nonlinear vibrations of finite-dimensional systems; stability of nonlinear vibrations; homoclinic and heteroclinic trajectories and chaotic behavior of dynamical systems; dynamics of vibro-impact systems; vibrations and waves in nonlinear elastic systems; rational fractional approximations in theory of non-linear vibrations.
New methods of analysis of nonlinear normal vibration modes of essentially nonlinear finite-dimensional conservative systems are developed. A problem of construction of normal vibrations in nonautonomous and self-oscillating finite-dimensional systems close to conservative ones is solved. Methods of the stability investigation of normal vibration modes in systems with a single and several equilibrium positions are developed. New methods of construction of homo- and heteroclinic trajectories as well of matching of local expansions with using fractional rational approximations, are developed. Normal vibrations and stationary traveling waves in some discrete and continuous vibro-impact systems are investigated. New results on problems of dynamics of nonlinear elastic systems (rods, shells, arches) are obtained.
Graduated from Faculty of Mathematics and Mechanics of Dnepropetrovsk State University (DSU) in 1970 (department of applied theory of elasticity). Ph.D. thesis was defended in 1974. D.Sci. thesis was defended in 1988. A list of my works contains more than 100 titles. Since 1996 I have led the research seminar at Kharkov Polytechnical University on nonlinear dynamics.
A. F. Vakakis, L. I. Manevich, Yu. V. Mikhlin,V. N. Pilipchuk, and A. A. Zevin. Normal Modes and Localization in Nonlinear Systems. NY: Wiley, 1996.
Yu. V. Mikhlin and B. I. Morgunov. Normal vibrations in near-conservative self-excited and viscoelastic nonlinear systems // Nonlinear Dynamics, 2001, 25, 33–48.
Yu. V. Mikhlin, A. M. Volok. Solitary transversal waves and vibro-impact motions in infinite chains and rods // Int. Journal of Solids and Structure, 2000, 37, 3403–3420.
Yu. V. Mikhlin and A. L. Zhupiev. An application of the Ince algebraization to the stability of non-linear normal vibration modes // Int. Journal of Nonlinear Mechanics, 1997, 32(1), 393–409.
Yu. V. Mikhlin, I. Adrianov, V. Astashev, V. Pilipchuk, O. Gendelman, J. Kaplunov, Yu. Starosvetsky, V. Smirnov, M. Kovaleva, “In memory of Professor Leonid I. Manevitch”, Rus. J. Nonlin. Dyn., 16:3 (2020), 527–528