Qualitative resonance of Shil'nikov-like strange attractors, Part II: Mathematical analysis

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

Volumn 14, Issue 3, 2004, Pages 893-912

  De Feo, Oscar ^a  

^a EPFL   (Switzerland)

Author keywords

Bifurcation analysis; Continuation techniques; 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;

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

RESONANCE;

EID: 13844298999     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127404009739     Document Type: Article

References (66)

1. Afraimovich, V., Verichev, N. & Rabinovich, M. [1986] "Stochastically synchronized oscillators in dissipative systems," Radiophys. Quant. Electron. 29, 795-803.
2. Basar, E. [1990] "Chaotic dynamics and resonance phenomena in brain function: Progress, perspectives, and thoughts," Chaos in Brain Function, ed. Basar, E. (Springer-Verlag, Berlin, Germany), pp. 1-30.
3. Bell, D. & Jacobson, D. [1975] Singular Optimal Control Problems (Academic Press, London, UK).
4. Belykh, V., Belykh, I., Colding-Jørgensen, M. & Mosekilde, E. [2000] "-Homoclinic bifurcations leading to the emergence of bursting oscillations in cell models," Europ. Phys. J. E3, 205-219.
5. Bittanti, S. & Colaneri, P. [1999] "Periodic control," Encyclopedya of Electrical and Electronics Engineering, ed. Webster, J. (John Wiley, NY), pp. 59-74.
6. ∞ Methods (Addison-Wesley, Menlo Park, CA).
7. Bykov, V. [1998] "On bifurcations leading to chaos in Chua's circuit," Int. J Bifurcation and Chaos 8, 685-699.
8. Callier, F. & Desoer, C. [1991] Linear System Theory (Springer-Verlag, NY).
9. ∞ Viewpoint (Academic Press, London, UK).
10. 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.
11. 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, Switzerland.
12. De Feo, O. [2002] "Qualitative resonance of Shil'nikovlike strange attractors, Part I: Experimental evidence," 'Int. J. Bifurcation and Chaos, 14, 873-891.
13. De Nicolao, G. [1994] "Cyclomonotonicity, Riccati equations and periodic receding horizon control," Automatica 30, 1375-1388.
14. De Nicolao, G., Ferrari-Trecate, G. & Pinzoni, S. [1998] "Zeros of continuous-time linear periodic systems," Automatica 34, 1651-1655.
15. de Souza, C. [1987] "Riccati differential equation in optimal filtering of periodic non-stabilizing systems," Int. J. Contr. 46, 1235-1250.
16. Dellnitz, M. & Hohmann, A. [1996] "The computation of unstable manifolds using subdivision and continuation," Nonlinear Dynamical Systems and Chaos, eds. Broer, H., van Gils, S., Hoveijn, I. & Takens, F. (Birkhauser, Boston, MA), pp. 449-459.
17. Dellnitz, M. & Junge, O. [1997] "Almost invariant sets in Chua's circuit," Int. J. Bifurcation and Chaos 7, 2475-2486.
18. Doedel, E., Keller, H. & Kernévez, J. [1991a] "Numerical analysis and control of bifurcation problems, Part I: Bifurcation in finite dimensions," Int. J. Bifurcation and Chaos 1, 493-520.
19. Doedel, E., Keller, H., & Kernévez, J. [1991b] "Numerical analysis and control of bifurcation problems, Part II: Bifurcation in infinite dimensions," Int. J. Bifurcation and Chaos 1, 745-772.
20. Doedel, E., Champneys, A., Fairgrieve, T., Kuznetsov, Y., Sandstede, B. & Wang, X. [1998] AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations with HomCont Computer Science Department, Concordia University, Montreal, Canada Motreal, Quebec, Canada.
21. Elbert, T., Ray, W., Kowalik, Z., Skinner, J., Graf, K. & Birbaumer, N. [1994] "Chaos and physiology: Deterministic chaos in excitable cell assemblies," Physiol. Rev. 74, 1-47.
22. Fan, Y. & Holden, A. [1993] "Bifurcations, burstings, chaos and crises in the Rose-Hindmarsh model for neuronal activity," Chaos Solit. Fract. 3, 439-449.
23. Freeman, W. [2000] "Characteristics of the synchronization of brain activity imposed by finite conduction velocities of axons," Int. J. Bifurcation and Chaos 10, 2307-2322.
24. Freire, E., Rodriguez-Luis, A., Gamero, E. & Ponce, E. [1993] "A case study for homoclinic chaos in an autonomous electronic circuit," Physica D62, 230-253.
25. Fujisaka, H. & Yamada, T. [1983] "Stability theory of synchronized motion in coupled oscillator systems," Progr. Theoret. Phys. 69, 32-46.
26. Gaspard, P. & Nicolis, G. [1983] "What can we learn from homoclinic orbits in chaotic dynamics?" J. Stat. Phys. 31, 499-518.
27. Gaspard, P., Kapral, R. & Nicolis, G. [1984] "Bifurcation phenomena near homoclinic systems: A two-parameter analysis," J. Stat. Phys. 35, 697-727.
28. Getting, P. [1989] "Emerging principles governing the operation of neural network," Ann. Rev. Neurosci. 12, 185-204.
29. Glendinning, G. & Sparrow, C. [1984] "Local and global behavior near homoclinic orbits," J. Stat. Phys. 35, 215-225.
