Geodesic
What is Quasi-Conservative Dynamics? On the Anniversary of A. D. Morozov
Russian journal of nonlinear dynamics, Tome 21 (2025) no. 1, pp. 5-13.

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We give a review of scientific results of the remarkable Russian mathematician Albert Dmitrievich Morozov who is the world’s recognized leader in the theory of nearly Hamiltonian systems and one of the founders of the modern mathematical theory of synchronization in oscillatory systems. This review was prepared in connection with the 80th birthday of A.D. Morozov and the authors wish him all the best, good health and creative success.
Keywords: nearly Hamiltonian system, nonlinear resonance, quasi-coservative system
Mots-clés : periodic perturbations 
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S. V. Gonchenko; K. E. Morozov. What is Quasi-Conservative Dynamics? On the Anniversary of A. D. Morozov. Russian journal of nonlinear dynamics, Tome 21 (2025) no. 1, pp. 5-13. http://geodesic.mathdoc.fr/item/ND_2025_21_1_a1/

[1] Morozov, A. D. and Shil'nikov, L. P., “To Mathematical Theory of Oscillatory Synchronization”, Dokl. Akad. Nauk SSSR, 223:6 (1975), 1340–1343 (Russian) | MR

[2] Morozov, A. D. and Shil'nikov, L. P., “On Sinchronization of Oscillations”, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1975, no. 5 (Russian)

[3] Prikl. Mat. Mekh., 47:3 (1983), 385–394 (Russian) | DOI | MR

[4] Arnol'd, V. I., Mathematical Methods of Classical Mechanics, Grad. Texts in Math., 60, 2nd ed., Springer, New York, 1997, 529 pp. | MR

[5] Dokl. Akad. Nauk SSSR (N. S.), 98 (1954), 527–530 (Russian) | DOI | MR | MR

[6] Uspekhi Mat. Nauk, 18:6(114) (1963), 91–192 (Russian) | DOI | MR

[7] Moser, J., “On Invariant Curves of Area-Preserving Mappings of an Annulus”, Nach. Akad. Wiss. Göttingen, Math. Phys. Kl. II, 1962:1 (1962), 1–20 | MR

[8] Smale, S., “Diffeomorphisms with Many Periodic Points”, Differential and Combinatorial Topology: A Symposium in Honor of Marston Morse, ed. S. S. Cairns, Princeton Univ. Press, Princeton, N.J., 1965, 63–80 | DOI | MR

[9] Mat. Sb. (N.S.), 74(116):3 (1967), 378–397 (Russian) | DOI | MR

[10] Tr. Mosk. Mat. Obs., 12 (1963), 3–52 (Russian)

[11] Morozov, A. D., To the Theory of the Duffing Type Equations Close to the Nonlinear Conservative, PhD Thesis, Gorky State Univ., Gorky, 1975, 157 pp. (Russian)

[12] Zh. Vychisl. Mat. Mat. Fiz., 13:5 (1973), 1134–1152 (Russian) | DOI | MR

[13] Morozov, A. D., “The Complete Qualitative Investigation of Duffing's Equation”, Differ. Uravn., 12:2 (1976), 241–255 (Russian) | MR

[14] Morozov, A. D., Quasi-Conservative Systems: Cycles, Resonances and Chaos, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 30, World Sci., River Edge, N.J., 1998, 340 pp. | MR

[15] Morozov, A. D., Resonances, Cycles and Chaos in Quasi-Conservative Systems, R Dynamics, Izhevsk, 2005, 420 pp. (Russian) | MR

[16] Morozov, A. D., “On Existence of Homoclinic Curves in Non-Autonomous Relay System”, Proc. of the 3rd All-Union Congress on Theoretical and Applied Mechanics (Moscow, 1968) (Russian)

[17] Morozov, A. D., “On Piecewise Smooth Systems Containing Homoclinic Curves”, Proc. of the 5th Internat. Conf. on Nonlinear Oscillations (Kiev, 1969): Vol. 2. Qualitative Methods of Nonlinear Oscillations, Naukova Dumka, Kiev, 1970, 341–346 (Russian)

