Dynamical Behavior of a Rigid Body with One Fixed Point (Gyroscope). Basic Concepts and Results. Open Problems: a Review

Document Type: Research Paper


1 Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. BonchevStr., Bl. 4, Bulgaria

2 University of Transport, G. Milev Str., 158, 1574 Sofia, Bulgaria


The study of the dynamic behavior of a rigid body with one fixed point (gyroscope) has a long history. A number of famous mathematicians and mechanical engineers have devoted enormous time and effort to clarify the role of dynamic effects on its movement (behavior) – stable, periodic, quasi-periodic or chaotic. The main objectives of this review are: 1) to outline the characteristic features of the theory of dynamical systems and 2) to reveal the specific properties of the motion of a rigid body with one fixed point (gyroscope).This article consists of six sections. The first section addresses the main concepts of the theory of dynamical systems. Section two presents the main theoretical results (obtained so far) concerning the dynamic behavior of a solid with one fixed point (gyroscope). Section three examines the problem of gyroscopic stabilization. Section four deals with the non-linear (chaotic) dynamics of the gyroscope. Section five is a brief analysis of the gyroscope applications in engineering. The final section provides conclusions and generalizations on why the theory of dynamical systems should be used in the study of the movement of gyroscopic systems.


Main Subjects

[1] Afraimovich, V., Gonchenko, S., Lerman, L., Shilnikov, A. and Turaev, D., “Scientific Heritage of L.P. Shilnikov”, Regular and Chaotic Dynamics, Vol. 19, No. 4, pp. 435-460, 2014.

[2] Alligood, K., Sauer, T. and Yorke, J., Chaos. An Introduction to Dynamical Systems, Springer, NewYork, 1996.

[3] Anchev, A., “On the Stability of Permanent Rotations of a Heavy Gyrostat”, J. of Appl. Math. and Mech., Vol. 26, No. 1, pp. 22-28, 1962.

[4] Andronov, A., Witt, A. and Chaikin, S., Theory of Oscillations, Addison-Wesley, Reading, MA, 1966.

[5] Аndronov, А. And Pontryagin, L., “Systemes grossieres”, DAN USSR, Vol. 14, pp. 247-251, 1937.

[6] Аrnold, V., Ordinary Differential Equations. Fourth Ed., Igevsk, 2000 (in Russian).

[7] Arnold, V., Afraimovich, V., Iliaschenko, Yu. and Shilnikov, L., Bifurcation Theory. Nauka, Moscow, 1986 (in Russian).

[8] Arrowsmith, D. and Place, C., Dynamical Systems: differential equations, maps and chaotic behaviour, Chapman & Hall, London, 1992.

[9] Aslanov, V. and Doroshin, A., “Chaotic Dynamics of an unbalanced Gyrostat”, J. of Applied Mathematics and Mechanics, Vol. 74, pp. 524-535, 2010.

[10] Bachvarov, S., Mechanics. Part I, Stand. Print, Sofia, 2001 (in Bulgarian).

[11] Banhi, V. and Savin, A., “Molecular Gyroscopes and Biological Affects of Weak Extremely Low-frequency Magnetic Fields”, Physical Review E, Vol. 65, pp. 051912, 2002.

[12] Bardin, B., “On the Orbital Stability of Pendulum Like Motions of Rigid Body in the Bobylev-Steklov Case”, Regular and Chaotic Dynamics, Vol. 15, No. 6, pp. 704-716, 2010.

[13] Barreira, L. and Valls, C., Dynamical Systems: An Introduction, Springer, London, 2013.

[14] Basak, I., “Explicit Solution of the Zhukovski-Volterra Gyrostat”, Regular and Chaotic Dynamics, Vol. 14, No. 2, pp. 223-236, 2009.

[15] Bautin, N. and Leontovich, E., Methods and Approaches for Qualitative Investigation of Two Dimensional Dynamical Systems, Nauka, Moscow, 1989 (in Russian).

