Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

Regular and Chaotic Dynamics

Volumn 17, Issue 6, 2012, Pages 512-532

  Borisov, Alexey V ^a   Jalnine, Alexey Yu ^a,b   Kuznetsov, Sergey P ^a,b   Sataev, Igor R ^b   Sedova, Julia V ^a,b  

^a UDMURT STATE UNIVERSITY   (Russian Federation)

^b INSTITUTE OF RADIO ENGINEERING AND ELECTRONICS   (Russian Federation)

Author keywords

bifurcation; fractal dimension; Lyapunov exponents; nonholonomic mechanics; rattleback; rigid body dynamics; strange attractor

Indexed keywords

EID: 84880597616     PISSN: 15603547     EISSN: 14684845     Source Type: Journal    
DOI: 10.1134/S1560354712060044     Document Type: Article

References (33)

1. Arnol'd, V. I., Mathematical Methods of Classical Mechanics, 2nd ed., Grad. Texts in Math., vol. 60, New York: Springer, 1989.
2. Abraham, R. and Marsden, J. E., Foundations of Mechanics, Providence, R. I.: AMS, Chelsea, 2008.
3. Borisov, A. V. and Mamaev, I. S., The Rolling Motion of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177-200.
4. Nonholonomic Dynamical Systems: Integrability, Chaos, Strange Attractors, A. V. Borisov, I. S. Mamaev (Eds.), Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2002 (Russian).
5. Borisov, A. V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 408-418 [Physics-Uspekhi, 2003, vol. 46, no. 4, pp. 393-403].
6. Borisov, A. V., Mamaev, I. S., and Kilin, A. A., Selected Problems of Nonholonomic Mechanics, preprint, Izhevsk: Institute of Computer Science, 2005 (Russian).
7. Walker, J., The Mysterious "Rattleback": A Stone That Spins in One Direction and Then Reverses, Sci. Amer., 1979, vol. 241, no. 4, pp. 144-149.
8. Gonchenko, A. S., Gonchenko, S. V., and Shilnikov, L. P., Towards Scenarios of Chaos Appearance in Three-Dimensional Maps, Nelin. Dinam., 2012, vol. 8, no. 1, pp. 3-28 (Russian).
9. Gonchenko, A. S., Gonchenko, S. V., and Kazakov, A. O., On Some Novel Aspects of Chaotic Dynamics of the Rattleback, Nelin. Dinam., 2012, vol. 8, no. 3, pp. 507-518 (Russian).
10. Rabinovich, M. I. and Trubetskov, D. I., Oscillations and Waves: In Linear and Nonlinear Systems, Dordrecht: Kluwer, 1989.
11. Lichtenberg, A. J. and Lieberman, M. A., Regular and Chaotic Dynamics, 2nd ed., Appl. Math. Sci., vol. 38, New York: Springer, 1992.
12. Schuster, H. G. and Just, W., Deterministic Chaos: An Introduction, 4th ed., Weinheim: Wiley-VCH, 2005.
13. Kuznetsov, S. P., Dynamical Chaos, Moscow: Fizmatlit, 2001 (Russian).
14. Kuznetsov, A. P., Kuznetsov, S. P., Sataev, I. R., and Chua, L. O., Multi-Parameter Criticality in Chua's Circuit at Period-Doubling Transition to Chaos, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1996, vol. 6, no. 1, pp. 119-148.
15. Kuznetsov, A. P., Sataev, I. R., and Turukina, L. V., Synchronization and Multi-Frequency Oscillations in the Chain of Phase Oscillators, Nelin. Dinam., 2010, vol. 6, no. 4, pp. 693-717 (Russian).
16. Kuznetsov, A. P., Sataev, I. R., and Turukina, L. V., On the Road Towards Multidimensional Tori, Commun. Nonlinear Sci. Numer. Simul., 2011, vol. 16, pp. 2371-2376.
17. Broer, H., Simó, C., and Vitolo, R., Bifurcations and Strange Attractors in the Lorenz-84 Climate Model with Seasonal Forcing, Nonlinearity, 2002, vol. 15, pp. 1205-1267.
18. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, V. T., Numerical Recipes in C: The Art of Scientific Computing, Cambridge: Cambridge Univ. Press, 1992.
19. Hénon, M., On the Numerical Computation of Poincaré Maps, Phys. D, 1982, vol. 5, nos. 2-3, pp. 412-414.
20. Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., Lyapunov Characteristic Exponents for Smooth Dynamical Systems and for Hamiltonian Systems: A Method for Computing All of Them: P. 1: Theory; P. 2: Numerical Application, Meccanica, 1980, vol. 15, pp. 9-30.
21. Cochran, W. G., Errors of Measurement in Statistics, Technometrics, 1968, vol. 10, no. 4, pp. 637-666.
22. Gallas, J. A. C., Dissecting Shrimps: Results for Some One-Dimensional Physical Systems, Phys. A, 1994, vol. 202, pp. 196-223.
23. Kuznetsov, S. P., Hyperbolic Chaos: A Physicist's View, Berlin: Springer, 2012.
24. Kaplan, J. L. and Yorke, J. A., A Chaotic Behavior of Multi-Dimensional Differential Equations, in Functional Differential Equations and Approximations of Fixed Points, H. O. Peitgen and H. O. Walther (Eds.), Lecture Notes in Math., vol. 730, Berlin: Springer, 1979, pp. 204-227.
25. Feigenbaum, M. J., The Universal Metric Properties of Nonlinear Transformations, J. Stat. Phys., 1979, vol. 21, pp. 669-706.
26. Kuznetsov, S. P., Kuznetsov, A. P., and Sataev, I. R., Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps, J. Stat. Phys., 2005, vol. 121, nos. 5-6, pp. 697-748.
27. Reick, C., Universal Corrections to Parameter Scaling in Period-Doubling Systems: Multiple Scaling and Crossover, Phys. Rev. A, 1992, vol. 45, pp. 777-792.
28. Reichl, L. E., The Transition to Chaos: Conservative Classical Systems and Quantum Manifestations, 2nd ed., New York: Springer, 2004.
29. Jakobson, M. V., Absolutely Continuous Invariant Measures for One-Parameter Families of One-Dimensional Maps, Comm. Math. Phys., 1981, vol. 81, no. 1, pp. 39-88.
30. Benediks, M. and Carleson, L., Dynamics of the Hénon Map, Ann. Math., 1991, vol. 133, pp. 73-169.
31. Jenkins, G. M. and Watts, D. G., Spectral Analysis and Its Application, San Francisco: Holden-Day, 1968.
32. Grassberger, P., Generalized Dimensions of Strange Attractors, Phys. Lett. A, 1983, vol. 97, pp. 227-230.
33. Grassberger, P. and Procaccia, I., Measuring the Strangeness of Strange Attractors, Phys. D, 1983, vol. 9, pp. 189-208.