Secrets of the Celtic stone dynamics

A.S. Gonchenko; S.V. Gonchenko; A.O. Kazakov

Secrets of the Celtic stone dynamics

A.S. Gonchenko¹, S.V. Gonchenko², A.O. Kazakov³

research article Pages: 25-32

^1,3 Ph.D. (Phys.-Math.), Junior Research Scientist, UNN, Nizhny Novgorod
² Dr. Sc. (Phys.-Math.), Head of Laboratory, UNN, Nizhny Novgorod

Keywords: Celtic stone nonholonomic model reverse chaos strange attractor Lorenz attractor Shilnikov attractor mixed dynamics

Abstract

This paper presents an overview of recent studies on dynamics of Celtic stone moving on a plane. Its main feature, which is well known for a long time, is the ability to change the direction of rotation to opposite one (reversal phenomenon). In this paper we con-sider the so-called nonholonomic model of Celtic stone that allows us not only to explain this phenomenon from a mathematical point of view but also many other very interesting properties of the Celtic stone dynamics associated, in particular, with the chaotic motion of the stone.

How to cite

Гонченко А.С., Гонченко С.В., Казаков А.О. Секреты динамики кельтского камня // Нелинейный мир. 2015. № 7. С. 25–32.

References

Walker G.T. On a curious dynamical property of celts // Proceedings of the Cambridge Philosophical Society. 1892.№8.R. 305-6.
Walker J. The Amateur Scientist: The mysterious "rattleback": A stone that spins in one direction and then reverses // ScientificAmerican. 1979. V. 241. P. 172-84.
Kane T.R., Levinson D.A. A realistic solution of the symmetric top problem // Journal of Applied Mechanics. 1978. V. 45.№. 4.R. 903−909.
Astapov I.S. Ob ustojjchivosti vrashhenija keltskogo kamnja//Vestnik MGU.Ser.Matematika,mekhanika. 1980.№2.S. 97−100.
Karapetjan A.V. O permanentnykh vrashhenijakh tjazhelogo tverdogo tela na absoljutno sherokhovatojj gorizontalnojj ploskosti PMM. 1981. T. 45. № 5. S. 808−814.
Markeev A.P. O dinamike tverdogo tela na absoljutno sherokhovatojj ploskosti//PMM. 1983.T. 47.№4.S. 575−582.
Borisov A.V., Mamaev I.S. Strannye attraktory v dinamike keltskikh kamnejj//Uspekhi fizicheskikh nauk. 2003.T. 173. № 4. S. 407−418.
Borisov A.V., Mamaev I.S.Strannye attraktory v dinamike keltskikh kamnejj//V kn.: Negolonomnye dinamicheskie sistemy. M.-Izhevsk. 2002. S. 293−316.
Gonchenko A.S., Gonchenko S.V., Kazakov A.O. O nekotorykh novykh aspektakh khaoticheskojj dinamiki«keltskogo kamnja» //Ne-linejjnaja dinamika. 2012. T. 8. № 3. S. 507−518.
Gonchenko A.S., Gonchenko S.V., Kazakov A.O. Richness of chaotic dynamics in nonholonomic models of a Celtic stone // Regularand Chaotic Dynamics. 2013. T. 18. №. 5. S. 521−538.
Borisov A.V., Kazakov A.O., Kuznecov S.P.Nelinejjnaja dinamika keltskogo kamnja:negolonomnaja model//Uspekhi fizicheskikh nauk. 2014. T. 184. №. 5. S. 493−500.
Kuznecov S.P., ZHalnin A.JU., Sataev I.R, Sedova JU.V.Fenomeny nelinejjnojj dinamiki dissipativnykh sistem v negolo-nomnojj mekhanike «keltskogo kamnja» // Nelinejjnaja dinamika. 2012. T. 8. № 4. S. 735−762.
Lamb J.S.W., Stenkin O.V. Newhouse Regions for Reversible Systems with Infinitely Many Stable, Unstable and Elliptic PeriodicOrbits // Nonlinearity. 2004. V. 17. № 4. R. 1217−1244.
Delshams A., Gonchenko S.V., Gonchenko A.S., Lazaro J.T., Stenkin O. Abundance of Attracting, Repelling and Elliptic PeriodicOrbits in Two-Dimensional Reversible Maps // Nonlinearity. 2013. V. 26. № 1. R. 1−33.
Kazakov A.O.Fenomeny khaoticheskojj dinamiki v zadache o kachenii rok-n-rollera bez verchenija//Nelinejjnaja dinamika.2013. T. 9. №. 2. S. 309−325.
Kazakov A.O. On the chaotic dynamics of a rubber ball with three internal rotors // Nonlinear Dynamics & Mobile Robotics. 2014.T. 2. №. 1. S. 73−97.
SHilnikov L.P. Teorija bifurkacijj i turbulentnost//V sb. «Metody kachestvennojj teorii differencialnykh uravnenijj». Gorkijj. 1986. S. 150−163.
Gonchenko A.S., Gonchenko S.V. O sushhestvovanii attraktorov lorencevskogo tipa v negolonomnojj modeli keltskogo kamnja. Nelinejjnaja dinamika. 2013. T. 9. № 1. S. 77−89.
Gonchenko A.S. Ob attraktorakh lorencevskogo tipa v modeli keltskogo kamnja//Vestnik Udmurtskogo universiteta.Ser.Matematika. Mekhanika. Kompjuternye nauki. 2013. №. 2. S. 3−11.

