What I Did in Dynamics

L. M. Lerman

Russian journal of nonlinear dynamics, Tome 21 (2025) no. 1, pp. 15-31.

Résumé

This text presents a loose journey on my activity over the scientific life on the background of the rapid development of the theory of dynamical systems during the last 50 years. I intentionally do not explain nor go into mathematical details, otherwise this would require too many pages.

Détail

Keywords: nonautonomous dynamics, Hamiltonian dynamics, integrable Hamiltonian systems, localized solutions, traveling waves, patterns of elliptic PDEs
Mots-clés : homoclinic solutions

Bibliographie

[1] Ablowitz, M. J. and Segur, H., Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, Penn., 1981, 435 pp. | MR

[2] Alfimov, G. L., Eleonsky, V. M., and Lerman, L. M., “Solitary Wave Solutions of Nonlocal Sine-Gordon Equations”, Chaos, 8:1 (1998), 257–271 | DOI | MR

[3] Alfimov, G. L., Eleonsky, V. M., Kulagin, N. E., Lehrmann, L. M., and Silin, V. P., “New Stationary Field Distributions in Nonlinear Optics and Mechasnics of Continuous Media”, Phys. Lett. A, 138:8 (1989), 443–446 | DOI

[4] Alfimov, G. L., Eleonsky, V. M., Kulagin, N. E., Lerman, L. M., and Silin, V. P., “On Existence of Nontrivial Solutions for the Equation $\Delta u - u + u^3 = 0$”, Phys. D, 44:1–2 (1990), 168–177 | DOI | MR

[5] Alfimov, G. L., Eleonsky, V. M., Kulagin, N. E., Lerman, L. M., and Silin, V. P., “On Some Types of Multidimensional Self-Localized Solutions of the Equation $\Delta u + f(u) = 0$”, Methods of Qualitative Theory and Theory of Bifurcations, eds. N. N. Bautin, L. P. Shil'nikov, Gorky Gos. Univ., Gorky, 1991, 154–169 (Russian) | MR

[6] Anosov, D. V., “Dynamical Systems in the 1960s: The Hyperbolic Revolution”, Mathematical Events of the Twentieth Century, eds. A. A. Bolibruch, Yu. S. Osipov, Ya. G. Sinai, Springer, Berlin, 2006, 1–17 | MR

[7] Aranson, I. S., Gorshkov, K. A., Lomov, A. S., and Rabinovich, M. I., “Stable Particle-Like Solutions of Multidimensional Nonlinear Fields”, Phys. D, 43:2–3 (1990), 435–453 | DOI | MR

[8] Belyakov, L. A. and Shilnikov, L. P., “Homoclinic Curves and Complex Solitary Waves”, Selecta Math. Soviet., 9:3 (1990), 219–228 | MR

[9] Belyakov, L. A., Glebsky, L. Yu., and Lerman, L. M., “Abundance of Stable Stationary Localized Solutions to the Generalized $1$D Swift – Hohenberg Equation”, Comput. Math. Appl., 34:2–4 (1997), 253–266 | DOI | MR

[10] Bobenko, A. I., “The Landau – Lifshits Equation. The “Dressing” Procedure. Elementary Excitations”, Zap. Nauchn. Semin. LOMI, 123 (1983), 58–66 (Russian) | MR

[11] Bolsinov, A. V., “Methods of Calculation of the Fomenko – Zieschang Invariant”, Topological Classification of Integrable Systems, Adv. Soviet Math., 6, AMS, Providence, R.I., 1991, 147–183 | MR

[12] Uspekhi Mat. Nauk, 45:2(272) (1990), 49–77, 240 (Russian) | DOI | MR

[13] Bonatti, C. and Grines, V., “Knots As Topological Invariants for Gradient-Like Diffeomorphisms of the Sphere $S^3$”, J. Dynam. Control Systems, 6:4 (2000), 579–602 | DOI | MR

[14] Bronshtein, I. U., Nonautonomous Dynamical Systems, Shtiintsa, Kishinev, 1984, 291 pp. (Russian) | MR

[15] Solitons, eds. R. K. Bullough, P. J. Caudrey, Springer, Berlin, 1980, xvii, 389 pp. | MR

[16] Calogero, F. and Degasperis, A., Spectral Transform and Solitons: Vol. 1. Tools to Solve and Investigate Nonlinear Evolution Equations, Stud. Math. Appl., 13, North-Holland, Amsterdam, 1982, xv, 516 pp. | MR

[17] Conley, Ch. C., “On the Ultimate Behavior of Orbits with Respect to an Unstable Critical Point: 1. Oscillating, Asymptotic, and Capture Orbits”, J. Differential Equations, 5 (1969), 136–158 | DOI | MR

