×

A new and simple chaos toy. (English) Zbl 1051.70013

Summary: We present two new, and perhaps the simplest yet, mechanical chaos demonstrations. They are both designed based on a recipe of competing nonlinear oscillations. One of these devices is simple enough that using the provided description, it can be built using a bicycle wheel, a piece of wood routed with an elliptical track, and a ball bearing. We provide a thorough Lagrangian mechanics based derivation of equations of motion, and a proof of chaos based on showing the existence of an embedded Smale horseshoe using Melnikov’s method. We conclude with discussion of a future application.

MSC:

70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
70H03 Lagrange’s equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abraham R., Foundations of Mechanics (1985)
[2] Arrowsmith D. K., An Introduction to Dynamical Systems (1990) · Zbl 0702.58002
[3] DOI: 10.1142/S0218127499001516 · Zbl 1089.37510 · doi:10.1142/S0218127499001516
[4] DOI: 10.1142/9789812798640 · doi:10.1142/9789812798640
[5] Goldstein H., Classical Mechanics (1980)
[6] DOI: 10.1007/978-1-4612-1140-2 · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[7] DOI: 10.4324/9780203214589 · doi:10.4324/9780203214589
[8] DOI: 10.1103/RevModPhys.64.795 · Zbl 1160.37302 · doi:10.1103/RevModPhys.64.795
[9] Moon F. C., Chaotic and Fractal Dynamics, An Introduction for Applied Scientists and Engineers (1992)
[10] DOI: 10.1103/PhysRevLett.64.1196 · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196
[11] DOI: 10.1007/BF01587686 · Zbl 0205.55001 · doi:10.1007/BF01587686
[12] DOI: 10.1119/1.16860 · doi:10.1119/1.16860
[13] S. Smale, Differential and Combinatorial Topology, ed. S. S. Cairns (Princeton University Press, Princeton, 1965) pp. 63–80.
[14] Strogatz S., Nonlinear Dynamics and Chaos, With Applications to Physics, Biology, Chemistry, and Engineering (1994)
[15] DOI: 10.1007/978-1-4757-4067-7 · doi:10.1007/978-1-4757-4067-7
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.