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 WJM  Vol.2 No.2 , April 2012
Oscillator with Random Mass
Abstract: We consider an oscillator with a random mass for which the particles of the surrounding medium adhere to the oscillator for some random time after the collision (Brownian motion with adhesion). This is another form of a stochastic oscillator, different from oscillator usually studied that is subject to a random force or having random frequency or random damping. We calculated first two moments for different form of a random force, and studied different resonance phenomena (stochastic resonance, vibration resonance and “erratic” behavior) interposed between order and chaos.
Cite this paper: nullM. Gitterman, "Oscillator with Random Mass," World Journal of Mechanics, Vol. 2 No. 2, 2012, pp. 113-124. doi: 10.4236/wjm.2012.22013.
References

[1]   M. Gitterman, “The Noisy Oscillator: The First Hundred Years, from Einstein until Now,” World Scientific, Singapore, 2005.

[2]   R. Mazo, “Brownian Motion: Fluctuations, Dynamics, and Applications,” Oxford Science Publication, Oxford, 2002.

[3]   A. Ishimaru, “Wave Propagation and Scattering in Random Media,” John Wiley and Sons, New York, 1999.

[4]   R. Kubo “A Stochastic Theory in Line Shape,” In: K. E. Shuler, Ed., Stochastic Processes in Chemical Physics, Wiley, New York, 1969, pp. 101-128.

[5]   O. M. Phillips, “The Dynamics of the Upper Ocean,” Cambridge University Press, Cambridge, 1977.

[6]   M. Turelli, “Theoretical Population Biology,” Academic, New York, 1977.

[7]   H. Takayasu, A.-H. Sato and M. Takayasu, “Stable Infinite Variance Fluctuations in Randomly Amplified Lan- gevin Systems,” Physical Review Letters, Vol. 79, No. 6, 1997, pp. 966-969. doi:10.1103/PhysRevLett.79.966

[8]   B. West and V. Seshadri, “Model of Gravity Wave Growth Due to Fluctuations in the Air-Sea Coupling Pa- rameter,” Journal of Geophysical Research, Vol. 86, No. 5, 1981, pp. 4293-4296. doi:10.1029/JC086iC05p04293

[9]   M. Gitterman, “Phase Transitions in Moving Systems,” Physical Review E, Vol. 70, No. 3, 2003, Article ID: 036116. doi:10.1103/PhysRevE.70.036116

[10]   A. Onuki, “Phase Transitions of Fluids in Shear Flow,” Journal of Physics: Condensed Matter, Vol. 9, No. 29, 1997, pp. 6119-6159. doi:10.1088/0953-8984/9/29/001

[11]   J. M. Chomaz and A. Couairon, “Against the Wind,” Physics of Fluids, Vol. 11, No. 10, 1999, pp. 2977-2984. doi:10.1063/1.870157

[12]   F. Hestol and A. Libchaber, “Unidirectional Crystal Growth and Crystal Anisotropy,” Physica Scripta, Vol. 1985, No. 9, 1985, pp. 126-129.

[13]   A. Saul and K. Showalter, “Propagating Reaction Diffu- sion Fronts,” In: R. J. Field and M. Burger, Eds., Oscillations and Traveling Waves in Chemical Systems, Wiley, New York, 1985, pp. 419-439.

[14]   M. Gitterrman, B. Ya. Shapiro and I. Shapiro, “Phase Transitions in Vortex Matter Driven by Bias Current,” Physical Review B, Vol. 65, No. 17, 2002, Article ID: 174510. doi:10.1103/PhysRevB.65.174510

[15]   M. Gitterman, “New Stochastic Equation for a Harmonic Oscillator: Brownian Motion with Adhesion,” Journal of Physics C: Conference Series, Vol. 248, No. 1, 2010, Article ID: 012049.

[16]   M. Gitterman “New Type of Brownian Motion,” Journal of Statistical Physics, Vol. 146, No. 1, 2010, pp. 239-243.

[17]   M. Gitterman and V. I. Kljatskin, “Brownian Motion with Adhesion: Harmonic Oscillator with Fluctuating Mass,” Physical Review E, Vol. 81, No. 5, 2010, Article ID: 051139.