30. Gonchenko, S., Shil'nikov, L. & Turaev, D. [1997] "Quasiattractors and homoclinic tangencies," Comput. Math. Appl. 34, 195-227.
31. Grasselli, O. & Longhi, S., [1991] "Robust tracking and regulation of linear periodic discrete-time systems," Int. J. Contr. 54, 613-633.
32. Hansen, K. [1993] Symbolic Dynamics in Chaotic Systems, Ph.D. thesis of the University of Oslo (Blindern, Norway).
33. Hansen, K. [1994] "Bifurcations of homoclinic orbits in bimodal maps," Physica E50, 1653-1656.
34. Hastings, H. & Powell, T. [1991] "Chaos in three species food chain," Ecology 72, 896-903.
35. Hindmarsh, J. & Rose, R. [1984] "A model of neuronal bursting using three coupled first order differential equations," Philos. Trans. Roy. Soc. London B221, 87-102.
36. Holden, A. & Fan, Y. [1992] "Crises-induced chaos in the Rose-Hindmarsh model for neuronal activity," Chaos Solit. Fract. 2, 583-595.
37. Hopfield, J. [1991] "Olfactory computation and object perception," Proc. Nat. Acad. Sci. USA 88, 6462-6466.
38. Hopfield, J. [1995] "Pattern recognition computation using action potential timing for stimulus representation," Nature 376, 33-36.
39. Hopfield, J. [1996] "Transforming neural computations and representing time," Proc. Nat. Acad. Sci. USA 93, 15440-15444.
40. Izhikevich, E. [2000] "Neural excitability, spiking and bursting," Int. J. Bifurcation and Chaos 10, 1171-1266.
41. Kaas-Peteresen, C. [1986] "Bifurcations in the Rose-Hindmarsh and Chay models," Chaos in Biological Systems, eds. Degen, H., Holden, A. & Olsen, L. (Plenum Press, NY), pp. 183-190.
42. Khibnik, A., Roose, D. & Chua, L. [1993] "On periodic orbits and homoclinic bifurcations in Chua's circuit with a smooth nonlinearity," Int. J. Bifurcation and Chaos 3, 363-384.
43. Kuznetsov, Y. & Rinaldi, S. [1996] "Remarks on food chain dynamics," Math. Biosci. 134, 1-33.
44. Kuznetsov, Y. & Levitin, V. [1997] "CONTENT: A multiplatform environment for continuation and bifurcation analysis of dynamical systems," Dynamical Systems Laboratory CWI, Centrum voor Wiskunde en Informatica, National Research Institute for Mathematics and Computer Science Amsterdam, the Netherlands.
45. Kuznetsov, Y. [1998] Elements of Applied Bifurcation Theory, 2nd edition (Springer-Verlag, NY).
46. Kuznetsov, Y., De Feo, O. & Rinaldi, S. [2001] "Belyakov homoclinic bifurcations in a tritrophic food chain model," SIAM J. Appl. Math. 62, 462-487.
47. Kwakernaak, H. & Sivan, R. [1972] Linear Optimal Control Systems (John Wiley, NY).
48. Lorenz, E. [1963] "Deterministic nonperiodic flow," J. Atmos. Sci. 20, 130-141.
49. Madan, R. [1993] Chua's Circuit: A Paradigm for Chaos, 4th edition (World Scientific, Singapore).
50. 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.
51. Mastumoto, T. [1993] Bifurcations: Sights, Sounds and Mathematics (Springer-Verlag, Tokyo, Japan).
52. Osinga, H. & Feudel, U. [2000] "Boundary crisis in quasiperiodically forced systems," Physica D141, 54-64.
53. Ott, E. [1993] Chaos in Dynamical Systems (Cambridge University Press, NY).
54. Pecora, L. & Carroll, T. [1990] "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824.
55. Rabinovich, M., Varona, P. & Abarbanel, H. [2000] "Nonlinear cooperative dynamics of living neurons," Int. J. Bifurcation and Chaos 10, 913-933.
56. Robert, C., Alligoodb, T., Ott, E. & Yorke, J. [2000] "Explosions of chaotic sets," Physica D144, 44-61.
57. Rössler, O. [1976] "An equation for continuous chaos," Phys. Lett. A57, 397-398.
58. Sedra, S. & Smith, K. [1998] Microelectronic Circuits, 4th edition (Oxford University Press, NY).
59. Shil'nikov, A., Nicolis, G. & Nicolis, C. [1995] "Bifurcation and predictability analysis of a low-order atmospheric circulation model," Int. J. Bifurcation and Chaos 5, 1701-1711.
60. Shil'nikov, L., Shil'nikov, A., Turaev, D. & Chua, L. [2000] Methods of Qualitative Theory in Nonlinear Dynamics: Part I (World Scientific, Singapore).
61. Shil'nikov, L., Shil'nikov, A., Turaev, D. & Chua, L. [2001] Methods of Qualitative Theory in Nonlinear Dynamics: Part II (World Scientific, Singapore).
62. Strauss, M. [1992] Introduction to Optimal Control Theory (Springer-Verlag, NY).
63. Svobodny, T. & Russell, D. [1989] "Phase identification in linear time-periodic systems," IEEE Trans. Automat. Contr. 34, 218-220.
64. ∞ filtering for a class of uncertain periodic systems," IEE Proc. Contr. Th. Appl. 138, 479-483.
65. ∞ state estimation for linear periodic systems," IEEE Trans. Automat. Contr. 38, 1704-1707.
66. Zhou, Z. & Peng, S. [2000] "Cyclic star products and universalities in symbolic dynamics of trimodal maps," Physica D140, 213-226.