[16] Bloch, A., Nonholonomic Mechanics and Control. Springer, New York, 2003.

[17] Bolotin, S., “The Hill Determinant of a Periodic Trajectories”, Mathematika, Mechanika, Vol. 3, pp. 30-34, 1988.

[18] Borisov, A. and Mamaev, I., Dynamics of Rigid Body, Moscow-Ijevsk, 2001.

[19] Borisov, A., Kilin, A. and Mamaev, I., “Absolute and Relative Choreographies in Rigid Body Dynamics”, Regular and Chaotic Dynamics, Vol. 13, No. 3, pp. 204-220, 2008.

[20] Borisov, A., Kilin, A. and Mamaev, I., “Chaos in a Restricted Problem of Rotation of a Rigid Body with a Fixed Point”, Regular and Chaotic Dynamics, Vol. 13, No. 3, pp. 221-233, 2008.

[21] Brizard, A., An Introduction to Lagrangian Mechanics, World Scientific, Singapore, 2008.

[22] Bulatovic, R., “The Stability of Linear Potential Gyroscopic Systems in Cases when the Potential Energy has a Maximum”, Prikl. Mat. Mekh., Vol. 61, No. 3, pp. 385-389, 1997.

[23] Burra, L., Chaotic Dynamics in Nonlinear Theory, Springer, New York, 2014.

[24] Butenin, N., Neimark, Yu. and Fufaev, N., Introduction in Theory of Nonlinear Oscillations, Nauka, Мoscow 1976 (in Russian).

[25] Carlson, J. and Doyle, J., “Highly Optimized Tolerance: robustness and design in complex systems”, Phys. Rev. Lett., Vol. 84, pp. 2529-2532, 2000.

[26] Chang, D. and Marsden, J., Gyroscopic forces and collision avoidance with convex obstacles. In: New Trends in Nonlinear Dynamics and Control and Their Application, Springer, Berlin, Vol. 295, pp. 145-159, 2003.

[27] Chen, H., “Chaos and Chaos Synchronization of a Symmetric Gyro with Linear Plus Cubic Damping”, J. of Sound and Vibrations, Vol. 255, No. 4, pp. 719-740, 2002.

[28] Chen, H. and Ge, Zh., “Bifurcations and Chaos of a Two-degree of Freedom Dissipative Gyroscope”, Chaos, Solitons and Fractals, Vol. 24, pp. 125-136, 2005.

[29] Chetayev, N., The Stability of Motion, Pergamon Press, New York 1961.

[30] Coddington, E. and Levinson, N., Theory of Ordinary Differential Equations, McGraw-Hill Inc, London, 1987.

[31] Coutinho, M.: Guide to Dynamics Simulations of Rigid Bodies and Particle Systems. Springer, London (2013)

[32] Deriglazov, A., Classical Mechanics: Hamiltonian and Lagrangian formalism, Springer, New York, 2010.

[33] Desloge, E., Classical Mechanics, Vol. I, John Wiley & Sons, New York, 1982.

[34] Elipe, A., Arribas, M. and Riaguas, A., “Complete Analysis of Bifurcations in the Axial Gyrostat Problem”, J. Phys. A: Math. Gen.,Vol. 30, pp. 587-601, 1997.

[35] Elmandouh, A., “New Integrable Problems in the Dynamics of Particle and Rigid Body”, Acta Mech., Vol. 226, No. 11, pp. 3749-3762, 2015.

[36] Elmandouh, A., “New Integrable Problems in Rigid Body Dynamics with Quartic Integrals”, Acta Mech., Vol. 226, No. 8, pp. 2461-2472, 2015.

[37] Eueliri, L., “Theoria Motus Corporum Solidorum seu Rigidorum”. Griefswald, A. F. Rose, 1785; or Eueleri, L., Opera Omnia Ser. 2 Teubner, 3, 1948 and 4, 1950.

