ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

The Method of Transfer to the Derivative Space: 40 Years of Evolution


I. M. Burkin

Professor, head of department of mathematical analysis,
Department of Mechanics and Mathematics,
Tula State University,
Tula, Russia


In 1975 the so called “method of transfer to the derivative space” was proposed. It is an efficiently verified frequency criterion of the existence of a nontrivial periodic solution in multidimensional models of automatic control systems with one differentiable nonlinear term. The method used the classical torus principle and refrained from any constructions in the phase space of the system under study. Moreover, the method allowed researchers to broaden the class of systems to that it may be applied. In this work we give a survey of the results presenting generalization and expansion of the method. We also show the connection between the method of transfer to the derivative space, well known the Poincare-Bendixon generalized principle proposed by R. A. Smith and the results of contemporary authors who are active in the theory of oscillations in multidimensional systems. During recent years the author obtained frequency criteria of the existence of orbitally stable cycles in multivariable automatic control systems (MIMO systems) and the methods of construction of multidimensional systems having the only equilibrium state and any given in advance number of orbitally stable cycles, which are described in the paper. The extension of the Poincare-Bendixon generalized principle to multidimensional systems with angular coordinate is presented. We show the application of described methods of investigation of oscillation processes in multidimensional dynamical systems to the solving S. Smale problem from the biological cells chemical kinetics theory and also to the finding hidden attractors of the Chua generalized system and minimal global attractor of a system with a polynomial nonlinear term. The publication is illustrated by numerous examples.



  1. Burkin I. M. Leonov G. A. Chastotnye usloviya suschestvovaniya netrivial'nogo periodicheskogo resheniya u nelineynoy sistemy s odnoy stazionarnoy nelineynost'yu. Metody i modeli upravleniya. 1975, vyp. 9. RPI, Riga, s. 175-177
  2. Burkin I. M., Leonov G. A. O suschestvovanii netrivial'nych periodicheskich resheniy v avtokolebatel'nych sistemach. Sib. mat. zhurnal. 1977, №2,. s. 251-262
  3. Friedrichs D. O. On nonlinear vibrations of third order. Studies in Nonlinear Vibrarion Theory. Institute of Mathematics and Mechanics, New York University Press, 1946, pp. 65-103
  4. Rauch L. L., Oscillation of a third order nonlinear autonomous system, Contribution to the Theory of Nonlinear Oscillations. , Univ. Press 1950, pp. 39-88
  5. Shirokorad B. V. O suschestvovanii zikla vne usloviy absolyutnoy ustoychivosti trechmernoy sistemy. Avtomatika i telemechanika. 1958. T. 15, №10, s. 953-967
  6. Sherman S. A third-order nonlinear system arising from a nuclear spin generator, Contr. Dif. Eqns. 1963. no. 2, pp. 197- 227
  7. Vaysbord E. M. O suschestvovanii periodicheskogo resheniya u nelineynogo dif-ferenzial'nogo uravneniya tret'ego poryadka. . Matem. sb. 1962, 56(98):1 , c. 43-58
  8. V. A. Pliss. Nelokal'nye problemy teorii kolebaniy. 1964. M. " Nauka", 367 s
  9. Vinogradov N. N. Nekotorye teoremy o suschestvovanii periodicheskich resheniy odnoy avtonomnoy sistemy shesti differenzial'nych uravneniy. Dif. uravne-niya. 1965, t. 1, №3, s. 330-334
  10. Mulholland R. J., Nonlinear oscillations of a third-order differential equation. . J. nonlinear Mech. 1971, no 6, pp. 279- 294
  11. Kamachrin A. M. Existence and uniqueness of a periodic solution to a relay system with hysteresis. . Diff. Eqns. 1972, no. 8, pp. 1505-1506
  12. Leonov G. A. Chastotnye usloviya suschestvovaniya netrivial'nych periodicheskich resheniy v avtonomnych sistemach.. Sib. mat. zhurnal. 1973, T. 14, № 6, s. 1505-1506
  13. Williamson D., Periodic motion in nonlinear systems . IEEE trans. Automat. Control 1975, AC-20, no. 4, pp. 479-486
  14. Noldus E. A frequency domain approach to the problem of the existence of periodic motion in autonomous nonlinear feedback systems, Z. Angew. math. Mech. 1969, no. 3. pp. 166-177
  15. Noldus E. A counterpart of Popov’s theorem for the existence of periodic solutions. Int. J. Control 1971, vol. 13, no. 4, pp. 705- 719
  16. Hastings S., The existence of periodic solutions to Nagumo’s equation, Q. Jl. Math., Oxford Ser. 1974, 2(25), pp. 369-378
  17. Hastings S. P., Murray J. D. The existence of oscillatory solutions in the field-noyes model for the Belousov-Zhabotinskii reaction. SIAM J. Appl. Math. 1975, 28(3), pp. 678-688