[18] Morozov, A. D., Problems of Theory Oscillations in Non-Conservative Systems, Close to Nonlinear Integrable Ones, PhD Dissertation, Moscow State Univ., Moscow, 1990, 283 pp. (Russian)

[19] Morozov, A. D., “Pendulum-Type Equations and Resonance”, Methods of the Qualitative Theory of Differential Equations, ed. E. A. Leontovich-Andronova, GGU, Gorki, 1980, 3–16 (Russian) | MR

[20] Morozov, A. D. and Fedorov, E. L., “On the Investigation of Equations with One Degree of Freedom, Close to Nonlinear Integrable Ones”, Differ. Uravn., 19:9 (1983), 1511–1516 (Russian) | MR

[21] Prikl. Mat. Mekh., 53:5 (1989), 721–730 (Russian) | DOI | MR

[22] Morozov, A. D., Mathematical Methods of the Theory of Oscillations, R Dynamics, Izhevsk, 2017, 144 pp. (Russian)

[23] Morozov, A. D., Introduction to Fractal Theory, R Dynamics, Izhevsk, 2004, 160 pp. (Russian) | MR

[24] Karabanov, A. A. and Morozov, A. D., “On Degenerate Resonances in Hamiltonian Systems with Two Degrees of Freedom”, Chaos Solitons Fractals, 69 (2014), 201–208 | DOI | MR

[25] Morozov, A. D. and Kondrashov, R. E., “On Resonances in Systems of Two Weakly Coupled Oscillators”, Regul. Chaotic Dyn., 14:2 (2009), 237–247 | DOI | MR

[26] Korolev, S. A. and Morozov, A. D., “On Periodic Perturbations of Self-Oscillating Pendulum Equations”, Nelin. Dinam., 6:1 (2010), 79–89 (Russian) | DOI

[27] Morozov, A. D. and Kostromina, O. S., “On Periodic Perturbations of Asymmetric Duffing – Van-der-Pol Equation”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24:5 (2014), Art. 1450061, 16 pp. | DOI | MR

[28] Morozov, A. D. and Morozov, K. E., “Transitory Shift in Pendular Type Equations”, Nelin. Dinam., 12:4 (2016), 577–589 (Russian) | DOI | MR

[29] Differ. Uravn., 53:12 (2017), 1607–1615 (Russian) | DOI | DOI | MR

[30] Morozov, A. D. and Morozov, K. E., “On Synchronization of Quasiperiodic Oscillations”, Russian J. Nonlinear Dyn., 14:3 (2018), 367–376 | MR

[31] Morozov, A. D. and Morozov, K. E., “Global Dynamics of Systems Close to Hamiltonian Ones under Nonconservative Quasi-Periodic Perturbation”, Russian J. Nonlinear Dyn., 15:2 (2019), 187–198 | MR

[32] Morozov, A. D. and Morozov, K. E., “Synchronization of Quasiperiodic Oscillations in Nearly Hamiltonian Systems: The Degenerate Case”, Chaos, 31:8 (2021), Paper No. 083109, 10 pp. | DOI | MR

[33] Morozov, A. D. and Morozov, K. E., “Degenerate Resonances and Synchronization in Nearly Hamiltonian Systems under Quasi-Periodic Perturbations”, Regul. Chaotic Dyn., 27:5 (2022), 572–585 | DOI | MR

[34] Morozov, A. D. and Morozov, K. E., “Quasi-Periodic Parametric Perturbations of Two-Dimensional Hamiltonian Systems with Nonmonotonic Rotation”, Regul. Chaotic Dyn., 29:1 (2024), 65–77 | DOI | MR

[35] Morozov, A. D. and Dragunov, T. N., Visualization and Analysis of Invariant Sets for Dynamical Systems, R Dynamics, Institute of Computer Science, Izhevsk, 2003, 304 pp. (Russian) | MR

[36] Morozov, A. D. and Dragunov, T. N., Using the WinSet Program to Visualize Dynamic Systems, NNGU, Nighny Novgorod, 2007, 102 pp. (Russian)