[18]   M. Gitterman and I. Shapiro, “Stochastic Resonance in a Harmonic Oscillator with Random Mass Subject to Asymmetric Dichotomous Noise” Journal of Statistical Physics, Vol. 144, No. 1, 2011, pp. 139-149.

[19]   M. Gitterman “Harmonic Oscillator with Fluctuating Mass,” Journal of Modern Physics, Vol. 2, 2010, pp. 1136-1140.

[20]   J. Portman, M. Khasin, S. W. Shaw and M. I. Dykman, “The Spectrum of an Oscillator with Fluctuating Mass and Nanomechanical Mass Sensing,” Bulletin of the American Physical Society March Meeting, Portland, 15-19 March 2010.

[21]   J. Luczka, P. Hanggi and A. Gadomski, “Diffusion of Clusters with Randomly Growing Masses,” Physical Review E, Vol. 51, No. 6, 1995, pp. 5762-5769. doi:10.1103/PhysRevE.51.5762

[22]   M. S. Abdalla, “Time-Dependent Harmonic Oscillator with Variable Mass under the Action of a Driving Force,” Physical Review A, Vol. 34, No. 6, 1986, pp. 4598-4605. doi:10.1103/PhysRevA.34.4598

[23]   R. Lambiotte and M. Ausloos, “Brownian Particle Having a Fluctuating Mass,” Physical Review E, Vol. 73, No. 1, 2005, Article ID: 011105.

[24]   A. Gadomski and J. Siódmiak, “A Kinetic Model of Protein Crystal Growth in Mass Convection Regime” Crystal Research and Technology, Vol. 37, No. 2-3, 2002, pp. 281-291. doi:10.1002/1521-4079(200202)37:2/3<281::AID-CRAT281>3.3.CO;2-4

[25]   M. Rub and A. Gadomski, “Nonequilibrium Thermodynamics versus Model Grain Growth: Derivation and Some Physical Implication,” Physica A, Vol. 326, No. 3-4, 2003, pp. 333-343. doi:10.1016/S0378-4371(03)00282-6

[26]   A. Gadomski, J. Siódmiak, I. Santamara-Holek, J. M. Rub and M. Ausloos, “Kinetics of Growth Process Controlled by Mass-Convective Fluctuations and Finite-Size Curvature Effects,” Acta Physica Polonica B, Vol. 36, No. 5, 2005, pp. 1537-1559.

[27]   A. T. Pérez, D. Saville and C. Soria, “Modeling the Electrophoretic Deposition of Colloidal Particles,” Europhysics Letters, Vol. 55, No. 3, 2001, pp. 425-431. doi:10.1209/epl/i2001-00431-5

[28]   I. Goldhirsch and G. Zanetti, “Clustering Instability in Dissipative Gases,” Physical Review Letters, Vol. 70, No. 11, 1993, pp. 1619-1622. doi:10.1103/PhysRevLett.70.1619

[29]   S. Luding and H. J. Herrmann, “Cluster-Growth in Freely Cooling Granular Media,” Chaos, Vol. 9, No. 3, 1999, pp. 673-682. doi:10.1063/1.166441

[30]   I. Temizer, “A Computational Model for Aggregation in a Class of Granual Materials,” Master Thesis, University of California, Berkeley, 2001.

[31]   W. Benz, “From Dust to Planets,” Spatium, Vol. 6, 2000, pp. 3-15.

[32]   J. Blum, et al., “Growth and Form of Planetary Seedlings: Results from a Microgravity Aggregation Experiment,” Physical Review Letters, Vol. 85, No. 12, 2000, pp. 2426- 2429.

