Development of the nonlinear dynamical systems theory from radio engineering to electronics

International Journal of Bifurcation and Chaos

Volumn 19, Issue 7, 2009, Pages 2131-2163

  Letellier, Christophe ^a   Ginoux, Jean Marc ^b  

^a UNIVERSITÉ DE ROUEN   (France)

^b UNIVERSITÉ DE TOULON   (France)

Author keywords

Chaos; Electronic circuits; Nonlinear oscillation

Indexed keywords

CHAOS THEORY; NETWORKS (CIRCUITS); NONLINEAR DYNAMICAL SYSTEMS; SYSTEM THEORY;

CHUA CIRCUIT; NONLINEAR OSCILLATION; RADIO ENGINEERING; THREE-BODY PROBLEM; VAN DER POL EQUATION;

DYNAMICAL SYSTEMS;

EID: 70350610905     PISSN: 02181274     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218127409023986     Document Type: Article

References (108)

1. Abraham, H. & Bloch, E. [1919] "Sur l'entretien des oscillations ḿecaniques au moyen des lampes ̀a trois ́electrodes," Comptes-Rendus de l'Acad́emie des Sciences 168, 1197-1198.
2. Andronov, A. A. [1929] "Les cycles limites de Poincaŕe et la th́eorie des oscillations auto-entretenues," Comptes-Rendus de l'Acad́emie des Sciences 189, 559-561.
3. Andronov, A. & Vitt, A. [1930] "Zur Theorie des Mitnehmens von van der Pol," Archiv fur Elektrotech. 24, 99.
4. Andronov, A. A. & Lyubiana, A. G. [1935] "Application of Poincaŕe's theory on 'bifurcation points' and on the 'change of stability' to simple self-oscillating systems," Zh. Eksp. Teor. Fiz. 5, 3-4.
5. Andronov, A. A., Vitt, A. A. & Khaikin, S. E. [1937] Theory of Oscillators, Traduction anglaise par [1966] F. Immirzi (Dover).
6. Andronov, A. A. & Maier, A. G. [1944] "Misesa v teorii pryamogo regulisovaniya i teoriya to technykh probrazovanii," Doklady AN SSSR 53, 58-62.
7. Armstrong, E. H. [1914] "Operating feature of the Audion - Explanation of its action as an ampli.er, as a detector of high-frequency and as a valve," Electrical World 64, 1149.
8. Arńeodo, A., Coullet, P. & Tresser, C. [1981] "Possible new strange attractors with spiral structure," Commun. Math. Phys. 79, 573-579.
9. Appleton, E. & van der Pol, B. [1921] "On a type of oscillation-hysteresis in a simple triode generator," The London, Edinburgh, and Dublin Philosophical Magazine and J. Sci., Series 6, 43, 177-193.
10. Appleton, E. [1922] "The automatic synchronisation of triode oscillators," Proc. Cambridge Philos. Soc. 21, 231-248.
11. Birkho., D. [1927] Dynamical Systems, American Math. Soc. Coll. Publ., Vol.9 (American Mathematical Society, Providence, RI).
12. Bissel, C. [1998] "A. A. Andronov and the development of Soviet control engineering," IEEE Contr. Syst. Mag. 18, 56-62.
13. Blair, K. W. & Loud, W. S. [1960] "Periodic solution ofx + c ÿ x + g(x) = Ef (t) under variation of certain parameters," J. Soc. Ind. Appl. Math. 8, 74-101.
14. Brockett, R. F. [1982] "On conditions leading to chaos in feedback systems," Proc. 21st IEEE Conf. Decision and Control 2, 932-936.
15. Butterworth, S. [1920] "The maintenance of a vibrating system by means of a triode valve," Proc. Roy. Soc. London 32, 345-360.
16. Cartwright, M. L. & Littlewood, J. E. [1947] "On non-linear differential equations of the second-order," Ann. Math. 48, 472-494.
17. Chua, L. O. [1968] "Synthesis of new nonlinear network elements," Proc. IEEE 56, 1325-11243
18. Chua, L. O. [1971] "Memristor - the missing circuit element," IEEE Trans. Circuit Th. 18, 507-519.
19. Chua, L. O. & Green, D. N. [1974] "Synthesis of nonlinear periodic systems," IEEE Trans. Circuits Syst. CAS-21, 286-294.
20. Chua, L. O. & Green, D. N. [1976] "A qualitative analysis of the behavior of dynamic nonlinear networks: Stability of autonomous networks," IEEE Trans. Circuits Syst. CAS-23, 355-379.
