×

zbMATH — the first resource for mathematics

Fermi-like acceleration and power-law energy growth in nonholonomic systems. (English) Zbl 1418.37106

MSC:
37J60 Nonholonomic dynamical systems
34A34 Nonlinear ordinary differential equations and systems
70F25 Nonholonomic systems related to the dynamics of a system of particles
Software:
Equator
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Bizyaev I A, Borisov A V and Kuznetsov S P 2017 Chaplygin sleigh with periodically oscillating internal mass Europhys. Lett.119 60008
[2] Bizyaev I A, Borisov A V and Mamaev I S 2016 Dynamics of the Chaplygin sleigh on a cylinder Regul. Chaotic Dyn.21 136-46 · Zbl 1346.70006
[3] Bizyaev I A, Borisov A V and Mamaev I S 2017 The Chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration Regul. Chaotic Dyn.22 955-75 · Zbl 1398.37056
[4] Bizyaev I A, Borisov A V and Mamaev I S 2018 Exotic dynamics of nonholonomic roller racer with periodic control Regul. Chaotic Dyn.23 983-94 · Zbl 1412.37064
[5] Bloch A M, Marsden J E and Zenkov D V 2009 Quasivelocities and symmetries in non-holonomic systems Dynam. Syst.24 187-222 · Zbl 1231.37034
[6] Bloch A M 2003 Nonholonomic Mechanics and Control (New York: Springer) · Zbl 1045.70001
[7] Borisov A V, Jalnine A Y, Kuznetsov S P, Sataev I R and Sedova Y V 2012 Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback Regul. Chaotic Dyn.17 512-32 · Zbl 1263.74021
[8] Borisov A V, Kazakov A O and Sataev I R 2014 The reversal and chaotic attractor in the nonholonomic model of Chaplygin’s top Regul. Chaotic Dyn.19 718-33 · Zbl 1358.70006
[9] Borisov A V, Kilin A A and Mamaev I S 2015 On the Hadamard-Hamel problem and the dynamics of wheeled vehicles Regul. Chaotic Dyn.20 752-66 · Zbl 1367.70034
[10] Borisov A V, Kilin A A and Mamaev I S 2012 How to control Chaplygin’s sphere using rotors Regul. Chaotic Dyn.17 258-72 · Zbl 1264.37016
[11] Borisov A V, Kilin A A and Mamaev I S 2013 How to control the Chaplygin ball using rotors. II Regul. Chaotic Dyn.18 144-58 · Zbl 1303.37021
[12] Borisov A V and Kuznetsov S P 2016 Regular and chaotic motions of a Chaplygin sleigh under periodic pulsed torque impacts Regul. Chaotic Dyn.21 792-803 · Zbl 1368.37075
[13] Borisov A V and Mamaev I S 2009 The dynamics of a Chaplygin sleigh J. Appl. Math. Mech.73 156-61
[14] Borisov A V and Mamaev I S 2003 Strangeattractors in rattleback dynamics Phys.—Usp.46 393-403
[15] Borisov A V, Mamaev I S and Bizyaev I A 2015 The Jacobi integral in nonholonomic mechanics Regul. Chaotic Dyn.20 383-400 · Zbl 1367.70036
[16] Brahic H 1971 Numerical study of a simple dynamical system. I. The associated plane area-preserving mapping Astron. Astrophys.12 98-110 · Zbl 0239.70001
[17] Carathéodory C 1933 Der Schlitten Z. Angew. Math. Mech.13 71-6 · Zbl 0006.37301
[18] Chaplygin S A 2008 On the theory of motion of nonholonomic systems. The reducing-multiplier theorem Regul. Chaotic Dyn.13 369-76 · Zbl 1229.37082
[19] see also: Chaplygin S A 1912 On the theory of motion of nonholonomic systems. The reducing-multiplier theorem Mat. Sb.28 303-14
[20] De Simoi J and Dolgopyat D 2012 Dynamics of some piecewise smooth Fermi-Ulam models Chaos22 026124 · Zbl 1331.37133
[21] Demidovich B P 1956 On the existence of a limiting regime of a certain nonlinear system of ordinary differential equations Uchen. Zap. Moskov. Gos. Univ.181 3-12 (In Russian)
[22] Fedonyuk V and Tallapragada P 2018 Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh Nonlinear Dyn.93 835-46 · Zbl 1398.34045
[23] Fermi E 1949 On the origin of the cosmic radiation Phys. Rev.75 1169-74 · Zbl 0032.09604
[24] Gonchenko A S, Gonchenko S V and Kazakov A O 2013 Richness of chaotic dynamics in nonholonomic models of a celtic stone Regul. Chaotic Dyn.18 521-38 · Zbl 1417.37222
[25] Guckenheimer J and Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (New York: Springer) · Zbl 0515.34001
