of periodic solutions and their bifurcations," Int. J. Bifurcation and Chaos 10, 935-958.
9. De Feo, O. [2001] Modeling Diversity by Strange Attractors with Application to Temporal Pattern Recognition, Ph.D. thesis of the Swiss Federal Institute of Technology Lausanne (Lausanne, Switzerland).
10. De Feo, O. [2004] "Qualitative resonance of Shil'nikovlike strange attractors, Part II: Mathematical analysis," Int. J. Bifurcation and Chaos 14, 893-912.
11. Fujisaka, H. & Yamada, T. [1983] "Stability theory of synchronized motion in coupled oscillator systems," Progr. Theoret. Phys. 69, 32-46.
12. Gaspard, P. & Nicolis, G. [1983] "What can we learn from homoclinic. orbits in chaotic dynamics?" J. Statist. Phys. 31, 499-518.
13. Gaspard, P., Kapral, R. & Nicolis, G. [1984] "Bifurcation phenomena near homoclinic systems: A two-parameter analysis," J. Statist. Phys. 35, 697-727.
14. Grewal, M. & Andrews, A. [1993] Kalman Filtering Theory and Practice (Prentice-Hall, Englewood Cliffs, NJ).
15. Hasler, M. [1994] "Synchronization principles and applications," in Circuits and Systems Tutorials, eds. Toumazou, C., Battersby, N. & Porta, S. (IEEE Press, NY), pp. 314-327.
16. Hastings, H. & Powell, T. [1991] "Chaos in three species food chain," Ecology 72, 896-903.
17. Kučera, V., Colaneri, P. & de Souza, C. [1998] "Output stabilizability of periodic systems: Necessary and sufficient conditions," Proc. American Control Conf. 1998 (IEEE Press, Piscataway, NJ).
18. Kuznetsov, Y. & Rinaldi, S. [1996] "Remarks on food chain dynamics," Math. Biosci. 134, 1-33.
19. Kuznetsov, Y. [1998] Elements of Applied Bifurcation Theory, 2nd edition (Springer-Verlag, NY).
20. Kuznetsov, Y., De Feo, O. & Rinaldi, S. [2001] "Belyakov homoclinic bifurcations in a tritrophic food chain model," SIAM J. Appl. Math. 62, 462-487.
21. Lakshmanan, M. & Murali, K. [1996] Chaos in Nonlinear Oscillators: Controlling and Synchronization (World Scientific, Singapore).
22. Maggio, G., De Feo, O. & Kennedy, M. [1999] "Nonlinear analysis of the Colpitts oscillator and applications to design," IEEE Trans. Circuit Syst.-I 46, 1118-1130.
23. Mastumoto, T. [1993] Bifurcations: Sights, Sounds and Mathematics (Springer-Verlag, Tokyo, Japan).
24. Pecora, L. & Carroll, T. [1990] "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824.
25. Petersen, I. & Savkin, A. [1999] Robust Kalman Filtering for Signals and Systems with Large Uncertainties (Birkhauser, Boston, MA).
26. Sedra, S. & Smith, K. [1998] Microelectronic Circuits, 4th edition (Oxford University Press, NY).
27. Shil'nikov, L., Shil'nikov, A., Turaev, D. & Chua, L. [2000] Methods of Qualitative Theory in Nonlinear Dynamics: Part I (World Scientific, Singapore).
28. Shil'nikov, L., Shil'nikov, A., Turaev, D. & Chua, L. [2001] Methods of Qualitative Theory in Nonlinear Dynamics: Part II (World Scientific, Singapore).
29. Tresser, C. [1984] "On some theorems of L. P. Shil'nikov and some applications," Ann. de l'Institute Henri Poincaré 40, 441-461.