[18] Devaney, R. L., “Homoclinic Orbits in Hamiltonian Systems”, J. Differential Equations, 21:2 (1976), 431–438 | DOI | MR

[19] Dodd, R. K. and Eilbeck, J. Ch., Gibbon, J. D., and Morris, H. C., Solitons and Nonlinear Wave Equations, Acad. Press, New York, 1982, x, 630 pp. | MR

[20] Eckhaus, W., Studies in Non-Linear Stability Theory, Springer Tracts in Natur. Phil., 6, Springer, New York, 1965, viii, 117 pp. | DOI | MR

[21] Zh. Èksper. Teoret. Fiz., 68:5 (1975), 1928–1936 (Russian) | MR

[22] Zh. Èksper. Teoret. Fiz., 77:1 (1979), 409–413 (Russian)

[23] Faddeev, L. D. and Takhtajan, L. A., Hamiltonian Methods in the Theory of Solitons, transl. from the 1986 Russian original by A. G. Reyman, reprint of the 1987 English ed., Classics Math., Springer, Berlin, 2007, x, 592 pp. | MR

[24] Izv. Akad. Nauk SSSR Ser. Mat., 50:6 (1986), 1276–1307 (Russian) | DOI | MR

[25] Gardner, C. S., Greene, J. M., Kruskal, M. D., and Miura, R. M., “Method for Solving the Korteweg – de Vries Equation”, Phys. Rev. Lett., 19:19 (1967), 1095–1097 | DOI | MR

[26] Gelfreich, V. G. and Lerman, L. M., “Almost Invariant Elliptic Manifold in a Singularly Perturbed Hamiltonian System”, Nonlinearity, 15:2 (2002), 447–457 | DOI | MR

[27] Gelfreich, V. G. and Lerman, L. M., “Long-Periodic Orbits and Invariant Tori in a Singularly Perturbed Hamiltonian System”, Phys. D, 176:3–4 (2003), 125–146 | DOI | MR

[28] Sovrem. Mat. Prilozh., 2003, no. 8, 85–107 (Russian) | DOI | MR

[29] Glebsky, L. Yu. and Lerman, L. M., “On Small Stationary Localized Solutions for the Generalized $1$D Swift – Hohenberg Equation”, Chaos, 5:2 (1995), 424–431 | DOI | MR

[30] Glebsky, L. Yu. and Lerman, L. M., “Instability of Small Stationary Localized Solutions to a Class of Reversible $1 + 1$ PDEs”, Nonlinearity, 10:2 (1997), 389–407 | DOI | MR

[31] Teoret. Mat. Fiz., 212:1 (2022), 15–32 (Russian) | DOI | DOI | MR

[32] Grines, V. Z. and Lerman, L. M., “Gradient-Like Diffeomorphisms and Periodic Vector Fields”, Mosc. Math. J., 23:4 (2023), 533–544 | DOI | MR

[33] Grines, V. Z. and Lerman, L. M., “Nonautonomous Dynamics: Classification, Invariants, and Implementation”, J. Math. Sci. (N. Y.), 283:1 (2024), 40–62 | DOI | MR

[34] Grines, V., Medvedev, T., and Pochinka, O., Dynamical Systems on $2$- and $3$-Manifolds, Dev. Math., 46, Springer, New York, 2016, xxvi, 295 pp. | MR

[35] Gonchenko, S. V., Lerman, L. M., Shilnikov, A. L., and Turaev, D. V., “Scientific Heritage of L. P. Shilnikov: Part 2. Homoclinic Chaos”, Regul. Chaotic Dyn., 30:2 (2025) | DOI | MR

[36] Haberman, R., “Slowly Varying Jump and Transition Phenomena Associated with Algebraic Bifurcation Problems”, SIAM J. Appl. Math., 37:1 (1979), 69–106 | DOI | MR

[37] Iooss, G. and Peroeme, M. C., “Perturbed Homoclinic Solutions in Reversible $1 : 1$ Resonant Vector Fields”, J. Differ. Equ., 102:1 (1993), 62–88 | DOI | MR

[38] Kharlamov, M. P., Topological Analysis of Integrable Problems of Rigid Body Dynamics, Leningr. Gos. Univ., Leningrad, 1988, 197 pp. (Russian) | MR

[39] Kirschgässner, K., “Wave-Solutions of Reversible Systems and Applications”, J. Differential Equations, 45:1 (1982), 113–127 | DOI | MR

