Regular and Chaotic Dynamics

, Volume 24, Issue 3, pp 329–352 | Cite as

A Parabolic Chaplygin Pendulum and a Paul Trap: Nonintegrability, Stability, and Boundedness

  • Alexey V. BorisovEmail author
  • Alexander A. KilinEmail author
  • Ivan S. MamaevEmail author


This paper is a small review devoted to the dynamics of a point on a paraboloid. Specifically, it is concerned with the motion both under the action of a gravitational field and without it. It is assumed that the paraboloid can rotate about a vertical axis with constant angular velocity. The paper includes both well-known results and a number of new results.

We consider the two most widespread friction (resistance) models: dry (Coulomb) friction and viscous friction. It is shown that the addition of external damping (air drag) can lead to stability of equilibrium at the saddle point and hence to preservation of the region of bounded motion in a neighborhood of the saddle point. Analysis of three-dimensional Poincaré sections shows that limit cycles can arise in this case in the neighborhood of the saddle point.


parabolic pendulum Paul trap rotating paraboloid internal damping external damping friction resistance linear stability Hill’s region bifurcational diagram Poincaré section bounded trajectory chaos integrability nonintegrability separatrix splitting 

MSC2010 numbers

37J25 37J05 


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The authors express their gratitude to V. V. Kozlov and I. A. Bizyaev for fruitful discussions and useful comments.


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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.A. A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  2. 2.Institute of Mathematics and Mechanics of the Ural Branch of RASEkaterinburgRussia
  3. 3.Udmurt State UniversityIzhevskRussia
  4. 4.Moscow Institute of Physics and TechnologyDolgoprudnyiRussia
  5. 5.Center for Technologies in Robotics and Mechatronics ComponentsInnopolis UniversityInnopolisRussia

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