21. Chua, L. O., Matsumoto, T. & Ichiraku, S. [1980] "Geometric properties of resistive nonlinear n-ports: Transversality, structural stability, reciprocity and anti-reciprocity," IEEE Trans. Circuits Syst. CAS- 27, 577-603.
22. Chua, L. O. & Ushida, A. [1981] "Algorithms for computing almost periodic steady-state response for nonlinear systems to multiple input frequencies," IEEE Trans. Circuits Syst. CAS-28, 953-971.
23. Chua, L. O., Hasler, M., Neyrinck, J. & Verburgh, P. [1982] "Dynamics of a piecewise linear resonant circuit," IEEE Trans. Circuits Syst. CAS-29, 535-547.
24. Chua, L. O. [1992] "The genesis of Chua's circuit," Archiv. Elektronik Ubertragungstechnik 46, 250-257.
25. Decaux, B. & Le Corbeiller, Ph. [1931] "Sur un syst̀eme ́electrique autoentretenu utilisant un tube ńeon," Comptes-Rendus de l'Acad́emie des Sciences (Paris) 193, 723.
26. Eccles, W. H. [1919] "The use of the triode valve in maintaining the vibration of a tuning fork," Proc. Physical Society of London 31, 269.
27. Eccles, W. H. & Jordan, F. [1919] "Sustaining the vibration of a tuning fork by a triode valve," The Electrician 82, 704.
28. Eddington, A. [1926] The Internal Constitution of Stars (Cambridge), p. 300.
29. Gao, J., Lu, H., He, D., Stone, L. & Zheng, Z. [2003] "Unexpected correspondence between noise-induced and master-slave complete synchronizations," Phys. Rev. E 68, 037202.
30. Glendinning, P. [1984] "Bifurcations near homoclinic orbits with symmetry," Phys. Lett. A 103, 163-166.
31. Grasiuk, A. Z. & Oraevsky, A. N. [1864] "The Dynamics of quantum oscillators," Quantum Electronics and Coherent Light, ed. Miles, P. A. (Academic Press), pp. 192-197.
32. Hayashi, C. [1953] Nonlinear Oscillations in Physical Systems (McGraw-Hill).
33. Hayashi, C., Shibayama, H. & Ueda, Y. [1963] "Quasiperiodic oscillations in a self-oscillatory system with external force," International Union in Theoretical Applied Mechanics Kiev 1, 495-509.
34. Hayashi, C., Ueda, Y., Akamatsu, N. & Itakura, H. [1970] "On the behavior of self-oscillatory systems with external force," Trans. Instit. Electron. Commun. Engin. A 53, 150-158.
35. Hayashi, C., Ueda, Y., Akamatsu, N. & Itakura, H. [1971] "On steady-state solutions of a nonlinear differential equation of the second order," Report of the Research Institute for Mathematical Sciences (universit ́e de Kyoto), 113, 1-27.
36. Kamke, E. [1944] differentialgleichungen I: Gewohnliche differentialgleichungen, Leipzig.
37. Khaikin, S. E. [1930] "Continuous and discontinuous oscillations," Zhurnal Prikladnoi Fiziki 7, 21.
38. Kiyashko, S. V., Pikovsky, A. S. & Rabinovich, M. I. [1980] "Self-exciting oscillator for the radio-frequency range with stochastic behavior," Radio Engin. Electron. Phys. 25, 74-79.
39. Komuro, M. [1983] "On embeddings of subshifts of finite type," Lecture Notes in Mathematics 1021, 299-301.
40. Komuro, M. [1985] "Lorenz attractors do not have the pseudo-orbit tracing property," J. Math. Soc. Japan 37, 489-514.
41. Kremann, [1913] Die periodischen Eischeinungen in der Chemie, Stuttgart, p. 124.
42. Krylo., N. & Bogoliubo., N. [1932] "Quelques exemples d'oscillations non lińeaires," Comptes-Rendus de l'Acad́emie des Sciences 194, 957;
43. "Sur le ph́enom̀ene d'entrainement en radiotechnique," Comptes-Rendus de l'Acad́emie des Sciences 194, 1064;
44. "Les ph́enom̀enes de d́emultiplication de fŕequence en radiotechnique," Comptes-Rendus de l'Acad́emie des Sciences 194, 1119.
45. Le Corbeiller, Ph. [1933] "Les syst̀emes autoentretenus et les oscillations de relaxation," Econometrica 1, 328-332.
46. Letellier, C. [2006] Le chaos dans la nature, Vuibert.
47. Letellier, C., Roulin, E. & Rossler, O. E. [2006] "Inequivalent topologies of chaos in simple equations," Chaos Solit. Fract. 28, 337-360.