[40] Kolonitskii, S. B., Lerman, L. M., and Nazarov, A. I., Entire Solutions to the Swift – Hohenberg Equation via Variational Approach, , 2024, 24 pp. arXiv:2404.05066 [math.AP]

[41] Koltsova, O. Yu. and Lerman, L. M., “Periodic and Homoclinic Orbits in a Two-Parameter Unfolding of a Hamiltonian System with a Homoclinic Orbit to a Saddle-Center”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 5:2 (1995), 397–408 | DOI | MR

[42] Koltsova, O. Yu. and Lerman, L. M., “New Criterion of Nonintegrability for an $n$-Degrees-of-Freedom Hamiltonian System”, Hamiltonian Systems with Three or More Degrees of Freedom (S'Agaró, 1995), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 533, ed. C. Simó, Kluwer, Dordrecht, 1999, 458–470 | MR

[43] Koltsova, O. Yu. and Lerman, L. M., “Families of Transverse Poincaré Homoclinic Orbits in $2n$-Dimensional Hamiltonian Systems Close to the System with a Loop to a Saddle-Center, Nonlinear Dynamics, Bifurcations and Chaotic Behavior”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 6:6 (1996), 991–1006 | DOI | MR

[44] Koltsova, O., Lerman, L., Delshams, A., and Gutiérrez, P., “Homoclinic Orbits to Invariant Tori near a Homoclinic Orbit to Center-Center-Saddle Equilibrium”, Phys. D, 201:3–4 (2005), 268–290 | DOI | MR

[45] Kovalev, A. M. and Chudnenko, A. N., “On the Stability of the Equilibrium Position of a Two-Dimensional Hamiltonian System in the Case of Equal Frequencies”, Dopov. Akad. Nauk Ukr. RSR, Ser. A, 1977, no. 11, 1011–1014, 1056 (Ukrainian) | MR

[46] Kristiansen, K. U. and Wulff, C., “Exponential Estimates of Symplectic Slow Manifolds”, J. Differential Equations, 261:1 (2016), 56–101 | DOI | MR

[47] Trudy Mat. Inst. Steklova, 327 (2024), 140–219 (Russian) | DOI | DOI | MR

[48] Zh. Vychisl. Mat. Mat. Fiz., 48:4 (2008), 693–712 (Russian) | DOI | MR

[49] Tr. Mat. Inst. Steklova, 261 (2008), 188–209 (Russian) | DOI | MR

[50] Kulagin, N. E. and Lerman, L. M., “Localized Solutions of a Piecewise Linear Model of the Stationary Swift – Hohenberg Equation on the Line and on the Plane”, J. Math. Sci. (N. Y.), 202:5 (2014), 684–702 | DOI | MR

[51] Kulagin, N., Lerman, L., and Malkin, A., “Solitons and Cavitons in a Nonlocal Whitham Equation”, Commun. Nonlinear Sci. Numer. Simul., 93 (2021), Paper No. 105525, 19 pp. | DOI | MR

[52] Kulagin, N. E., Lerman, L. M., and Trifonov, K. N., “Twin Heteroclinic Connections of Reversible Systems”, Regul. Chaotic Dyn., 29:1 (2024), 40–64 | DOI | MR

[53] Kulagin, N. E. and Lerman, L. M., “Spatial Dynamics in the Family of Sixth-Order Differential Equations from the Theory of Partial Formation”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 32:6 (2024), 878–896 (Russian)

[54] Lamb, G. D., Elements of Soliton Theory, Wiley, New York, 1980, 304 pp. | MR

[55] Lerman, L. M. and Shilnikov, L. P., “On the Existence and Stability of Almost Periodic Tubes”, Proc. of the 5th Internat. Conf. on Nonlinear Oscillations (Kiev, 1969): Vol. 2. Qualitative Methods of Nonlinear Oscillations, Naukova Dumka, Kiev, 1970, 292–297 (Russian)

[56] Lerman, L. M., On Nonautonomous Dynamical Systems of the Morse – Smale Type, PhD Thesis, Gorky State Univ., Gorky, 1975, 150 pp. (Russian)

[57] Dokl. Akad. Nauk SSSR, 209:3 (1973), 544–547 (Russian) | MR

[58] Lerman, L. M. and Shil'nikov, L. P., “Homoclinical Structures in Nonautonomous Systems: Nonautonomous Chaos”, Chaos, 2:3 (1992), 447–454 | DOI | MR

[59] Lerman, L. M. and Umanskiǐ, Ya. L., “Structure of the Poisson Action of $\mathbf{R}^2$ on a Four-Dimensional Symplectic Manifold: 1”, Selecta Math. Sov., 6:4 (1987), 365–396 | MR

