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First integrals and asymptotic trajectories. (English. Russian original) Zbl 1444.34053
Sb. Math. 211, No. 1, 29-54 (2020); translation from Mat. Sb. 211, No. 1, 32-59 (2020).
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
58K05 Critical points of functions and mappings on manifolds
Full Text: DOI
[1] V. V. Kozlov 1992 Linear systems with a quadratic integral Prikl. Mat. Mekh.56 6 900-906
[2] English transl. in V. V. Kozlov 1992 J. Appl. Math. Mech.56 6 803-809 · Zbl 0792.70014
[3] V. V. Kozlov and A. A. Karapetyan 2005 On the stability degree Differ. Uravn.41 2 186-192
[4] English transl. in V. V. Kozlov and A. A. Karapetyan 2005 Differ. Equ.41 2 195-201 · Zbl 1090.34564
[5] V. I. Arnol’d, V. A. Vasil’ev, V. V. Goryunov and O. V. Lyashko 1988 Singularities. Local and global theory Dynamical systems. VI. Singularity theory I Sovr. Probl. Mat. Find. Napravl. 6 VINITI, Moscow 5-250 · Zbl 0787.58001
[6] English transl. in V. I. Arnol’d, V. A. Vasil’ev, V. V. Goryunov and O. V. Lyashko 1993 Encyclopaedia Math. Sci. 6 Springer-Verlag, Berlin 1-245
[7] T. Poston and I. Stewart 1978 Catastrophe theory and its applications Surveys Reference Works Math. 2 Pitman, London-San Francisco, CA-Melbourne xviii+491 pp.
[8] V. V. Kozlov 2008 On the instability of equilibria of conservative systems under typical degenerations Differ. Uravn.44 8 1033-1040
[9] English transl. in V. V. Kozlov 2008 Differ. Equ.44 8 1064-1071 · Zbl 1189.34097
[10] J. Williamson 1940 An algebraic problem involving the involutory integrals of linear dynamical systems Amer. J. Math.62 1 881-911 · Zbl 0024.13301
[11] H. Kocak 1982 Linear Hamiltonian systems are integrable with quadratics J. Math. Phys.23 12 2375-2380 · Zbl 0507.70015
[12] V. V. Kozlov 2018 Linear Hamiltonian systems: quadratic integrals, singular subspaces and stability Regul. Chaotic Dyn.23 1 26-46 · Zbl 1400.37061
[13] V. V. Kozlov and S. D. Furta 2009 Asymptotic solutions of strongly nonlinear systems of differential equations Regulyarnaya i Khaotichskaya Dinamika, Moscow-Izhevsk 2nd ed., 312 pp.
[14] English transl. V. V. Kozlov and S. D. Furta 2013 Springer Monogr. Math. Springer, Heidelberg xx+262 pp. · Zbl 1322.34003
[15] A. M. Molčanov (Molchanov) 1961 Subdivision of motions and asymptotic methods in the theory of nonlinear oscillations Dokl. Akad. Nauk SSSR136 5 1030-1033
[16] English transl. in A. M. Molčanov (Molchanov) 1961 Soviet Math. Dokl.2 162-165
[17] L. G. Khazin and È. È Shnol’ 1985 Stability of critical equilibrium states Department of Scientific and Technical Information of the Scientific Centre for Biological research of the USSR Academy of Sciences, Pushchino 216 pp.
[18] English transl. L. G. Khazin and È. È Shnol’ 1991 Nonlinear Sci. Theory Appl. Manchester Univ. Press, Manchester xii+208 pp.
[19] A. N. Kuznetsov 1989 Existence of solutions entering at a singular point of an autonomous system having a formal solution Funktsional. Anal. Prilozhen.23 4 63-74 · Zbl 0717.34004
[20] English transl. in A. N. Kuznetsov 1989 Funct. Anal. Appl.23 4 308-317 · Zbl 0717.34004
[21] V. V. Kozlov 1982 Asymptotic solutions of equations of classical mechanics Probl. Mat. Mekh.46 4 573-577
[22] English transl. in V. V. Kozlov 1982 J. Appl. Math. Mech.46 4 454-457 · Zbl 0522.70020
[23] S. Bolotin and P. Negrini 1995 Asymptotic solutions of Lagrangian systems with gyroscopic forces NoDEA Nonlinear Differential Equations Appl.2 4 417-444 · Zbl 0839.58024
[24] M. Brunella 1998 Instability of equilibria in dimension three Ann. Inst. Fourier (Grenoble)48 5 1345-1357 · Zbl 0930.37025
[25] V. V. Kozlov and D. V. Treschev 2018 Instability, asymptotic trajectories and dimension the phase space Mosc. Math. J.18 4 681-692 · Zbl 1416.37030
[26] V. V. Kozlov and D. V. Treschev 1999 Instability of isolated equilibria of dynamical systems with invariant measure in spaces of odd dimension Mat. Zametki65 5 674-680
[27] English transl. in V. V. Kozlov and D. V. Treschev 1999 Math. Notes65 5 565-570 · Zbl 0962.34036
[28] C. L. Siegel and J. K. Moser 1971 Lectures on celestial mechanics Grundlehren Math. Wiss. 187 Springer-Verlag, New York-Heidelberg xii+290 pp.
[29] V. V. Kozlov 2008 Gyroscopic stabilization of degenerate equilibria and the topology of real algebraic varieties Dokl. Ross. Akad. Nauk420 4 447-450 · Zbl 1152.37022
[30] English transl. in V. V. Kozlov 2008 Dokl. Math.77 3 412-415 · Zbl 1152.37022
[31] H. Poincaré 1883 Sur les formes cubiques ternaires et quaternaires J. École Polytech.50 199-253 · JFM 15.0097.01
[32] H. Matsumura and P. Monsky 1963/1964 On the automorphisms of hypersurfaces J. Math. Kyoto Univ.3 3 347-361 · Zbl 0141.37401
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