[1] Walker G.T. On a curious dynamical property of celts // Proceedings of the Cambridge Philosophical Society. 1892.№8.R. 305-6.

[2] Walker J. The Amateur Scientist: The mysterious "rattleback": A stone that spins in one direction and then reverses // ScientificAmerican. 1979. V. 241. P. 172-84.

[3] Kane T.R., Levinson D.A. A realistic solution of the symmetric top problem // Journal of Applied Mechanics. 1978. V. 45.№. 4.R. 903−909.

[4] Astapov I.S. Ob ustojjchivosti vrashhenija keltskogo kamnja//Vestnik MGU.Ser.Matematika,mekhanika. 1980.№2.S. 97−100.

[5] Karapetjan A.V. O permanentnykh vrashhenijakh tjazhelogo tverdogo tela na absoljutno sherokhovatojj gorizontalnojj ploskosti PMM. 1981. T. 45. № 5. S. 808−814.

[6] Markeev A.P. O dinamike tverdogo tela na absoljutno sherokhovatojj ploskosti//PMM. 1983.T. 47.№4.S. 575−582.

[7] Borisov A.V., Mamaev I.S. Strannye attraktory v dinamike keltskikh kamnejj//Uspekhi fizicheskikh nauk. 2003.T. 173. № 4. S. 407−418.

[8] Borisov A.V., Mamaev I.S.Strannye attraktory v dinamike keltskikh kamnejj//V kn.: Negolonomnye dinamicheskie sistemy. M.-Izhevsk. 2002. S. 293−316.

[9] Gonchenko A.S., Gonchenko S.V., Kazakov A.O. O nekotorykh novykh aspektakh khaoticheskojj dinamiki«keltskogo kamnja» //Ne-linejjnaja dinamika. 2012. T. 8. № 3. S. 507−518.

[10] Gonchenko A.S., Gonchenko S.V., Kazakov A.O. Richness of chaotic dynamics in nonholonomic models of a Celtic stone // Regularand Chaotic Dynamics. 2013. T. 18. №. 5. S. 521−538.

[11] Borisov A.V., Kazakov A.O., Kuznecov S.P.Nelinejjnaja dinamika keltskogo kamnja:negolonomnaja model//Uspekhi fizicheskikh nauk. 2014. T. 184. №. 5. S. 493−500.

[12] Kuznecov S.P., ZHalnin A.JU., Sataev I.R, Sedova JU.V.Fenomeny nelinejjnojj dinamiki dissipativnykh sistem v negolo-nomnojj mekhanike «keltskogo kamnja» // Nelinejjnaja dinamika. 2012. T. 8. № 4. S. 735−762.

[13] Lamb J.S.W., Stenkin O.V. Newhouse Regions for Reversible Systems with Infinitely Many Stable, Unstable and Elliptic PeriodicOrbits // Nonlinearity. 2004. V. 17. № 4. R. 1217−1244.

[14] Delshams A., Gonchenko S.V., Gonchenko A.S., Lazaro J.T., Stenkin O. Abundance of Attracting, Repelling and Elliptic PeriodicOrbits in Two-Dimensional Reversible Maps // Nonlinearity. 2013. V. 26. № 1. R. 1−33.

[15] Kazakov A.O.Fenomeny khaoticheskojj dinamiki v zadache o kachenii rok-n-rollera bez verchenija//Nelinejjnaja dinamika.2013. T. 9. №. 2. S. 309−325.

[16] Kazakov A.O. On the chaotic dynamics of a rubber ball with three internal rotors // Nonlinear Dynamics & Mobile Robotics. 2014.T. 2. №. 1. S. 73−97.

[17] SHilnikov L.P. Teorija bifurkacijj i turbulentnost//V sb. «Metody kachestvennojj teorii differencialnykh uravnenijj». Gorkijj. 1986. S. 150−163.

[18] Gonchenko A.S., Gonchenko S.V. O sushhestvovanii attraktorov lorencevskogo tipa v negolonomnojj modeli keltskogo kamnja. Nelinejjnaja dinamika. 2013. T. 9. № 1. S. 77−89.

[19] Gonchenko A.S. Ob attraktorakh lorencevskogo tipa v modeli keltskogo kamnja//Vestnik Udmurtskogo universiteta.Ser.Matematika. Mekhanika. Kompjuternye nauki. 2013. №. 2. S. 3−11.

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