[38] Fan, Y. and Chay T., “Crisis and topological entropy”, Physical Review E, Vol. 51, pp. 1012-1019, 1995.

[39] Farivar, F., Shoorehdeli, M., Nekoui, M. and Teshnehlab, M., “Chaos Control and Generalized Projective Synchronization of Heavy Symmetric Chaotic Gyroscope Systems via Gaussian Radial Basis Adaptive Variable Structure Control”, Chaos, Solitons and Fractals, Vol. 45, pp. 80-97, 2012.

[40] Ge, Z., Chen, H. and Chen, H., “The Regular and Chaotic Motions of Symmetric Heavy Gyroscope with Harmonic Excitation”, J. of Sound and Vibration, Vol. 198, No. 2, pp. 131-147, 1996.

[41] Gluhovsky, A., “Nonlinear Systems that are Superpositions of Gyrostats”, Sov. Phys. Dokl., Vol. 27, pp. 823-825, 1982.

[42] Gluhovsky, A., “The structure of Energy Conserving Low-order Models”, Physics of Fluids, Vol. 11, No. 2, pp. 334-343, 1999.

[43] Gonchenko, S. and Ovsyannikov, I., “On Bifurcations of Three-dimensional Diffeomorphisms with a Non-transversal Heteroclinic Cycle Containing Saddle-foci”, Nonlinear Dynamics, Vol. 6, No. 1, pp. 61-77, 2010 (in Russian).

[44] Gradwell, G., Khonsari, M. and Ram, Y., “Stability Boundaries of a Conservative Gyroscopic System”, J. of Applied Mechanics, Vol. 70, pp. 561-567, 2003.

[45] Gray, G., Kammer, D., Dobson, I. and Miller, A., “Heteroclinic Bifurcations in Rigid Bodies Containing Internally Moving Parts and a Viscous Damper”, ASME Applied Mechanics, Vol. 66, 720-728, 1999.

[46] Guckenheimer, J. and Holmes, Ph., Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields, Springer, New York, 1992.

[47] Han, Zh. and Wang, S., “Multiple Solutions for Nonlinear Systems with Gyroscopic Terms”, Nonlinear Analysis, Vol. 75, pp. 5756-5764, 2012.

[48] Hartman, Ph., Ordinary Differential Equations, John Willey & Sons, London, 1964.

[49] Hauger, W., “Stability of a Gyroscopic Nonconservative System”, J. of Applied Mechanics, Vol. 42, pp. 739-740, 1975.

[50] Hill G., “On the Part of the Motion of the Lunar Perigee which is a Function of the Mean Motion of the Sun and Moon”, Acta Math., Vol. 8, pp. 1-36, 1886.

[51] Idowu, B., Vincent, U. and Njah, A., “Synchronization of Chaos in Non-identical Parametrically Excited Systems”, Chaos, Solitons and Fractals, Vol. 39, pp. 2322-2331, 2009.

[52] Kirillov, O., “Gyroscopic Stabilization in the Presence of Nonconservative Forces”, Dokl. Acad. Nauk, Vol. 416, pp. 451-456, 2007.

[53] Kliem, W. and Pommer, Ch., “Indefinite Damping in Mechanical Systems and Gyroscopic Stabilization”, Z. Angew. Math. Phys. (ZAMP), Vol. 60, pp. 785-795, 2009.

[54] Kowalevski, S., “Sur le Probleme de la Rotation d’un corps Solide Antor d’un Point Fixe”, Acta Math., Vol. 12, pp. 177-232, 1889.

[55] Kozlov, V., “Gyroscopic Stabilization and Parametric Resonance”, J. of Applied Maths and Mechs., Vol. 65, No. 5, pp.715-721, 2001.

[56] Krechetnikov, R. and Marsden J., “On Destabilizing Effects of Two Fundamental Nonconservative Forces”, Physica D, Vol. 214, pp. 25-32, 2006.

