Estimation of chaotic thresholds for the recently proposed rotating pendulum. (English) Zbl 1270.70063
Summary: We investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.
MSC:
| 70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |
| 70K50 | Bifurcations and instability for nonlinear problems in mechanics |
| 70K44 | Homoclinic and heteroclinic trajectories for nonlinear problems in mechanics |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
Keywords:
rotating pendulum; SD oscillator; irrational nonlinearity; chaotic thresholds; singular closed orbitsReferences:
| [1] | DOI: 10.1142/S0218127499000341 · Zbl 0947.70015 · doi:10.1142/S0218127499000341 |
| [2] | Bayly P. V., Phil. Trans. R. Soc. A 443 pp 391– (1993) |
| [3] | DOI: 10.1016/0167-2789(93)90263-Z · Zbl 0778.70020 · doi:10.1016/0167-2789(93)90263-Z |
| [4] | Cao Q. J., Phys. Rev. E 74 (2006) |
| [5] | DOI: 10.1098/rsta.2007.2115 · Zbl 1153.34329 · doi:10.1098/rsta.2007.2115 |
| [6] | DOI: 10.1016/j.ijnonlinmec.2008.01.003 · doi:10.1016/j.ijnonlinmec.2008.01.003 |
| [7] | Cao Q. J., Chin. Phys. Lett. 28 (2011) |
| [8] | DOI: 10.1007/s11071-006-9059-3 · Zbl 1180.70045 · doi:10.1007/s11071-006-9059-3 |
| [9] | DOI: 10.1007/BF01314257 · doi:10.1007/BF01314257 |
| [10] | DOI: 10.1007/s00332-002-0514-0 · Zbl 1021.37039 · doi:10.1007/s00332-002-0514-0 |
| [11] | DOI: 10.1016/j.jappmathmech.2006.03.010 · doi:10.1016/j.jappmathmech.2006.03.010 |
| [12] | DOI: 10.1023/A:1012990329884 · Zbl 1011.70021 · doi:10.1023/A:1012990329884 |
| [13] | DOI: 10.1103/PhysRevLett.50.346 · doi:10.1103/PhysRevLett.50.346 |
| [14] | DOI: 10.1007/s11433-012-4880-9 · doi:10.1007/s11433-012-4880-9 |
| [15] | DOI: 10.1016/j.phpro.2011.06.023 · doi:10.1016/j.phpro.2011.06.023 |
| [16] | DOI: 10.1016/0005-1098(78)90036-5 · Zbl 0385.93028 · doi:10.1016/0005-1098(78)90036-5 |
| [17] | Hopf E., S”ch. Akad.Wiss. Leipz. Math.-Nat. 94 pp 3– (1942) |
| [18] | DOI: 10.1023/A:1008356204490 · Zbl 0963.70013 · doi:10.1023/A:1008356204490 |
| [19] | DOI: 10.1007/s11370-008-0024-5 · doi:10.1007/s11370-008-0024-5 |
| [20] | DOI: 10.1063/1.166077 · doi:10.1063/1.166077 |
| [21] | DOI: 10.1007/s10846-005-9022-4 · doi:10.1007/s10846-005-9022-4 |
| [22] | DOI: 10.1016/0020-7462(95)00068-2 · Zbl 0865.70018 · doi:10.1016/0020-7462(95)00068-2 |
| [23] | DOI: 10.1137/0511048 · Zbl 0459.33001 · doi:10.1137/0511048 |
| [24] | DOI: 10.1023/A:1008256920441 · Zbl 0907.70015 · doi:10.1023/A:1008256920441 |
| [25] | DOI: 10.1007/s11071-005-9001-0 · Zbl 1170.70322 · doi:10.1007/s11071-005-9001-0 |
| [26] | DOI: 10.1142/S0218127407017756 · Zbl 1185.37048 · doi:10.1142/S0218127407017756 |
| [27] | DOI: 10.1007/s10778-007-0068-9 · doi:10.1007/s10778-007-0068-9 |
| [28] | DOI: 10.1023/A:1012299219996 · doi:10.1023/A:1012299219996 |
| [29] | DOI: 10.1016/j.scient.2012.02.014 · doi:10.1016/j.scient.2012.02.014 |
| [30] | Melnikov V. K., Trans. Moscow. Math. 12 pp 1– (1963) |
| [31] | Miles W. J., Q. Appl. Math. 20 pp 21– (1962) · Zbl 0108.18601 · doi:10.1090/qam/133521 |
| [32] | DOI: 10.1016/0167-2789(84)90013-7 · Zbl 0577.70025 · doi:10.1016/0167-2789(84)90013-7 |
| [33] | DOI: 10.1016/j.fusengdes.2008.08.036 · doi:10.1016/j.fusengdes.2008.08.036 |
| [34] | Narkis Y., J. Appl. Math. Phys. 28 pp 211– (1997) |
| [35] | DOI: 10.1007/BF01314256 · doi:10.1007/BF01314256 |
| [36] | DOI: 10.1142/S0218127400001365 · Zbl 0965.70036 · doi:10.1142/S0218127400001365 |
| [37] | DOI: 10.1142/S0218127402004231 · Zbl 1051.70016 · doi:10.1142/S0218127402004231 |
| [38] | Tian R. L., Chin. Phys. Lett. 27 (2010) |
| [39] | DOI: 10.1142/S0218127412501088 · Zbl 1258.34109 · doi:10.1142/S0218127412501088 |
| [40] | DOI: 10.1088/0143-0807/7/3/003 · doi:10.1088/0143-0807/7/3/003 |
| [41] | Wan S. D., Appl. Math. Mech. 19 pp 541– (1988) |
| [42] | Whitaker R. J., Educat. 13 pp 401– (2004) |
| [43] | DOI: 10.1142/S0218127497001904 · Zbl 0925.93346 · doi:10.1142/S0218127497001904 |
| [44] | DOI: 10.1016/j.cnsns.2004.06.004 · Zbl 1078.34019 · doi:10.1016/j.cnsns.2004.06.004 |
| [45] | DOI: 10.1023/A:1017512317443 · Zbl 1050.70014 · doi:10.1023/A:1017512317443 |
| [46] | DOI: 10.1016/0375-9601(92)90716-Y · doi:10.1016/0375-9601(92)90716-Y |
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