  18. Tyson J. J., On the existence of oscillatory solutions in negative feedback cellular control process, . Math. Biol. 1975, no. 1, pp. 311-315
  19. Smith R. A. The Poincare-Bendixson theorem for certain differential equations of higher order. Proc. Roy. Soc. Edinburgh 1979, Sect. A 83, pp. 63-79
  20. Smith R. A. Existence of periodic orbits of autonomous ordinary differential equations. Proc. Roy. Soc. Edinburgh. 1980, Sect. A 85, pp. 153-172
  21. Smith R. A. An index theorem and Bendixson's negative criterion for certain differential equations of higher dimension. . Proc. Roy. Soc. Edinburgh. 1981, A91, pp. 63-77
  22. Smith R. A. Certain differential equations have only isolated periodic orbits. Ann. Mat. Pura. Appl. 1984, vol. 137, pp. 217-244
  23. Smith R. A. Poincaré index theorem concerning periodic orbits of differential equations. Proc. London Math. Soc. 1984, vol. 48, pp. 341-362
  24. Smith R. A. Massera's Convergence Theorem for Periodic Nonlinear Differential Equations. J. Math. Analysis and Appl.. 1986 vol. 120, pp. 679-708
  25. Smith R. A. Orbital stability for ordinary differential equations. J. Diff. Eq. 1987, vol. 69. no. 2, pp. 265 -287
  26. Smith R. A. Poincaré -Bendixson theory for certain retarded functional-differential equations. Diff. Int. Eq. 1992. no. 5, pp. 213-240
  27. Smith R. A. Some modified Michaelis-Menten equations having stable closed trajectories. Proc. Roy. Soc. Edinburgh, 1988, 109A, pp. 341-359
  28. Smith R. A. Orbital stability and inertial manifolds for certain reaction diffusion systems. Proc. London Math. Soc. 1994, vol. 69, no. 3, pp. 91-120
  29. Mallet-Paret J., Smith H. L., The Poincaré -Bendixson theorem for monotone feedback systems, J. Dynam. Diff. Eq. 1990, vol. 2, no. 4, , pp. 367-421
  30. Mallet-Paret J., Sell G. R., The Poincaré -Bendixson theorem for monotone cyclic feedback systems with delay, J. Diff. Eq. 1996, 125, pp. 441-489
  31. Sanchez L. A. Cones of rank 2 and the Poincaré -Bendixson property for a new class of monotone systems, J. Diff. Eq. 2009, 216, pp. 1170-1190
  32. Sanchez L. A. Existence of periodic orbits for high-dimensional autonomous systems. J. Math. Anal. Appl. 2010, 363 , pp. 409-418
  33. Leonov G. A., Burkin I. M., Shepelijavyi A. I Frequency Methods in Oscillation Theory. Kluwer Academic Publishers. 196б, 403 р
  34. Gelig A. Ch., Leonov G. A., Yakubovich V. A. Ustoychivost' nelineynych sistem s ne-edinstvennym sostoyaniem ravnovesiya. M. " Nauka", 1978, 400 s
  35. Heiden U. an der, Existence of periodic solutions of a nerve equation, Biol. Cybern. 1976, 21, pp. 37-39
  36. Hastings S. P, Tyson J. J., Webster D., Existence of periodic solutions for negative feedback cellular control systems, J. Diff Eqns. 1977, vol. 25, pp. 39-64
  37. Lorenz E. N. Deterministic non-periodic flow. J. Atmos. Sci. 1963. vol. 20, pp. 130-141
  38. Smale S. Diffeomorphfisms with many periodic points. Combin. Topology. Princeton. Univ. Press. 1965, pp. 21-30/
  39. MatsumotoT. A chaotic attractor from Chua's circuit. IEEE Trans. CAS-31, 1984, pp. 1055-1058
  40. Hirsch M. W. Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Math. Anal. 1982, vol. 13, pp. 167-179
  41. Hirsch M. W. Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere. SIAM J. Math. Anal. 1985, vol. 16, pp. 423-439
  42. Hirsch M. W. Systems of differential equations which are competitive or cooperative. III: Competing species. Nonlinearity. 1988, vol. 1, pp. 51-71
  43. Hirsch M. W. Systems of differential equations that are competitive or cooperative. IV: Structural stability in three dimensional systems. SIAM J. Math. Anal. 1990, vol. 21, pp. 1225-1234
  44. Hirsch M. W. Systems of differential equations that are competitive or cooperative. V. Convergence in 3-dimensional systems, J. Differential Equations, 1989, vol. 80, pp. 94-106
  45. Hirsch M. W. Systems of differential equations that are competitive or cooperative. VI. A local Crclosing lemma for 3-dimensional systems, Ergodic Theory Dynam. Systems, 1991, vol. 11, pp. 443-454
  46. Hirsch M. W., Smith H., Monotone dynamical systems, in: Handbook of Differential Systems (Ordinary Differential Equations). Elsevier, Amsterdam, 2005, vol. 2, pp. 239-358
  47. Smith H. L., Monotone Dynamical Systems, Amer. Math. Soc., Providence, 1995
  48. P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Res. Notes Math. Ser., 1991. vol. 247, Longman Scientifi c and Technical, Harlow
  49. Angeli D., P. de Leenheer, Sontag E. D. Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. J. Mathematical Biology, 2010, vol. 61, pp. 581-616
  50. Craciun G., Pantea C., Sontag E. D. Graph-theoretic analysis of multistability and monotonicity for biochemical reaction networks. Design and Analysis of Biomolecular Circuits , Springer-Verlag, 2011, pp. 63-72.