[57] Krechetnikov, R. and Marsden J., “Dissipation Induced Instabilities in Finite Dimensions”, Reviews of Modern Physics, Vol. 79, pp. 519-553, 2007.

[58] Krechetnikov, R. and Marsden J.” Dissipation Induced Instability Phenomena in Infinite Dimensional Systems”, Arch. Rational Mech. Anal., Vol. 194, pp. 611-668, 2009.

[59] Kuznetsov, Yu., Elements of Applied Bifurcation Theory, 2 ed., Springer, New York, 1998.

[60] Lakhadanov, V., “On Stabilization of Potential Systems”, PMM, Vol. 39, No. 1, pp. 45-50, 1975.

[61] Lancaster, P., “Stability of Linear Gyroscopic Systems: A review”, Linear Algebra and its Applications, Vol. 439, pp. 686-706, 2013.

[62] Landa, P., Nonlinear Oscillations and Waves in Dynamical Systems, Kluwer Academic Publishers, Dordrecht, 1996.

[63] Leimanis, E., The General Problem of Motion of Coupled Rigid Bodies about a Fixed Point, Springer, Berlin, 1965.

[64] Leipnik, R. and Newton, T., “Double Strange Attractor in Rigid Body Motion with Linear Feedback Control”, Phys. Lett., Vol. 86, No. 2, pp. 63-67, 1981.

[65] Leonhardt, U. and Piwnicki, P., “Ultrahigh Sensitivity of Slow-light Gyroscope”, Physical Review A, Vol. 62, pp. 055801, 2000.

[66] Leontovich, Е. and Мayer, A., “Оn Trajectories which Definite the Qualitative Separation of Sphere of Trajectories”, DAN USSR, Vol. 14, pp. 251-254, 1937.

[67] Liu, Y. and Rimrott, F., “Global Motion of a Dissipative Asymmetric Gyrostat”, Archive of Applied Mechanics Vol. 62, pp. 329-337, 1992.

[68] Luo, J., Nie, Y., Zhang, Y. and Zhou, Z., “Null Result for Violation of the Equivalence Principle with Free-fall Rotating Gyroscopes”, Physical Review D, Vol. 65, pp. 042005, 2002.

[69] Markeev, A., “The Dynamics of a Rigid Body Colliding with a Rigid Surface”, Regular and Chaotic Dynamics, Vol. 13, No. 2, pp. 96-129, 2008.

[70] Marsden, J., Lectures on Mechanics, CambridgeUniversity Press, London Math. Society Lecture Notes Series 174., 1992.

[71] Merkin D., Gyroscopic Systems, Nauka, Moscow, 1924 (in Russian).

[72] Metelicin, I., Theory of Gyroscope. Theory of stability, Nauka, Moscow, 1977 (in Russian).

[73] Morozov, V. and Kalenova, V., “Stability of Non-autonomous Mechanical Systems with Gyroscopic and Dissipative Forces”, XII Russian Control Conference, Moscow 16-19 June 2014, pp. 1888-1894, 2014.

[74] Neimark, Yu., Method of Points Map in Theory of Nonlinear Oscillations, Nauka, Moscow, 1972 (in Russian).

[75] Neimark, Yu. and Landa, P., Stochastic and Chaotic Oscillations, Kluwer Acad. Publishers, Singapore, 1992.

[76] Nikolov, S., and Nedkova, N., “Gyrostat Model Regular and Chaotic Behaviour”, J. of Theoretical and Applied Mechanics, Vol. 45, No 4, pp. 15-30, 2015.

[77] Panchev, S.: Theory of Chaos. Sofia, Acad. Publ.“prof. Marin Drinov”, Sofia, 2001 (in Bulgarian).

[78] Pakniyat, A., Salarieh, H. and Alasty, A., “Stability Analysis of a New Class of MEMS Gyroscopes with Parametric Resonance”, Acta Mech. Vol. 223, No. 6, pp. 1169-1185, 2012.

