Abstract
Invariant surfaces play a crucial role in the dynamics of mechanical systems separating regions filled with chaotic behavior. Cases where such surfaces can be found are rare enough. Perhaps the most famous of these is the so-called Hess case in the mechanics of a heavy rigid body with a fixed point.
We consider here the motion of a non-autonomous mechanical pendulum-like system with one degree of freedom. The conditions of existence for invariant surfaces of such a system corresponding to non-split separatrices are investigated. In the case where an invariant surface exists, combination of regular and chaotic behavior is studied analytically via the Poincaré-Mel’nikov separatrix splitting method, and numerically using the Poincaré maps.
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References
Hess, W., Über die Euler’schen Bewegungsgleichungen und über eine neue particuläre Lösung des Problems der Bewegung eines starren Körpers um einen festen Punkt, Math. Ann., 1890, vol. 37, no. 2, pp. 153–181.
Appel’rot, G. G., Concerning Section 1 of the Memoir of S. V. Kovalevskaya “Sur le problème de la rotation d’un corps solide autour d’un point fixe”, and the Appendix to This Paper, Mat. Sb., 1892, vol. 16, no. 3, pp. 483–507 (Russian).
Nekrassov, P. A., Zur Frage von der Bewegung eines schweren starren Körpers um einen festen Punkt, Mat. Sb., 1892, vol. 16, no. 3, pp. 508–517 (Russian).
Zhukovsky, N. E., Hess’ Loxodromic Pendulum, in Collected Works: Vol. 1, Moscow: Gostekhizdat, 1937, pp. 332–348 (Russian).
Mlodzieiowski, B. K. and Nekrasov, P. A., Conditions for the Existence of Asymptotic Periodic Motions in the Hess Problem, Tr. Otdel. Fiz. Nauk Obsch. Lyubit. Estestvozn., 1893, vol. 6, no. 1, pp. 43–52 (Russian).
Chaplygin, S. A., Concerning Hess’ Loxodromic Pendulum, in Collected Works: Vol. 1, Moscow: Gostekhizdat, 1948, pp. 133–135 (Russian).
Nekrassov, P. A., Étude analytique d’un cas du mouvement d’un corps pesant autour d’un point fixe, Mat. Sb., 1896, vol. 18, no. 2, pp. 161–274 (Russian).
Demin, V. G. and Stepanova, L. A., Construction and Analysis of Exact Solutions for the Equations of Rigid-Body Dynamics, Soviet Appl. Mech., 1976, vol. 12, no. 9, pp. 875–887; see also: Prikl. Mekh., 1976, vol. 12, no. 9, pp. 3–17.
Kozlov, V. V., Splitting of the Separatrices in the Perturbed Euler - Poinsot Problem, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1976, vol. 31, no. 6, pp. 99–104 (Russian).
Ziglin, S. L., Splitting of Separatrices, Branching of Solutions and Nonexistence of an Integral in the Dynamics of a Solid Body, Trans. Moscow Math. Soc., 1982, no. 1, pp. 283–298; Tr. Mosk. Mat. Obs., 1980, vol. 41, pp. 287–303.
Dovbysh, S. A., The separatrix of an unstable position of equilibrium of a Hess-Appelrot gyroscope, J. Appl. Math. Mech., 1992, vol. 56, no. 4, pp. 534–545.
Emel’yanova, I. S., A Case of the Solution of the Hess Problem in Trigonometric Functions, Russian Math. (Iz. VUZ), 1998, vol. 42, no. 3, pp. 7–12; see also: Izv. Vyssh. Uchebn. Zaved. Mat., 1998, no. 3, pp. 10–15.
Dragović, V. and Gajić, B., An L-A Pair for the Hess-Apel’rot System and a New Integrable Case for the Euler - Poisson Equations on so(4) x so(4), Proc. Roy. Soc. Edinburgh Sect. A, 2001, vol. 131, no. 4, pp. 845–855.
Belyaev, A. V., Analytic Properties of Solutions of the Euler -Poisson Equations in the Hess Case, Ukr. Math. Bull., 2005, vol. 2, no. 3, pp. 301–321; see also: Ukr. Mat. Visn., 2005, vol. 2, no. 3, pp. 297–317.
Lubowiecki, P. and Żołądek, H., The Hess - Appelrot System: 1. Invariant Torus and Its Normal Hyperbolicity, J. Geom. Mech., 2012, vol. 4, no. 4, pp. 443–467.
Lubowiecki, P. and Żołądek, H., The Hess - Appelrot System: 2. Perturbation and Limit Cycles, J. Differential Equations, 2012, vol. 252, no. 2, pp. 1701–1722.
Belyaev, A. V., On the General Solution of the Problem of the Motion of a Heavy Rigid Body in the Hess Case, Sb. Math., 2015, vol. 206, nos. 5–6, pp. 621–649; see also: Mat. Sb., 2015, vol. 206, no. 5, pp. 5–34.
Belyaev, A. V., On the Representation of Solutions of the Problem of a Heavy Rigid Body’s Motion in the Kovalevskaya Case by ζ-and \(\wp \)-Weierstrass Functions and Nonintegrability in Quadratures of the Hess Case, Sb. Math., 2016, vol. 207, nos. 7–8, pp. 889–914; see also: Mat. Sb., 2016, vol. 207, no. 7, pp. 3–28.
Kurek, R., Lubowiecki, P. and Żołądek, H., The Hess - Appelrot System: 3. Splitting of Separatrices and Chaos, Discrete Contin. Dyn. Syst. A, 2018, vol. 38, no. 4, pp. 1955–1981.
Żołądek, H., Perturbations of the Hess - Appelrot and the Lagrange Cases in the Rigid Body Dynamics, J. Geom. Phys., 2019, vol. 142, pp. 121–136.
