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The method of transfer to the derivative space: 40 years of evolution. (Russian. English summary) Zbl 1357.34062

Summary: 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 Poincaré-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 Poincaré-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.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
37C27 Periodic orbits of vector fields and flows
93D10 Popov-type stability of feedback systems
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[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 · Zbl 0216.11502
[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 · Zbl 0334.93020
[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 · Zbl 0175.10302
[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 · Zbl 0219.93014
[16] Hastings S., The existence of periodic solutions to Nagumo’s equation, Q. Jl. Math., Oxford Ser. 1974, 2(25), pp. 369-378 · Zbl 0305.34058
[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 · Zbl 0301.34044
[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 · Zbl 0408.34042
[20] Smith R. A. Existence of periodic orbits of autonomous ordinary differential equations. Proc. Roy. Soc. Edinburgh. 1980, Sect. A 85, pp. 153-172 · Zbl 0429.34040
[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 · Zbl 0499.34026
[22] Smith R. A. Certain differential equations have only isolated periodic orbits. Ann. Mat. Pura. Appl. 1984, vol. 137, pp. 217-244 · Zbl 0561.34028
[23] Smith R. A. Poincaré index theorem concerning periodic orbits of differential equations. Proc. London Math. Soc. 1984, vol. 48, pp. 341-362 · Zbl 0509.34046
[24] Smith R. A. Massera’s Convergence Theorem for Periodic Nonlinear Differential Equations. J. Math. Analysis and Appl.. 1986 vol. 120, pp. 679-708 · Zbl 0603.34033
[25] Smith R. A. Orbital stability for ordinary differential equations. J. Diff. Eq. 1987, vol. 69. no. 2, pp. 265 -287 · Zbl 0632.34054
[26] Smith R. A. Poincaré -Bendixson theory for certain retarded functional-differential equations. Diff. Int. Eq. 1992. no. 5, pp. 213-240 · Zbl 0754.34070
[27] Smith R. A. Some modified Michaelis-Menten equations having stable closed trajectories. Proc. Roy. Soc. Edinburgh, 1988, 109A, pp. 341-359 · Zbl 0652.34036
[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 · Zbl 0806.35082
[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 · Zbl 0712.34060
[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 · Zbl 0849.34056
[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 · Zbl 1203.34051
[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 · Zbl 0316.92001
[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 · Zbl 0361.34038
[37] Lorenz E. N. Deterministic non-periodic flow. J. Atmos. Sci. 1963. vol. 20, pp. 130-141 · Zbl 1417.37129
[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 · Zbl 0551.94020
[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 · Zbl 0494.34017
[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 · Zbl 0658.34023
[42] Hirsch M. W. Systems of differential equations which are competitive or cooperative. III: Competing species. Nonlinearity. 1988, vol. 1, pp. 51-71 · Zbl 0658.34024
[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 · Zbl 0734.34042
[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 · Zbl 0712.34045
[45] Hirsch M. W. Systems of differential equations that are competitive or cooperative. VI. A local C^{r}closing lemma for 3-dimensional systems, Ergodic Theory Dynam. Systems, 1991, vol. 11, pp. 443-454 · Zbl 0747.34027
[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 · Zbl 0731.35050
[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 · Zbl 1204.92038
[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. \[ \] · Zbl 1288.93070
[53] Ortega R., Sanchez L. A., Abstract competitive systems and orbital stability in R^{3}, Proc. Amer. Math. Soc., 2000, vol. 128, pp. 2911-2919\(. \) · Zbl 0951.34032
[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 · Zbl 1351.37268
[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 · Zbl 1203.37033
[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 · Zbl 0826.58020
[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) · Zbl 1226.34050
[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) · Zbl 1202.93099
[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) · Zbl 1247.34063
[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 · Zbl 1266.93072
[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 · Zbl 1242.34102
[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) · Zbl 1277.34052
[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 · Zbl 1308.93107
[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 · Zbl 1270.34003
[88] Leonov G. A., Kuznetsov N. V. \[ Analytical-numerical methods for hidden attractors localization: The 16th Hilbert problem, Aizerman and Kalman conjectures, and Chua circuit. Numerical Methods for Differential Equations, Optimization, and Technological Problems, Computational Methods in Applied Sciences, 2013, vol. 27, Part 1 (Springer), pp. 41-64 \] · Zbl 1267.65200
[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 · Zbl 1284.34064
[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 · Zbl 1255.37013
[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 · Zbl 1238.34122
[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 · Zbl 1284.34022
[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 · Zbl 1296.34113
[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 · Zbl 1296.34137
[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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