48. Li, T. Y. & Yorke, J. A. [1975] "Period-three implies chaos," Amer. Math. Monthly 82, 985.
49. Líenard, A. [1928] "Etude des oscillations autoentretenues," Revue Ǵeńerale de l'Electricit́e 23, 901-946.
50. Linsay, P. S. [1981] "Period-doubling and chaotic behavior in a driven anharmonic oscillator," Phys. Rev. Lett. 47, 1349-1352.
51. Lord Rayleigh [1894] The Theory of the Sound, Londres, Vol.1, p. 212.
52. Lorenz, E. N. [1963] "Deterministic nonperiodic flow," J. Atmosph. Sci. 20, 130-141.
53. Lotka, A. [1920] "Analytical note on certain rhythmic relations in organic systems," Proc. Nat. Acad. Sci. 6, 410-415.
54. Lotka, A. [1925] Elements of Physical Biology (Baltimore), p. 88.
55. Lyapunov, A. [1892] The General Problem of the Stability of Motion, Doctoral dissertation, University of Kharkov (Taylor & Francis Ltd), 1992.
56. Mandel'shtam, L. & Papaleksi, N. [1934] "Uber die Erregung von elektrischen Schwingungen durch Parameteranderung, J. Zeitschrift fur Tech. Phys. 4, 5.
57. Matsumoto, T., Chua, L. O., Kawakami, H. & Ichiraku, S. [1981] "Geometric-properties of dynamic non-linear networks - transversality, local-solvability and eventual passivity," IEEE Trans. Circuits Syst. CAS-28, 406-428.
58. Matsumoto, T., Ikegami, G. & Chua, L. O. [1983] "Strong structural stability of resistive nonlinear n-ports," IEEE Trans. Circuits Syst. CAS-30, 197-222.
59. Matsumoto, T., Chua, L. O. & Tanaka, S. [1984] "Simplest chaotic nonautonomous circuit," Phys. Rev. A 30, 1155-1157.
60. Matsumoto, T. [1984a] "A chaotic attractor from Chua's circuit," Memorandum No. UCB/ ERLM84/36, University of California, Berkeley.
61. Matsumoto, T. [1984b] "A chaotic attractor from Chua's circuit," IEEE Trans. Circuits Syst. CAS-31, 1055-1058.
62. Matsumoto, T., Chua, L. O. & Komuro, M. [1985] "The double scroll," IEEE Trans. Circuits Syst. CAS-32, 798-818.
63. Maxwell, J. C. [1877] Matter and Motion; [1991] reprinted (Dover).
64. Mees, A. I. & Chua, L. O. [1979] "The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems," IEEE Trans. Circuits Syst. CAS-26, 235-254.
65. Mees, A. I. & Chua, L. O. [1983] "Synchronization and chaos," IEEE Trans. Circuits Syst. CAS-30, 620-626.
66. Mesmy, R. [1935] Radio-́electricit́e Ǵeńerale - Tome i : Etude des Circuits et de la Propagation, E. Chiron, ́editeur.
67. Moore, D. W. & Spiegel, E. A. [1966] "A thermally excited nonlinear oscillator," Astrophys. J. 143, 871-887.
68. Parker, T. S. & Chua, L. O. [1983] "A computer-assisted study of forced relaxation oscillations," IEEE Trans. Circuits Syst. CAS-30, 518-533.
69. Pikovskii, A. S. & Rabinovich, M. I. [1978] "A simple autogenerator with stochastic behavior," Sov. Phys. Dokl. 23, 183-185.
70. Pikovsky, A. S. & Rabinovich, M. I. [1981] "Stochastic oscillations in dissipative systems," Physica D 2, 8-24.
71. Poincaŕe, H. [1881-1886] "Sur les courbes d́e.nies par une ́equations di.́erentielles," J. de Math́ematiques Pures et Appliqúees, Śerie iii 7, 375-422;
72. Śerie iii 8, 251-296;
73. Śerie iv 1, 167-244;
74. Śerie iv 2, 151-217.
75. Poincaŕe, H. [1891] "Le Probl̀eme des trois corps," Revue Ǵeńerale des Sciences 2, 1-5.
76. Poincaŕe, H. [1892] Les Ḿethodes Nouvelles de la Ḿecanique Ćeleste, i, Chap. 2 (Gauthier-Vilars, Paris).
77. Poincaŕe, H. [1899] Les Ḿethodes Nouvelles de la Ḿecanique Ćeleste, Tome iii, 389.
78. Rikitake, T. [1958] "Oscillations of a system of discs dynamos," Proc. Cambridge Philos. Soc. 54, 89.
79. Robinson, F. N. H. [1990] "Analogue electronic model of the Lorenz equations," Int. J. Electro. 68, 803-819.
80. Rocard, Y. [1932] "Sur les oscillateurs ̀a lampes ŕegĺes pr̀es de la limite d'entretien," Comptes-Rendus de l'Acad́emie des Sciences 194, 1325.
