Phase control in the mass-spring model with nonsmooth stiffness and external excitation. (English) Zbl 1284.34072
Summary: The control of chaotic dynamics in a nonlinear mass-spring model with nonsmooth stiffness is analyzed here. This is carried out by applying the phase control technique, which uses a periodic perturbation of a suitable phase \(\phi\). For this purpose, we take as prototype model a system consisting of a double-well potential with an additional spring component, which acts into the system only for large enough displacements. The crucial role of the phase is evidenced by using numerical simulations and also by using analytical methods, such as the Melnikov analysis. We expect that our results might be fruitful with implications in some mechanical problems such as suspension of vehicles, among others, where similar models are extensively used.
MSC:
34C60 | Qualitative investigation and simulation of ordinary differential equation models |
34A36 | Discontinuous ordinary differential equations |
70F40 | Problems involving a system of particles with friction |
70E55 | Dynamics of multibody systems |
34H10 | Chaos control for problems involving ordinary differential equations |
Keywords:
phase control; nonlinear oscillations; Melnikov criterion; chaos; nonsmooth dynamical systemsSoftware:
DynamicsReferences:
[1] | DOI: 10.1016/S0167-2789(02)00565-1 · Zbl 1008.37011 · doi:10.1016/S0167-2789(02)00565-1 |
[2] | DOI: 10.1103/PhysRevE.69.016203 · doi:10.1103/PhysRevE.69.016203 |
[3] | DOI: 10.1103/RevModPhys.81.333 · doi:10.1103/RevModPhys.81.333 |
[4] | Almendral J. A., Recent Res. Dev. Sound Vib. 2 pp 115– |
[5] | DOI: 10.1016/S0167-2789(01)00329-3 · Zbl 0994.37009 · doi:10.1016/S0167-2789(01)00329-3 |
[6] | Baltanás J. P., Recent Res. Dev. Sound Vib. 1 pp 29– |
[7] | Burden R. L., Numerical Analysis (1997) |
[8] | Duffing G., Erzwungene Schwingungen bei Veränderlicher Eigenfrequenz (1918) |
[9] | DOI: 10.1007/978-1-4612-1140-2 · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2 |
[10] | DOI: 10.1098/rsta.1979.0068 · Zbl 0423.34049 · doi:10.1098/rsta.1979.0068 |
[11] | DOI: 10.1016/S0375-9601(98)00095-4 · Zbl 0949.37013 · doi:10.1016/S0375-9601(98)00095-4 |
[12] | DOI: 10.1007/978-3-642-56589-2_19 · doi:10.1007/978-3-642-56589-2_19 |
[13] | DOI: 10.1103/PhysRevA.41.726 · doi:10.1103/PhysRevA.41.726 |
[14] | DOI: 10.1016/j.chaos.2005.11.026 · doi:10.1016/j.chaos.2005.11.026 |
[15] | DOI: 10.1016/j.cnsns.2007.01.003 · Zbl 1221.70037 · doi:10.1016/j.cnsns.2007.01.003 |
[16] | DOI: 10.1142/S021812741250006X · Zbl 1270.34129 · doi:10.1142/S021812741250006X |
[17] | DOI: 10.1103/PhysRevE.49.R2528 · doi:10.1103/PhysRevE.49.R2528 |
[18] | DOI: 10.1103/PhysRevLett.55.1439 · doi:10.1103/PhysRevLett.55.1439 |
[19] | Nusse H. C., Dynamics: Numerical Explorations (1997) |
[20] | DOI: 10.1103/PhysRevLett.64.1196 · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196 |
[21] | DOI: 10.1016/j.jsv.2007.04.044 · Zbl 1242.70046 · doi:10.1016/j.jsv.2007.04.044 |
[22] | DOI: 10.1103/PhysRevLett.74.1736 · doi:10.1103/PhysRevLett.74.1736 |
[23] | DOI: 10.1002/3527602585 · doi:10.1002/3527602585 |
[24] | Robinson R. C., Dynamical Systems: Continuous and Discrete (2004) · Zbl 1073.37001 |
[25] | DOI: 10.1103/PhysRevE.78.016205 · doi:10.1103/PhysRevE.78.016205 |
[26] | Sprott J. C., Chaos and Time Series Analysis (2003) · Zbl 1012.37001 |
[27] | DOI: 10.1177/107754630000600706 · doi:10.1177/107754630000600706 |
[28] | DOI: 10.1016/S0020-7462(03)00039-8 · Zbl 1348.74056 · doi:10.1016/S0020-7462(03)00039-8 |
[29] | DOI: 10.1103/PhysRevE.74.016202 · doi:10.1103/PhysRevE.74.016202 |
[30] | DOI: 10.1088/1367-2630/10/7/073030 · doi:10.1088/1367-2630/10/7/073030 |
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.