April 2018, 38(4): 1955-1981. doi: 10.3934/dcds.2018079

The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos

1. 

Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

2. 

Institute of Mathematics and Cryptology, Faculty of Cybernetics, Military Technical Academy, ul. Kaliskiego 2, 02-950 Warsaw, Poland

Received  December 2016 Revised  November 2017 Published  January 2018

We consider a special situation of the Hess-Appelrot case of the Euler-Poisson system which describes the dynamics of a rigid body about a fixed point. One has an equilibrium point of saddle type with coinciding stable and unstable invariant 2-dimensional separatrices. We show rigorously that, after a suitable perturbation of the Hess-Appelrot case, the separatrix connection is split such that only finite number of 1-dimensional homoclinic trajectories remain and that such situation leads to a chaotic dynamics with positive entropy and to the non-existence of any additional first integral.

Citation: Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, The Benjamin/Cummings Publ. Comp., London, 1978.

[2]

G. G. Appelrot, The problem of motion of a rigid body about a fixed point, Uchenye Zap. Moskov. Univ. Otdel Fiz. Mat. Nauk, 11 (1894), 1-112 [Russian].

[3]

G. G. Appelrot, Incompletely symmetric heavy gyroscope, Motion of a Solid Body around a Fixed Point, Izdat. AN SSSR, Moscow-Leningrad, 1940, 61-155.

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989 [Russian: Nauka, Moskva, 1974].

[5]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of the Mathematical and Celestial Mechanics, Encyclopaedia of Math. Sci., Dynamical Systems, 3, Springer, Berlin-Heidelberg-New York, 1988 2nd ed., 1993 [Russian: Itogi Nauki i Tekhniki, Fundamentalnye Napravlenya, Dinamicheskiye Sistemy, 3, VINITI, Moskva, 1985].

[6]

A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, Topology and stability of integrable systems, Russian Math. Surveys, 65 (2010), 259-317; [Russian: Uspekhi Mat. Nauk, 65 (2010), 71-132].

[7]

A. V. Borisov and I. S. Mamaev, The Hess case in the dynamics of a rigid body, J. Appl. Math. Mech., 67 (2003), 227-235 [Russian: Prikl. Mat. Mekh., 67 (2003), 256-265].

[8]

R. Devaney, Homoclinic orbits in Hamiltonian systems, J. Differential Equations, 21 (1976), 431-438; Transversal homoclinic orbits in integrable system, American J. Math., 100 (1978), 631-642. doi: 10.1016/0022-0396(76)90130-3.

[9]

S. A. Dovbysh, Some new dynamical effects in the perturbed Euler-Poincot problem associated with the splitting of separatrices, Prikl. Mat. Mekh., 53 (1989), 215-225 [Russian].

[10]

S. A. Dovbysh, Splitting of separatrices of unstable steady rotations and the non-integrability of the perturbed Lagrange problem, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., 3 (1990), 70-77 [Russian].

[11]

S. A. Dovbysh, The separatrix of an unstable position of equilibrium of the Hess-Appelrot gyroscope, Prikl. Mat. Mekh., 56 (1992), 534-545 [Russian].

[12]

V. Dragović and B. Gajić, An $L-A$ pair for the Hess-Appelrot system and a new integrable case for the Euler-Poisson equations on $so(4)× so(4)$, Roy. Soc. Edinburgh: Proc. A, 131 (2001), 845-855. doi: 10.1017/S0308210500001141.

[13]

V. V. Golubev, Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, State Publ. House of Theoret. Techn. Literat., Israel Program Sci. Transl., Jerusalem, 1960 [Russian: Gosud. Izdat. Tekhnik. -Teoret. Literat., Moskva, 1953].

[14]

W. Hess, Über die Euler'schen Bewegungsgleichungen und über eine neue particuäre Losung des Problems der Bewegung eines starren Körpers um einen festen Punkt, (German)Math. Ann., 37 (1890), 153-181. doi: 10.1007/BF01200234.

[15]

V. V. Kozlov, Non-existence of univalent integrals and ramification of solutions in the rigid body dynamics, Prikl. Mat. Mekh., 42 (1978), 400-406 [Russian].

[16]

V. V. Kozlov, Methods of Qualitative Analysis in the Rigid Body Dynamics, Izdat. Moskov. Univers., Moskva, 1980 [Russian].

[17]

V. V. Kozlov and D. V. Treshchev, Non-integrability of the general problem of rotation of a dynamically symmetric heavy rigid body with fixed point. Ⅰ, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1985), 73-81; Ⅱ, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1986), 39-44. [Russian].