[26] Hadamard J 1897 Sur certaines proprietes des trajectoires en dynamique J. Math. Ser. 5 3 331-87 · JFM 28.0643.01
[27] Hammersley J M 1961 Proc. 4th Berkeley Symp. on Mathematical Statistics and Probability(University of California Press,)
[28] Ito A 1979 Successive subharmonic bifurcations and chaos in a nonlinear Mathieu equation Prog. Theor. Phys.61 815-24 · Zbl 1171.34315
[29] Izrailev F M, Rabinovich M I and Ugodnikov A D 1981 Approximate description of three-dimensional dissipative systems with stochastic behaviour Phys. Lett. A 86 321-5
[30] Jung P, Marchegiani G and Marchesoni F 2016 Nonholonomic diffusion of a stochastic sled Phys. Rev. E 93 012606
[31] Karlis A K, Papachristou P K, Diakonos F K, Constantoudis V and Schmelcher P 2006 Hyperacceleration in a stochastic Fermi-Ulam model Phys. Rev. Lett.97 194102
[32] Kilin A A, Pivovarova E N and Ivanova T B 2015 Spherical robot of combined type: dynamics and control Regul. Chaotic Dyn.20 716-28 · Zbl 1339.93081
[33] Kozlov V V 2004 On a Uniform Distribution on a Torus Moscow Univ. Math. Bull.59 23-31 · Zbl 1076.37500
[34] see also: Kozlov V V 2004 Vestn. Mosk. Univ. Ser. 1 Mat. Mekh.22-9 78
[35] Kozlov V V 2015 The dynamics of systems with servoconstraints. I Regul. Chaotic Dyn.20 205-24 · Zbl 1353.70036
[36] Kozlov V V 2015 The dynamics of systems with servoconstraints. II Regul. Chaotic Dyn.20 401-27 · Zbl 1353.70012
[37] Krishnaprasad P S and Tsakiris D P 2001 Oscillations, SE(2)-snakes and motion control: a study of the roller – racer Dyn. Syst.16 347-97 · Zbl 1127.70316
[38] Leonard N E 1995 Periodic forcing, dynamics and control of underactuated spacecraft and underwater vehicles Proc. 34th IEEE Conf. on Decision and Control(December,) pp 3980-5
[39] Lichtenberg A J and Lieberman M A 1983 Regular and Stochastic Morion (New York: Springer)
[40] Lichtenberg A J, Lieberman M A and Cohen R H 1980 Fermi acceleration revisited Phys. D: Nonlinear Phenom.1 291-305 · Zbl 1194.37092
[41] Lieberman M A and Lichtenberg A J 1972 Stochastic and adiabatic behavior of particles accelerated by periodic forces Phys. Rev. A 5 1852-66
[42] Murray R M and Sastry S S 1993 Nonholonomic motion planning: steering using sinusoids IEEE Trans. Autom. Control38 700-16 · Zbl 0800.93840
[43] Neimark Ju I and Fufaev N A 1972 Dynamics of Nonholonomic Systems (Providence, RI: American Mathematical Society) · Zbl 0245.70011
[44] Oldham K B, Myland J and Spanier J 2010 An Atlas of Functions: with Equator, the Atlas Function Calculator (New York: Springer)
[45] Osborne J M and Zenkov D V 2005 Steering the Chaplygin sleigh by a moving mass 44th IEEE Conference on. IEEE Decision and Control, 2005 and 2005 European Control Conf. pp 1114-8
[46] Pustyl’nikov L D 1977 Stable and oscillating motions in nonautonomous dynamical systems: 2 Trudy MMO34 3-103 (In Russian) · Zbl 0412.58024
[47] Pustyl’nikov L D 1988 A new mechanism for particle acceleration and a relativistic analogue of the Fermi-Ulam model Theor. Math. Phys.77 1110-5
[48] see also: Pustyl’nikov L D 1988 Teoret. Mat. Fiz.77 154-60
[49] Sprott J C 2010 Elegant Chaos: Algebraically Simple Chaotic Flows (Singapore: World Scientific)
[50] Ulam S M 1961 On some statistical properties of dynamical systems Proc. 4th Berkeley Symp. on Mathematical Statistics and Probabilityvol 3 pp 315-20 · Zbl 0106.43102
[51] Weyl H 1938 Mean motion Am. J. Math.60 889-96 · JFM 64.0256.02
[52] Zaslavsky G M 1970 Statistical Irreversibility in Nonlinear Systems (Moscow: Nauka) (In Russian)
[53] Zaslavskii G M and Chirikov B V 1972 Stochastic instability of nonlinear oscillations Phys.—Usp.14 549-68 · Zbl 1156.34335
[54] Zehnder E 2010 Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities (Zurich: European Mathematical Society) · Zbl 1195.37001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.