[60] Lerman, L. M. and Umanskiǐ, Ya. L., “Structure of the Poisson Action of $\mathbf{R}^2$ on a Four-Dimensional Symplectic Manifold: 2”, Selecta Math. Sov., 7:1 (1988), 39–48 | MR

[61] Lerman, L. M. and Umanskiǐ, Ya. L., “Integrable Hamiltonian Systems and Poisson Actions”, Selecta Math. Sov., 9:1 (1990), 59–67 | MR

[62] Nguyen Tien Zung, “Symplectic Topology of Integrable Hamiltonian Systems: 1. Arnold – Liouville with Singularities”, Compositio Math., 101:2 (1996), 179–215 | MR

[63] Mat. Sb., 183:12 (1992), 141–176 (Russian) | MR

[64] Mat. Sb., 184:4 (1993), 105–138 (Russian) | MR

[65] Mat. Sb., 186:10 (1995), 89–102 (Russian) | DOI | MR

[66] Lerman, L. M. and Umanskiy, Ya. L., Four-Dimensional Integrable Hamiltonian Systems with Simple Singular Points (Topological Aspects), transl. from the Russian manuscript by A. Kononenko and A. Semenovich, Transl. Math. Monogr., 176, AMS, Providence, R.I., 1998, xii, 177 pp. | MR

[67] Lerman, L. M., “More about the Structure of Integrable Stationary Waves for the Landau – Lifshits Equation”, Selecta Math., 12:4 (1993), 333–351 | MR

[68] Pis'ma v Zh. Èksper. Teoret. Fiz., 51:6 (1990), 336–338 (Russian)

[69] Prikl. Mat. Mekh., 47:3 (1983), 395–401 (Russian) | DOI | MR

[70] Lerman, L. M. and Umanskii, Ya. L., “Melnikov Function Method for Finding Chaos”, Nonlinear World: Proc. of 4th Internat. Workshop on Nonlinear and Turbulent Processes in Physics (Kiev, USSR, Oct 1989): In 2 Vols.: Vol. 2, eds. A. G. Sitenko, V. E. Zakharov, V. M. Chernousenko, et al., Naukova Dumka, Kiev, 1989, 387–391

[71] Lerman, L., “Isoenergetical Structure of Integrable Hamiltonian Systems in Extended Neighborhoods of Simple Singular Points: Three Degrees of Freedom”, Methods of Qualitative Theory of Differential Equations and Related Topics: Supplement, Amer. Math. Soc. Transl. Ser. 2, 200, eds. L. M. Lerman, G. Polotovskii, L. P. Shilnikov, AMS, Providence, R.I., 2000, 219–242 | MR

[72] Lerman, L. M., “Complex Dynamics and Bifurcations in a Hamiltonian System Having a Transversal Homoclinic Orbit to a Saddle Focus”, Chaos, 1:2 (1991), 174–180 | DOI | MR

[73] Lerman, L. M., “Dynamical Phenomena near a Saddle-Focus Homoclinic Connection in a Hamiltonian System”, J. Statist. Phys., 101:1–2 (2000), 357–372 | DOI | MR

[74] Lerman, L. M., “Homo- and Heteroclinic Orbits, Hyperbolic Subsets in a One-Parameter Unfolding of a Hamiltonian System with Heteroclinic Contour with Two Saddle-Foci”, Regul. Chaotic Dyn., 2:3–4 (1997), 139–155 | MR

[75] Lerman, L. M., “Hamiltonian Systems with Loops of a Separatrix of a Saddle-Center”, Selecta Math. Soviet., 10 (1991), 297–306 | MR

[76] Lerman, L. M., Behavior of Multidimensional Hamiltonian Systems in the Neighborhood of Homoclinic Orbits to Singular Points, PhD Dissertation, Nighni Novgorod, National Research Lobachevsky State University of Nizhni Novgorod, 1999, 301 pp. (Russian)

[77] Lerman, L. M., Lectures on Ordinary Differential Equations, R Dynamics, Institute of Computer Science, Izhevsk, 2016, 280 pp. (Russian)

[78] Lerman, L. and Markova, A., “On Symplectic Dynamics near a Homoclinic Orbit to $1$-Elliptic Fixed Point”, Trans. Moscow Math. Soc., 76:2 (2015), 271–299 | DOI | MR

[79] Gelfreich, V. and Lerman, L., “Separatrix Splitting at a Hamiltonian $O^2 i\omega$ Bifurcation”, Regul. Chaotic Dyn., 19:6 (2014), 635–655 | DOI | MR