Kozlov, V. V., Integrability and Non-Integrability in Hamiltonian Mechanics, Russian Math. Surveys, 1983, vol. 38, no. 1, pp. 1–76; see also: Uspekhi Mat. Nauk, 1983, vol. 38, no. 1(229), pp. 3–67.
Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).
Borisov A. V., Mamayev I. S. The Hess Case in Rigid-Body Dynamics, J. Appl. Math. Mech., 2003, vol. 67, no. 2, pp. 227–235; see also: Prikl. Mat. Mekh., 2003, vol. 67, no. 2, pp. 256–265.
Burov, A. A., Nonintegrability of the Equation of Plane Oscillations of a Satellite in an Elliptic Orbit, Mosc. Univ. Mech. Bull., 1984, vol. 39, no. 1, pp. 38–41; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1984, no. 1, pp. 71–73.
Koch, B.-P. and Bruhn, B., Chaotic and Periodic Motions of Satellites in Elliptic Orbits, Z. Naturforsch. A, 1989, vol. 44, no. 12, pp. 1155–1162.
Maciejewski, A. J., Non-Integrability of the Planar Oscillations of a Satellite, Acta Astron., 1995, vol. 45, no. 1, pp. 333–344.
Teofilatto, P. and Graziani, F., On Librational Motion of Spacecraft, Chaos Solitons Fractals, 1996, vol. 7, no. 10, pp. 1721–1744.
Cherry, T. M., The Asymptotic Solutions of Analytic Hamiltonian Systems, J. Differential Equations, 1968, vol. 4, no. 2, pp. 142–159.
Kozlov, V. V., Oscillations of One-Dimensional Systems with Periodic Potential, Mosc. Univ. Mech. Bull., 1980, vol. 35, nos. 5–6, pp. 74–78; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1980, no. 6, pp. 104–107, 120.
Burov, A. and Kosenko, I., On Planar Oscillations of a Body with a Variable Mass Distribution in an Elliptic Orbit, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci., 2011, vol. 225, no. 10, pp. 2288–2295.
Zhuravskii, A. M., Handbook of Elliptical Functions, Moscow: Akad. Nauk SSSR, 1941 (Russian).
Sretenskii, L. N., Some Integrability Cases for the Equations of Gyrostat Motion, Dokl. Akad. Nauk SSSR, 1963, vol. 149, no. 2, pp. 292–294 (Russian).
Sretensky, L. N., On Some Cases of Motion of a Heavy Rigid Body with a Gyroscope, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1963, no. 3, pp. 60–71 (Russian).
Kozlov V. V., Onishchenko D. A. Nonintegrability of Kirchhoff’s Equations, Sov. Math. Dokl., 1982, vol. 26, pp. 495–498; see also: Dokl. Akad. Nauk SSSR, 1982, vol. 266, no. 6, pp. 1298–1300.
Burov, A. A., Partial Integrals in the Problem on the Motion of a Body Suspended from a String, Izv. Akad. Nauk. SSSR. Mekh. Tverd. Tela, 1987, no. 2, p. 84 (Russian).
Gorr, G. V. and Rubanovskii, V. N., On a New Class of Motions of a System of Heavy Hinged Rigid Bodies, J. Appl. Math. Mech., 1988, vol. 52, no. 5, pp. 551–555; see also: Prikl. Mat. Mekh., 1988, vol. 52, no. 5, pp. 707–712.
Burov, A. A., Particular Integrals of the Equations of Motion of a Rigid Body over a Smooth Horizontal Plane, Mech. Solids, 1986, vol. 21, no. 5, pp. 75–76; see also: Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela., 1986, no. 5, pp. 72–73.
Burov, A. A., On the Motion of a Heavy Rigid Body with Points Contiguous to a Smooth Surface, in Problems of Analytical Mechanics and Stability Theory: Collection of Papers Dedicated to the Memory of Academician V. V. Rumyantsev, V. V. Kozlov et al. (Eds.), Moscow: Fizmatlit, 2009, pp. 42–48 (Russian).
Burov, A. A. and Karapetyan, A. V., On the Motion of a Rigid Body in a Particle Flow, J. Appl. Math. Mech., 1993, vol. 57, no. 2, pp. 295–299; see also: Prikl. Mat. Mekh., 1993, vol. 57, no. 2, pp. 77–81.
Vecheslavov, V. V., Dynamics of Hamiltonian Systems under Piecewise Linear Force, JETP, 2005, vol. 100, no. 4, pp. 811–819; see also: Zh. Eksper. Teoret. Fiz., 2005, vol. 127, no. 4, pp. 915–924.
Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., The Hess-Appelrot System and Its Nonholonomic Analogs, Proc. Steklov Inst. Math., 2016, vol. 294, pp. 252–275; see also: Tr. Mat. Inst. Steklova, 2016, vol. 294, pp. 268–292.
Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., The Hess - Appelrot Case and Quantization of the Rotation Number, Regul. Chaotic Dyn., 2017, vol. 22, no. 2, pp. 180–196.
Beletskii, V. V., Essays on the Motion of Celestial Bodies, Basel: Birkhäuser, 2001.
Funding
This research is partially supported by RFBR, grants 18-01-00335, project EMaDeS (Centro-01-0145-FEDER-000017), and the Portuguese Foundation for Science and Technologies via the Centre for Mechanical and Aerospace Science and Technologies, C-MAST, POCI-01-0145-FEDER-007718.
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Burov, A.A., Guerman, A.D. & Nikonov, V.I. Asymptotic Invariant Surfaces for Non-Autonomous Pendulum-Type Systems. Regul. Chaot. Dyn. 25, 121–130 (2020). https://doi.org/10.1134/S1560354720010104
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DOI: https://doi.org/10.1134/S1560354720010104