81. Rossler, O. E. [1974a] "A synthetic approach to exotic kinetics (with examples)," in Physics and Mathematics of the Nervous System: Proc. Summer School Physics and Mathematics of the Nervous System (Trieste, August 21-31, 1973), eds. Conrad, M., Guttinger, W. & Dal Cin, M. (Springer-Verlag, Berlin), pp. 546-582.
82. Rossler, O. E. [1974b] "Chemical automata in homogeneous and reaction-di.usion kinetics," Lecture Notes in Biomath. 4, 399-418.
83. Rossler, O. E. [1974c] "A synthetic approach to exotic kinetics (with examples)," Lecture Notes in Biomath. 4, 546-582.
84. Rossler, O. E. [1976a] "Chaotic behavior in simple reaction systems," Zeitschrift fur Naturforschung A 31, 259-264.
85. Rossler, O. E. [1976b] "An equation for continuous chaos," Phys. Lett. A 57, 397-398.
86. Rossler, O. E. [1977] "Continuous chaos," Synergetics, ed. Haken, H., Proc. Int. Workshop on Synergetics at Schloss Elmau (Bavaria, May 2-7, 1977) (Springer- Verlag), pp. 184-197.
87. Rossler, O. E. & Ortoleva, P. J. [1978] "Strange attractors in 3-variable reaction systems," Lecture Notes in Biomath. 21, 67-73.
88. Rossler, R., Gotz, F. & Rossler, O. E. [1979] "Chaos in endocrinology," Biophys. J. A 25, 216.
89. Rossler, O. E. [August 2002] Private Communication, Rouen.
90. Ruelle, D. & Takens, F. [1971] "On the nature of turbulence," Commun. Math. Phys. 20, 167-192.
91. Sparrow, C. T. [1981] "Chaos in a three-dimensional single loop feedback system with a piecewise linear feedback function," J. Math. Anal. Appl. 83, 275-291.
92. Strukov, D. B., Snider, G. S., Stewart, D. R. & Williams, R. S. [2008] "The missing memristor found," Nature 453, 80-83.
93. Ueda, Y. [1965] Some Problems in the Theory of Nonlinear Oscillations, Th̀ese de l'universit́e de Tokyo, 1965, reproduite en anglais dans The Road to Chaos (Aerial Press).
94. Ueda, Y. [1978] "Random Phenomena resulting from nonlinearity: In the system described by Du.ng's equation," Trans. Instit. Electron. Commun. Engin. A 98, 167-173;
95. [1985] Reprinted in English in Int. J. Nonlin. Mech. 20, 481-491.
96. Ueda, Y. [1979] "Steady motions exhibited by Du.ing's equation - a picture book of regular and chaotic motions," in Conf. New Approaches to Nonlinear Problems in Dynamics, Monterey, CA, December 9-14.
97. Ueda, Y. [1992] "Strange attractors and the origin of chaos," Nonlin. Sci. Today 2.
98. van der Pol, B. [1934] "The nonlinear theory of electric oscillations," Proc. Instit. Radio Engin. 22, 1051-1086.
99. van der Pol, B. [1920] "A theory of free and forced triode vibrations," Tijdschrift Ned. Radiogenoot 1, 3-31.
100. van der Pol, B. [1922] "On oscillations hysteresis in a triode generator with two degrees of freedom," Philos. Mag. 43, 700.
101. van der Pol, B. [1926] "On relaxation-oscillations," Philos. Mag. 2, 978-992.
102. van der Pol, B. [1927] "Forced oscillations in a circuit with nonlinear resistance (reception with reactive triode)," Philos. Mag. 3, 65.
103. van der Pol, B. & van der Mark, J. [1927] "Frequency demultiplication," Nature 120, 363-364.
104. Vincent, J. H. [1920] "On some experiments in which two neighbouring maintained oscillatory circuits a.ect a resonating circuit," Proc. Roy. Soc. London 32, 84-92.
105. Volterra, V. [1926] "Variazoni e fluttuazioni del numero d'individui in specie animali conviventi," Mem. Acad. Lincei 2, 31-113.
106. Williams, R. F. [1979] "The structure of Lorenz attractors," Publication de l'Institut des Hautes-Etudes Scientifiques 50, 73-99.
107. Zeeman, E. C. [1972] "differential equations for the heartbeat and nerve impulse," Towards a Theoretical Biology, Vol.4, ed. Waddington, Ch. (Edinburgh University Press), pp. 8-67.
108. Zhong, G.-Q. & Ayrom, F. [1985] "Experimental confirmation of chaos from Chua's circuit," Circuit Th. Appl. 13, 93-98.