  51. Angeli D. Sontag E. D.. Behavior of responses of monotone and sign-definite systems. Mathematical System Theory, 2013, Create Space, pp. 51-64.
  52. Angeli D., Enciso G. A., Sontag E. D.. A small-gain result for orthant-monotone systems under mixed feedback. Systems and Control Letters, 2014, vol. 68, pp. 9-19.
  53. Ortega R., Sanchez L. A., Abstract competitive systems and orbital stability in R3, Proc. Amer. Math. Soc., 2000, vol. 128, pp. 2911-2919.
  54. Burkin I. M., Leonov G. A. O suschestvovanii netrivial'nych periodicheskich resheniy u odnoy nelineynoy sistemy tret'ego poryadka. Diff. uravneniya, 1984, t. 20, № 12, s. 1430-1435.
  55. Burkin I. M., Soboleva D. V. O strukture global'nogo attraktora mnogosvyaznych sistem avtomaticheskogo regulirovaniya. Izvestiya TulGU. Estestvennye nauki.. Izd. -vo TulGU, 2012, vyp. 1, s. 5-16
  56. Burkin I. M., Burkina L. I. Chastotnyy kriteriy suschestvovaniya ziklov u mnogo-svyaznych sistem avtomaticheskogo regulirovaniya. Vestnik TulGU. Seriya «Diffe-renzial'nye uravneniya i prikladnye zadachi», 2010, vyp. 1. TulGU,. s. 3-14
  57. Burkin I. M., Nguen N. K. Analytical-Numerical Methods of Finding Hidden Oscillations in Multidimensional Dynamical Systems. Differential Equations, 2014, vol. 50, no. 13, pp. 1695-1717
  58. Shalfeev V. D., Matrosov V. V. Nelineynaya dinamika sistem fazovoy sinchronizazii. Izd. -vo Nizhegorodskogo universiteta, 2013, 366 s
  59. Burkin I. M. Chastotnyy kriteriy orbital'noy ustoychivosti predel'nych ziklov vtorogo roda. Diff. uravneniya, 1993, t. 29, №6, s. 1061-1063
  60. Burkin I. M. Obobschennyy prinzip Puankare-Bendiksona dlya dinamicheskich sistem s zilindricheskim fazovym prostranstvom. Vestnik TulGU. Seriya " Differenzial'nye uravneniya i prikladnye zadachi", 2009, vyp. 1. Tula, s. 3-20
  61. Leonov G. A., Smirnova V. B. Matematicheskie problemy teorii fazovoy sinchronizazii. Sankt-Peterburg. Nauka, 2000, 400 s
  62. Bobylev N. A., Bulatov A. V, Korovin S. K, Kutuzov A. A. Ob odnoy scheme issledo-vaniya ziklov nelineynych sistem. Diff. uravneniya, t. 32, № 1, s 3-8
  63. Byrnes C. I. Topological Methods for Nonlinear Oscillations . Notices of the AMS, 2010, vol. 57, no 9, pp. 1080-1090
  64. Burkin I. M, Burkina L. I., Leonov G. A. Problema Barbashina v teorii fazovych sistem. Diff. uravneniya, 1981, t. 17, №11, s. 1932-1944
  65. Burkin I. M, Soboleva D. V. O mnogomernych sistemach s needinstvennym ziklom i metode garmonicheskogo balansa. Izvestiya TulGU. Estestvennye nauki, Izd-vo TulGU, 2011, vyp. 3, s. 5-21
  66. Burkin I. M. O strukture minimal'nogo global'nogo attraktora mnogomernych sistem s edinstvennym polozheniem ravnovesiya. Diff. uravneniya, 1997, t. 33, №3, s. 418-420
  67. Burkin I. M. O yavlenii bufernosti v mnogomernych dinamicheskich sistemach. Diff. uravneniya, 2002, t. 38, №5, s. 585 - 595
  68. Burkin I. M., Yakushin O. A. O mnogomernom variante trinadzatoy problemy Smeyla. Izvestiya TulGU. Seriya " Differenzial'nye uravneniya i prikladnye zadachi", 2004, vyp. 1, s. 12-29
  69. Burkin I. M., Yakushin O. A. Kolebaniya s zhestkim vozbuzhdeniem i fenomen bu-fernosti v mnogomernych modelyach reguliruemych sistem. Izvestiya TulGU. Seriya " Differenzial'nye uravneniya i prikladnye zadachi, 2005, vyp. 1, s. 24-31