[33]   J. Blum and G. Wurm, “Experiments on Sticking, Restructuring and Fragmentation of Preplanetary Dust Aggregates,” Icarus, Vol. 143, No. 1, 2000, pp. 138-146. doi:10.1006/icar.1999.6234

[34]   S. J. Weidenschilling, D. Spaute, D. R. Davis, F. Marzari and K. Ohtsuki, “Accretional Evolution of a Planetesimal Swarm,” Icarus, Vol. 128, No. 2, 1997, pp. 429-455. doi:10.1006/icar.1997.5747

[35]   N. Kaiser, “Review of the Fundamentals of Thin-Film Growth,” Applied Optics, Vol. 41, No. 16, 2002, pp. 3053-3060. doi:10.1364/AO.41.003053

[36]   T. Nagatani, “Kinetics of Clustering and Acceleration in 1D Traffic Flow,” Journal of Physical Society Japan, Vol. 65, 1996, pp. 3386-3389. doi:10.1143/JPSJ.65.3386

[37]   E. Ben-Naim, P. L. Krapivsky and S. Redner, “Kinetics of Clustering in Traffic Flows,” Physical Review E, Vol. 50, No. 2, 1994, pp. 822-829. doi:10.1103/PhysRevE.50.822

[38]   M. Ausloos and K. Ivanova, “Mechanistic Approach to Generalized Technical Analysis of Share Prices and Stock Market Indices,” European Journal of Physics B, Vol. 27, No. 4, 2002, pp. 177-187. doi:10.1007/s10051-002-9018-9

[39]   M. Ausloos and K. Ivanova, “Generalized Technical Analysis. Effects of Transaction Volume and Risk,” In: H. Takayasu, Ed., The Applications of Econophysics, Sprin- ger Verlag, Berlin, 2004, pp. 117-124.

[40]   V. E. Shapiro and V. M. Loginov, “Formulae of Differentiation” and Their Use for Solving Stochastic Equations,” Physica A, Vol. 91, 1978, pp. 563-574.

[41]   M. Gitterman, “Classical Harmonic Oscillator with Mul- tiplicative Noise,” Physica A, Vol. 352, No. 2-4, 2005, pp. 309-334. doi:10.1016/j.physa.2005.01.008

[42]   L. Zhang, S. C. Zhong, H. Peng and M. K. Luo, “Sto- chastic Multi-Resonance in a Linear System Driven by Multiplicative Polynomial Dichotomous Noise,” Chinese Physics Letters, Vol. 28, No. 9. 2011, Article ID: 090505.

[43]   M. Gitterman “Order and Chaos: Are They Contradictory or Complementary?” European Journal of Physics, Vol. 23, No. 2, 2002, pp. 119-122. doi:10.1088/0143-0807/23/2/304

[44]   M. I. Dykman, R. Mannela, P. V. E. McClintock, F. Moss, and M. Soskin, “Spectral Density of Fluctuations of a Double-Well Duffing Oscillator Driven by White Noise,” Physical Review A, Vol. 37, No. 4, 1988, pp. 1303-1313. doi:10.1103/PhysRevA.37.1303

[45]   A. Fulinski, “Relaxation, Noise-Induced Transitions, and Stochastic Resonance Driven by Non-Markovian Dichotomic Noise,” Physical Review E, Vol. 52, No. 4, 1995, pp. 4523-4526. doi:10.1103/PhysRevE.52.4523

[46]   V. Berdichevsky and M. Gitterman, “Multiplicative Stochastic Resonance in Linear Systems: Analytical Solution,” Europhysical Letters, Vol. 36, No. 3, 1996, pp. 161-166. doi:10.1209/epl/i1996-00203-9

[47]   R. Benzi, S. Sutera and A. Vulpani, “The Mechanism of Stochastic Resonance,” Journal of Physics A, Vol. 14, No. 11, 1981, pp. 453-458. doi:10.1088/0305-4470/14/11/006

[48]   G. Nicolis, “Stochastic Aspects of Climatic Transitions —Response to a Periodic Forcing,” Tellus, Vol. 34, No. 1, 1982, pp. 1-9. doi:10.1111/j.2153-3490.1982.tb01786.x

[49]   L. Gammaitoni, P. Hanggi, P. Jung and F. Marchesoni, “Stochastic Resonance,” Review of Modern Physics, Vol. 70, No. 1, 1998, pp. 223-287. doi:10.1103/RevModPhys.70.223

[50]   N. G. Stokes, N. D. Stein and V. P. E. McClintock, “Stochastic Resonance in Monostable Systems,” Journal of Physics A, Vol. 26, No. 7, 1993, pp. 385-390. doi:10.1088/0305-4470/26/7/007