[18]

P. Lubowiecki and H. Żołądek, The Hess-Appelrot system. Ⅰ. Invariant torus and its normal hyperbolicity, J. Geometric Mechanics, 4 (2012), 443-467.

[19]

P. Lubowiecki and H. Żołądek, The Hess-Appelrot system. Ⅱ. Perturbation and limit cycles, J. Differential Equations, 252 (2012), 1701-1722. doi: 10.1016/j.jde.2011.06.012.

[20]

A. J. Maciejewski and M. Przybylska, Differential Galois theory approach to the non-integrability of the heavy top, Ann. Fac.Sci. Toulouse Math.(6), 14 (2005), 123-160. doi: 10.5802/afst.1090.

[21]

N. A. Nekrasov, Analytic investigation of a particular case of motion of a heavy rigid body about fixed point, Matem. Sbornik, 18 (1895), 162-174 [Russian].

[22]

T. V. Salnikova, Non-integrability of the perturbed Lagrange problem, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1984), 62-66 [Russian].

[23]

D. V. Turaev and L. P. Shilnikov, On Hamiltonian systems with homoclinic saddle curves, Dokl. Akad. Nauk USSR, 304 (1989), 811-814 [Russian].

[24]

Yu. P. Varkhalev and G. V. Gorr, Asymptotically pendular motions of the Hess-Appelrot gyroscope, Prikl. Mat. Mekh., 48 (1984), 490-493 [Russian].

[25]

N. E. Zhukovski, Geometrische Interpretation des Hess'schen Falles der Bewegung eines Schweren Starken Korpers um eine Festen Punkt, Jahr. Berichte Deutschen Math. Verein., 3 (1894), 62-70.

[26]

S. L. Ziglin, Dichotomy of separatrices, bifurcation of solutions and nonexistence of an integral in the dynamics of a rigid body, Trudy Mosk. Mat. Obshch., 41 (1980), 287-303 [Russian].

[27]

S. L. Ziglin, Bifurcation of solutions and non-existence of first integrals in Hamiltonian mechanics. Ⅰ, Funct. Anal. Appl., 16 (1983), 181-189; Ⅱ, Funct. Anal. Appl., 17 (1983), 6-17; [Russian: Funkts. Anal. Prilozh., 16 (1982), 30-41; 17 (1983), 8-23].

[28]

H. Żołądek, The Monodromy Group, Birkhäuser, Basel, 2006.

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, The Benjamin/Cummings Publ. Comp., London, 1978.

[2]

G. G. Appelrot, The problem of motion of a rigid body about a fixed point, Uchenye Zap. Moskov. Univ. Otdel Fiz. Mat. Nauk, 11 (1894), 1-112 [Russian].

[3]

G. G. Appelrot, Incompletely symmetric heavy gyroscope, Motion of a Solid Body around a Fixed Point, Izdat. AN SSSR, Moscow-Leningrad, 1940, 61-155.

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989 [Russian: Nauka, Moskva, 1974].

[5]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of the Mathematical and Celestial Mechanics, Encyclopaedia of Math. Sci., Dynamical Systems, 3, Springer, Berlin-Heidelberg-New York, 1988 2nd ed., 1993 [Russian: Itogi Nauki i Tekhniki, Fundamentalnye Napravlenya, Dinamicheskiye Sistemy, 3, VINITI, Moskva, 1985].

[6]

A. V. Bolsinov, A. V. Borisov and I. S. Mamaev, Topology and stability of integrable systems, Russian Math. Surveys, 65 (2010), 259-317; [Russian: Uspekhi Mat. Nauk, 65 (2010), 71-132].

[7]

A. V. Borisov and I. S. Mamaev, The Hess case in the dynamics of a rigid body, J. Appl. Math. Mech., 67 (2003), 227-235 [Russian: Prikl. Mat. Mekh., 67 (2003), 256-265].

[8]

R. Devaney, Homoclinic orbits in Hamiltonian systems, J. Differential Equations, 21 (1976), 431-438; Transversal homoclinic orbits in integrable system, American J. Math., 100 (1978), 631-642. doi: 10.1016/0022-0396(76)90130-3.

[9]

S. A. Dovbysh, Some new dynamical effects in the perturbed Euler-Poincot problem associated with the splitting of separatrices, Prikl. Mat. Mekh., 53 (1989), 215-225 [Russian].