  70. Matveev A. S. Yakubovich V. A. Optimal'nye sistemy upravleniya: Obyknovennye differenzial'nye uravneniya. Spezial'nye zadachi, 2003. Izd. -vo S. -Peterburgskogo un. -ta, 540 s
  71. Voronov A. A. Osnovy teorii avtomaticheskogo upravleniya. Osobye lineynye i nelineynye sistemy. M., 1981, 303 s
  72. Leonov G. A. Ob odnoy gipoteze Voronova. Avtomatika i telemechanika, 1984. №5, c. 53-58
  73. Leonov G. A., Ponomarenko D. V., Smirnova V. B. Local instability and localization of attractors. Acta Applicandae Mathematicae, 1995, vol. 40, pp. 179-243
  74. Burkin I. M., Burkina L. I. O chisle ziklov trechmernoy sistemy i shestnadzatoy probleme Gil'berta. Izvestiya Rossiyskoy akademii estestvennych nauk. Diff. uravneniya, 2001, №5, s. 37-40
  75. Smeyl S. Matematicheskaya model' vzaimodeystviya dvuch kletok, ispol'zuyuschaya uravneniya T'yuringa. Bifurkaziya rozhdeniya zikla i ee prilozheniya. M., 1980, 360 s
  76. Burkin I. M, Soboleva D. V. On a Smale Problem. Diff. Equations, 2011, vol. 47, no. 1, pp. 1-9
  77. Leonov G. A,. Vagaitsev V. I, Kuznetsov N. V. Algorithm for localizing Chua attractors based on the harmonic linearization method. Doklady Mathematics, 2010, vol. 82, no. 1, pp. 663-666. (doi:10. 1134/S1064562410040411)
  78. Leonov G. A, Bragin V. O., Kuznetsov N. V. Algorithm for constructing counterexamples to the Kalman problem. Doklady Mathematics, 2010, vol. 82, no. 1, pp. 540-542. (doi:10. 1134/S1064562410040101)
  79. Leonov G. A., Kuznetsov N. V. Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems. Doklady Mathematics, 2011, vol. 8, no. 1, pp. 475-481. (doi:10. 1134/S1064562411040120)
  80. Leonov G. A., Kuznetsov N. V. Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems". IFAC Proceedings Volumes (IFAC-PapersOnline), 2011, vol. 18, no1, pp. 2494-2505. (doi:10. 3182/20110828-6-IT-1002. 03315)
  81. Kuznetsov N. V., Leonov G. A., Seledzhi S. M. Hidden oscillations in nonlinear control systems, IFAC Proceedings Volumes" IFAC Proceedings Volumes (IFAC-PapersOnline), 2011, vol. 18, no1, pp. 2506-2510. (doi: 10. 3182/20110828-6-IT-1002. 03316)
  82. Bragin V. O., Vagaitsev V. I., Kuznetsov N. V., Leonov G. A. Algorithms for fi nding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J. Comput. Syst. Sci. Int. 2011, vol. 50, no. 4, pp. 511-543
  83. Leonov G. A., Kuznetsov N. V., Vagaitsev, V. I. Localization of hidden Chua’s attractors. Phys. Lett. A 2011, vol. 375, pp. 2230-2233
  84. Leonov G. A., Kuznetsov N. V., Kuznetsova O. A., Seledzhi S. M., Vagaitsev, V. I. Hidden oscillations in dynamical systems, Trans Syst. Contr., 2011, . no. 6, pp. 54-67
  85. Leonov G. A, Kuznetsov N. V., Vagaitsev V. I. Hidden attractor in smooth Chua systems. Physica D: Nonlinear Phenomena, 2012, 241(18), pp. 1482-1486. (doi: 10. 1016/j. physd. 2012. 05. 016)
  86. Kuznetsov N. V., Kuznetsova O. A., Leonov G. A., Vagaitsev V. I. Analytical-numerical localization of hidden attractor in electrical Chua’s circuit. Lecture Notes in Electrical Engineering, 2013, 174, pp. 149-158