[51]   F. Marchesoni, “Comment on Stochastic Resonance in Washboard Potentials,” Physics Letters A, Vol. 352, No. 1-2, 1997, pp. 61-64. doi:10.1016/S0375-9601(97)00232-6

[52]   M. Gitterman, “Classical Harmonic Oscillator with Multi- plicative Noise,” Physica A, Vol. 352, No. 2-4, 2005, pp. 309-334. doi:10.1016/j.physa.2005.01.008

[53]   S.-Q. Jiang, B. Wu and T.-X. Gu, “Stochastic Resonance in a Harmonic Oscillator Fluctuating Intrinsic Frequency by Asymmetric Dichotomous Noise,” Journal of Electronic Science and Technology, Vol. 5, No. 4, 2007, pp. 344-347.

[54]   S. Jiang, F. Guo, Y. Zhow and T. Gu, “Stochastic Resonance in a Harmonic Oscillator with Randomizing Dam- ping by Asymmetric Dichotomous Noise,” International Conference on Communications, Circuits and Systems, Kokura, 11-13 July 2007, pp. 1044-1047.

[55]   P. S. Landa and P. V. E. McClintock, “Vibrational Resonance,” Journal of Physics A, Vol. 33, No. 45, 2000, pp. 433-438. doi:10.1088/0305-4470/33/45/103

[56]   Y. Braiman and I. Goldhirsch, “Taming Chaotic Dynamics With Weak Periodic Perturbations,” Physical Review Letters, Vol. 66, No. 20, 1991, pp. 2545-2548. doi:10.1103/PhysRevLett.66.2545

[57]   Y. Kim, S. Y. Lee and S. Y. Kim, “Experimental Observation of Dynamic Stabilization in a Double-Well Duffing Oscillator,” Physics Letters A, Vol. 275, No. 4, 2000, pp. 254-259. doi:10.1016/S0375-9601(00)00572-7

[58]   S. Rajasekar, S. Jeyakumari and M. A. F. Sanjuan, “Role of Depth and Location of Minima of a Double-Well Potential on Vibrational Resonance,” Journal of Physics A, Vol. 43, No. 46, 2010, Article ID: 465101.

[59]   I. Blekhmam and P. S. Landa, “Conjugate Resonances and Bifurcations in Nonlinear Systems under Biharmoni- cal Excitation,” International Journal of Non-Linear Me- chanics, Vol. 39, No. 3, 2004, pp. 421-426. doi:10.1016/S0020-7462(02)00201-9

[60]   J. P. Baltanas, et al., “Experimental Evidence, Numerics, and Theory of Vibrational Resonance in Bistable Sys- tems,” Physical Review E, Vol. 67, No. 6, 2003, Article ID: 066119.

[61]   V. N. Chizhevsky and G. Giacomelli, “Improvement of Signal-to-Noise Ratio in a Bistable Optical System: Comparison between Vibrational and Stochastic Resonance,” Physical Review A, Vol. 71, No. 1, 2005, Article ID: 011801.

[62]   S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M. A. F. Sanjuan, “Single and Multiple Vibrational Resonance in a Quintic Oscillator with Monostable Potentials,” Physical Review E, Vol. 80, No. 4, 2009, Article ID: 046608.

[63]   S. Jeyakumari, V. Chinnathambi, S. Rajasekar and M. A. F. Sanjuan, “Analysis of Vibrational Resonance in a Quintic Oscillator,” Chaos, Vol. 19, No. 4, 2009, Article ID: 043128.

[64]   J. C. Chedjou, H. B. Fotsin and P. Woafo, “Behavior of the Van der Pol Oscillator with Two External Periodic Forces,” Physica Scripta, Vol. 55, No. 4, 1997, pp. 390- 393. doi:10.1088/0031-8949/55/4/002

[65]   H. G. Schuster and W. Just, “Deterministic Chaos: An Introduction,” Wiley, New York, 2005. doi:10.1002/3527604804

[66]   E. Ott, “Chaos in Dynamical Systems,” Cambridge University Press, Cambridge, 2002.

 
 
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