[10]

S. A. Dovbysh, Splitting of separatrices of unstable steady rotations and the non-integrability of the perturbed Lagrange problem, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., 3 (1990), 70-77 [Russian].

[11]

S. A. Dovbysh, The separatrix of an unstable position of equilibrium of the Hess-Appelrot gyroscope, Prikl. Mat. Mekh., 56 (1992), 534-545 [Russian].

[12]

V. Dragović and B. Gajić, An $L-A$ pair for the Hess-Appelrot system and a new integrable case for the Euler-Poisson equations on $so(4)× so(4)$, Roy. Soc. Edinburgh: Proc. A, 131 (2001), 845-855. doi: 10.1017/S0308210500001141.

[13]

V. V. Golubev, Lectures on Integration of the Equations of Motion of a Rigid Body about a Fixed Point, State Publ. House of Theoret. Techn. Literat., Israel Program Sci. Transl., Jerusalem, 1960 [Russian: Gosud. Izdat. Tekhnik. -Teoret. Literat., Moskva, 1953].

[14]

W. Hess, Über die Euler'schen Bewegungsgleichungen und über eine neue particuäre Losung des Problems der Bewegung eines starren Körpers um einen festen Punkt, (German)Math. Ann., 37 (1890), 153-181. doi: 10.1007/BF01200234.

[15]

V. V. Kozlov, Non-existence of univalent integrals and ramification of solutions in the rigid body dynamics, Prikl. Mat. Mekh., 42 (1978), 400-406 [Russian].

[16]

V. V. Kozlov, Methods of Qualitative Analysis in the Rigid Body Dynamics, Izdat. Moskov. Univers., Moskva, 1980 [Russian].

[17]

V. V. Kozlov and D. V. Treshchev, Non-integrability of the general problem of rotation of a dynamically symmetric heavy rigid body with fixed point. Ⅰ, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1985), 73-81; Ⅱ, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1986), 39-44. [Russian].

[18]

P. Lubowiecki and H. Żołądek, The Hess-Appelrot system. Ⅰ. Invariant torus and its normal hyperbolicity, J. Geometric Mechanics, 4 (2012), 443-467.

[19]

P. Lubowiecki and H. Żołądek, The Hess-Appelrot system. Ⅱ. Perturbation and limit cycles, J. Differential Equations, 252 (2012), 1701-1722. doi: 10.1016/j.jde.2011.06.012.

[20]

A. J. Maciejewski and M. Przybylska, Differential Galois theory approach to the non-integrability of the heavy top, Ann. Fac.Sci. Toulouse Math.(6), 14 (2005), 123-160. doi: 10.5802/afst.1090.

[21]

N. A. Nekrasov, Analytic investigation of a particular case of motion of a heavy rigid body about fixed point, Matem. Sbornik, 18 (1895), 162-174 [Russian].

[22]

T. V. Salnikova, Non-integrability of the perturbed Lagrange problem, Vestnik Moskov. Univers., Ser. I. Mat. Mekh., (1984), 62-66 [Russian].

[23]

D. V. Turaev and L. P. Shilnikov, On Hamiltonian systems with homoclinic saddle curves, Dokl. Akad. Nauk USSR, 304 (1989), 811-814 [Russian].

[24]

Yu. P. Varkhalev and G. V. Gorr, Asymptotically pendular motions of the Hess-Appelrot gyroscope, Prikl. Mat. Mekh., 48 (1984), 490-493 [Russian].

[25]

N. E. Zhukovski, Geometrische Interpretation des Hess'schen Falles der Bewegung eines Schweren Starken Korpers um eine Festen Punkt, Jahr. Berichte Deutschen Math. Verein., 3 (1894), 62-70.

[26]

S. L. Ziglin, Dichotomy of separatrices, bifurcation of solutions and nonexistence of an integral in the dynamics of a rigid body, Trudy Mosk. Mat. Obshch., 41 (1980), 287-303 [Russian].

[27]

S. L. Ziglin, Bifurcation of solutions and non-existence of first integrals in Hamiltonian mechanics. Ⅰ, Funct. Anal. Appl., 16 (1983), 181-189; Ⅱ, Funct. Anal. Appl., 17 (1983), 6-17; [Russian: Funkts. Anal. Prilozh., 16 (1982), 30-41; 17 (1983), 8-23].

[28]

H. Żołądek, The Monodromy Group, Birkhäuser, Basel, 2006.

Figure 1.  Ovals of elliptic curves
Figure 2.  The scheme of the separatrix connection and of the sections
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