  87. Leonov G. A., Kuznetsov N. V. Hidden attractors in dynamical systems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits. Int. J. Bifurcation and Chaos, 2013, vol. 23, no. 1. 1330002
  88. Leonov G. A., Kuznetsov N. V. Analytical-numerical methods for hidden attractors localization: The 16th Hilbert problem, Aizerman and Kalman conjectures, and Chua circuit. Numerical Methods for Differential Equations, Optimization, and Technological Problems, Computational Methods in Applied Sciences, 2013, vol. 27, Part 1 (Springer), pp. 41-64
  89. Andrievsky B. R., Kuznetsov N. V., Leonov G. A., Pogromsky A. Yu. Hidden Oscillations in Aircraft Flight Control System with Input Saturation. IFAC Proceedings Volumes (IFAC-PapersOnline), 2013, vol. 5, no. 1, pp. 75-79. (doi: 10. 3182/20130703-3-FR-4039. 00026)
  90. Andrievsky B. R., Kuznetsov N. V., Leonov G. A., Seledzhi S. M. Hidden oscillations in stabilization system of flexible launcher with saturating actuators. IFAC Proceedings Volumes (IFAC-PapersOnline), 2013, vol. 19, no. 1, pp. 37-41. (doi: 10. 3182/20130902-5-DE-2040. 00040)
  91. Chua L. O. A zoo of Strange Attractors from the Canonical Chua's Circuits. Proc. Of the IEEE 35th Midwest Symp. on Circuits and Systems (Cat. No. 92CH3099-9). Washington, 1992, vol. 2, pp. 916 - 926
  92. Jafari S., Sprott J. C., Golpayegani S. M. R. H. Elementary quadratic chaotic flows with no equilibria. Phys. Lett. A, 2013, vol. 377, pp. 699-702
  93. Molaie M., Jafari S., Sprott J. C., Golpayegani, S. M. R. H. Simple chaotic flows with one stable equilibrium. Int. J. Bifurcation and Chaos. 2013, vol. 23, no. 11. 1350188
  94. Wangand X., Chen G. A chaotic system with only one stable equilibrium communications in Nonlinear Science and Numerical Simulation. 2012, vol. 17, no. 3, pp. 1264-1272
  95. Wei Z. Dynamical behaviors of a chaotic system with no equilibria. Phys. Lett. A, 2011, vol. 376, pp. 102-108
  96. Wei Z. Delayed feedback on the 3-D chaotic system only with two stable node-foci. Comput. Math. Appl. 2011, vol. 63, pp. 728-738
  97. Wang X. , Chen G. Constructing a chaotic system with any number of equilibria. Nonlinear Dyn., 2013, vol. 71, pp. 429-436
  98. Seng-Kin Lao, Shekofteh Y., Jafari . S, Sprott, J. C. Cost Function Based on Gaussian Mixture Model for Parameter Estimation of a Chaotic Circuit with a Hidden Attractor. Int. J. Bifurcation Chaos, 2014, vol. 24, no. 01. 1450010
  99. Pham, V. -T., Rahma, F., Frasca, M., Fortuna, L. Dynamics and synchronization of a novel hyperchaotic system without equilibrium. International Journal of Bifurcation and Chaos, 2014, 24(06). art. num. 1450087
  100. Zhao H., Lin Y., Dai Y. Hidden attractors and dynamics of a general autonomous van der Pol-Duffing oscillator. International Journal of Bifurcation and Chaos, 201424(06). art. num. 1450080
  101. Q. Li, H. Zeng, X. -S. Yang On hidden twin attractors and bifurcation in the Chua’s circuit . Nonlinear Dyn., 2014, 77 (1-2) , pp. 255-266
  102. Burkin I. M., Nguen Ngok Chien. O strukture minimal'nogo global'nogo attraktora obobschennoy sistemy L'enara s polinomial'noy nelineynost'yu. Izvestiya TulGU. Estestvennye nauki. Izd. -vo TulGU, 2014, vyp. 2. s. 46-52

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