Summary
We study the Melnikov criterion for a global homoclinic bifurcation and possible transition to chaos for a single degree of freedom nonlinear oscillator. This provides a systematic method of treatment for an arbitrary potential expressed as a fourth order polynomial. The equation of motion has external excitation and a Duffing type nonlinearity with one or two unsymmetric potential wells.
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References
Y. Ueda (1979) ArticleTitleRandomly transitional phenomena in the system governed by Duffing's equation J. Stat. Phys. 20 181–196 Occurrence Handle523641 Occurrence Handle10.1007/BF01011512
F. C. Moon P. J. Holmes (1979) ArticleTitleA magetoelastic strange attractor J. Sound Vibr. 65 275–296 Occurrence Handle10.1016/0022-460X(79)90520-0 Occurrence Handle0405.73082
W. Szemplińska-Stupnicka J. Rudowski (1993) ArticleTitleSteady state in twin-well potential oscillator: computer simulations and approximate analytical studies Chaos 3 173–186
W. Szemplińska-Stupnicka (1995) ArticleTitleThe analytical predictive criteria for chaos and escape in nonlinear oscillators: a survey Nonlinear Dynamics 7 129–147 Occurrence Handle1372774 Occurrence Handle10.1007/BF00053705
S. Lenci G. Rega (2004) ArticleTitleA unified control framework on the non-regular dynamics of mechanical oscillators J. Sound Vibr. 278 1051–1080 Occurrence Handle2101895 Occurrence Handle10.1016/j.jsv.2003.12.010
E. Tyrkiel (2005) ArticleTitleOn the role of chaotic saddles in generating chaotic dynamics in nonlinear driven oscillators Int. J. Bifurcation and Chaos 15 1215–1238 Occurrence Handle1089.37030 Occurrence Handle2152071 Occurrence Handle10.1142/S0218127405012727
J. J. Thompsen (2003) Vibrations and stability Springer Berlin
F. C. Moon (1980) ArticleTitleExperiments on chaotic motions of a forced nonlinear oscillator: strange attractors ASME J. Appl. Mech. 47 638–644 Occurrence Handle10.1115/1.3153746
T. Kapitaniak (1993) ArticleTitleAnalytical method of controling chaos in Duffing oscillator J. Sound Vibr. 163 182–187 Occurrence Handle0958.70504 Occurrence Handle1221218 Occurrence Handle10.1006/jsvi.1993.1158
F. C. Moon (1987) Chaotic vibrations Wiley New York Occurrence Handle0745.58003
V. K. Melnikov (1963) ArticleTitleOn the stability of the center for time periodic perturbations Trans. Moscow Math. Soc. 12 1–57 Occurrence Handle0135.31001
J. Guckenheimer P. Holmes (1983) Nonlinear oscillations, dynamical systems and bifurcations of vectorfields Springer New York
S. Wiggins (1990) Introduction to applied nonlinear dynamical systems and chaos Spinger New York Occurrence Handle0701.58001
A. Steindl H. Troger (1991) Chaotic motion in mechanical and engineering systems W. Szemplińska-Stupnicka H. Troger (Eds) Engineering applications of dynamics of chaos Springer Wien 150–223
S. Smale (1967) ArticleTitleDifferentiable dynamical systems Bull. Amer. Math. Soc 73 747 Occurrence Handle0202.55202 Occurrence Handle228014 Occurrence Handle10.1090/S0002-9904-1967-11798-1
J. M. T. Thompson (1989) ArticleTitleChaotic phenomena triggering the escape from a potential well Proc. Roy. Soc. London A 421 195–225 Occurrence Handle0674.70035 Occurrence Handle10.1098/rspa.1989.0009
B. Bruhn B.-P. Koch G. Schmidt (1994) ArticleTitleOn the onset of chaotic dymanics in asymmetric oscillators Z. Angew. Math. Mech. 74 325–331 Occurrence Handle1293717 Occurrence Handle0812.70015
Litak, G., Borowiec, M., Syta, A., Szabelski, K.: Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing. Submitted to International Journal of Non-linear Mechanics (2005).
Y. A. Brychkov O. I. Marichev A. P. Prudnikov (1989) Tables of indefinite integrals Gordon and Breach Amsterdam Occurrence Handle0728.26001
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Litak, G., Borowiec, M. Oscillators with asymmetric single and double well potentials: transition to chaos revisited. Acta Mechanica 184, 47–59 (2006). https://doi.org/10.1007/s00707-006-0340-9
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DOI: https://doi.org/10.1007/s00707-006-0340-9