[80] Lerman, L. M. and Yakovlev, E. I., “Geometry of Slow-Fast Hamiltonian Systems and Painlevé Equations”, Indag. Math. (N.S.), 27:5 (2016), 1219–1244 | DOI | MR

[81] Lerman, L., “Breaking Hyperbolicity for Smooth Symplectic Toral Diffeomorphisms”, Regul. Chaotic Dyn., 15:2–3 (2010), 194–209 | DOI | MR

[82] Lerman, L. M. and Meiss, J. D., “Mixed Dynamics in a Parabolic Standard Map”, Phys. D, 315 (2016), 58–71 | DOI | MR

[83] Lerman, L. M. and Trifonov, K. N., “Symplectic Partially Hyperbolic Automorphisms of $6$-Torus”, J. Geom. Phys., 195 (2024), Paper No. 105038, 21 pp. | DOI | MR

[84] Sibirsk. Mat. Zh., 29:3 (1988), 92–103, 220 (Russian) | DOI | MR

[85] Lerman, L. M. and Gubina, E. V., “Nonautonomous Gradient-Like Vector Fields on the Circle: Classification, Structural Stability and Autonomization”, Discrete Contin. Dyn. Syst. Ser. S, 13:4 (2020), 1341–1367 | MR

[86] Lerman, L. M., Naryshkin, P. E., and Nazarov, A. I., “Abundance of Entire Solutions to Nonlinear Elliptic Equations by the Variational Method”, Nonlinear Anal., 190 (2020), Art. 111590, 21 pp. | DOI | MR

[87] Meyer, K. R. and Zhang, X., “Stability of Skew Dynamical Systems”, J. Differential Equations, 132:1 (1996), 66–86 | DOI | MR

[88] Lloyd, D. J. B., Sandstede, B., Avitabile, D., and Champneys, A. R., “Localized Hexagon Patterns of the Planar Swift – Hohenberg Equation”, SIAM J. Appl. Dyn. Syst., 7:3 (2008), 1049–1100 | DOI | MR

[89] Meyer, K. R. and Sell, G. R., “Mel'nikov Transforms, Bernoulli Bundles, and Almost Periodic Perturbations”, Trans. Amer. Math. Soc., 314:1 (1989), 63–105 | MR

[90] Mielke, A., “A Reduction Principle for Nonautonomous Systems in Infinite-Dimensional Spaces”, J. Differential Equations, 65:1 (1986), 68–88 | DOI | MR

[91] Nguyen Tien Zung, “Symplectic Topology of Integrable Hamiltonian Systems: 2. Topological Classification”, Compositio Math., 138:2 (2003), 125–156 | DOI | MR

[92] Pontrjagin, L. S., “Les fonctions presque périodiques et l'analysis situs”, Comptes rendus hebdomadaires des séances de l'Académie des Sciences, 196 (1933), 1201–1203 | MR

[93] Robinson, C., “Horseshoes for Autonomous Hamiltonian Systems Using the Mel'nikov Integral”, Ergodic Theory Dynam. Systems, 8* (1988), 395–409 | MR

[94] Scheurle, J., “Chaotic Solutions of Systems with Almost Periodic Forcing”, Z. Angew. Math. Phys., 37:1 (1986), 12–26 | DOI | MR

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[96] Shilnikov, L., Selected Papers, eds. V. S. Afraimovich, L. A. Belyakov, S. V. Gonchenko, L. M. Lerman, D. V. Turaev, A. L. Shilnikov, National Research Lobachevsky State University of Nizhny Novgorod, Nighny Novgorod, 2017, 429 pp. (Russian)

[97] Shil'nikov, L. P., “On the Question of the Structure of the Neighborhood of a Homoclinic Tube of an Invariant Torus”, Dokl. Akad. Nauk SSSR, 180:2 (1968), 286–289 (Russian) | MR

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[100] Stoffer, D., “Transversal Homoclinic Points and Hyperbolic Sets for Non-Autonomous Maps: 1”, Z. Angew. Math. Phys., 39 (1988), 518–549 | DOI | MR

[101] Turaev, D., “Hyperbolic Sets near Homoclinic Loops to a Saddle for Systems with a First Integral”, Regul. Chaotic Dyn., 19:6 (2014), 681–693 | DOI | MR

[102] Dokl. Akad. Nauk SSSR, 304:4 (1989), 811–814 (Russian) | MR

[103] Mat. Zametki, 23:5 (1978), 739–752 (Russian) | DOI | MR

[104] Veselov, A. P., “The Landau – Lifshits Equation and Integrable Systems of Classical Mechanics”, Dokl. Akad. Nauk SSSR, 270:5 (1983), 1094–1097 (Russian) | MR

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