[79] Pali, J. and Smale, S., “Structure Stability Theorems”, Mathematics, Vol. 13, No. 2, pp. 145-155, 1969.

[80] Poston, T. and Stewart, I., Catastrophe Theory and its Applications, PITMAN, London, 1978.

[81] Rumiantsev, V., “On the Stability of Motion of Gyrostat”, J. Appl. Math. Mech., Vol. 25, pp. 9-19, 1961.

[82] Ruelle, D. and Takens, F., “On the Nature of Turbulence”, Commun. in Math. Phys., Vol. 20, pp. 167-192, 1971.

[83] Seyranian, A., Stoustrup, J. and Kliem, W., “On Gyroscopic Stabilization”, Zangew Math Phys (ZAMP), Vol. 46, pp. 255-267, 1995.

[84] Sheu, L., Chen, H., Chen, J., Tam, L., Chen, W. and et al., Chaos in the Newton-Leipnik System with Fractional Order”, Chaos, Solitons and Fractals, Vol. 36, pp. 98-103, 2008.

[85] Shilnikov, L., “On a New Type of Bifurcation of Multi-dimensional Dynamical Systems”, DAN USSR, Vol. 189, No 1, pp. 59-62, 1969.

[86] Shilnikov, L., On a New Type Bifurcation of Multidimensional Dynamical Systems”, Sov. Math., Vol. 10, pp. 1368-1371, 1969.

[87] Shilnikov, L., “A Contribution to the Problem of the Structure of an Extended Neighborhood of a Rough Equilibrium State of Saddle-focus Type”, Math.USSR Sbornik, Vol. 81(123), pp. 92-103, 1970.

[88] Shilnikov, L., Shilnikov, A., Turaev, D. and Chua, L., Methods of Qualitative Theory in Nonlinear Dynamics. Part II, World Scientific, Singapore, 2001.

[89] Sonechkin, D., Stochasticity in the Model of General Circulation Atmosphere, Gidrometeoizdat, Leningrad, 1984 in Russian.

[90] Stringari, S., “Superfluid Gyroscope with Cold Atomic Gases”, Physical Review Letters, Vol. 86, No. 21, pp. 4725-4728, 2001.

[91] Thomson, W. and Tait, P., Treatise on Natural Philosophy. vol. I, part I, Cambridge University Press, 1879.

[92] Tong, Chr. and Gluhovsky, A., “Gyrostatic Extensions of the Howard-Krishnamutri Model of the Thermal Convection with Shear”, Nonlin. Processes Geophys., Vol. 15, pp. 71-79, 2008.

[93] Tong, Chr., “Lord Kelvin’s Gyrostat and its Analogs in Physics, Including the Lorenz Model”, Am. J. Phys. Vol. 77, No.6, pp. 526-537, 2009.

[94] Tsai, N., Wu, B., “Nonlinear Dynamics and Control for Single-axis Gyroscope Systems”, Nonlinear Dyn., Vol. 51, pp. 355-364, 2008.

[95] Uitni, H., “Peculiarities of Map in Euclidian Space”, Мathematics, Vol. 13, No. 2, pp. 105-123, 1969.

[96] Wang, Ch. and Yau, H., “Nonlinear Dynamic Analysis and Sliding Mode Control for a Gyroscope System”, Nonlinear Dyn., Vol. 66, pp. 53-65, 2011.

[97] Will, C., “Covariant Calculation of General Relativistic Effects in an Orbiting Gyroscope Experiment”, Physical Review D, Vol. 67, pp. 062003, 2003.

[98] Woodman, O.: An Introduction to Internal Navigation. University of Cambridge Press (2007)

[99] Zhuravlev, V., Grounding in Theoretical Mechanics, Fizmatlit, Moscow, 2001 (in Russian).