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Harmonic Oscillator with Fluctuating Mass  [PDF]
Moshe Gitterman
Journal of Modern Physics (JMP) , 2011, DOI: 10.4236/jmp.2011.210140
Abstract: We generalize the previously considered cases of a harmonic oscillator subject to a random force (Brownian motion), or having random frequency, or random damping. We consider here a random mass which corresponds to an oscillator where the particles of the surrounding medium adhere to the oscillator for some (random) time after collision, thereby changing the oscillator mass. Such a model is appropriate to chemical and biological solutions as well as to some nano-technological devices. The first moment and stability conditions for white and dichotomous noise are analyzed.
Oscillator Subject to Periodic and Random Forces  [PDF]
Moshe Gitterman
Journal of Modern Physics (JMP) , 2013, DOI: 10.4236/jmp.2013.41015

We illustrate the influence of an external periodic force and noise on a physical system by the example of an oscillator. These two forces seem to be the reverse of each other, since the latter leads to disorder while the former works in an orderly fashion. Nevertheless, it is shown that they may influence a system in a similar way, sometime even substituting for one another. These examples serve to illustrate one of the main achievements of twentieth-century physics, which has established that deterministic and random phenomena complement rather than contradict each other.

Quartic Anharmonic Oscillator and Random Matrix Theory  [PDF]
G. M. Cicuta,S. Stramaglia,A. G. Ushveridze
Physics , 1995, DOI: 10.1142/S021773239600014X
Abstract: In this paper the relationship between the problem of constructing the ground state energy for the quantum quartic oscillator and the problem of computing mean eigenvalue of large positively definite random hermitean matrices is established. This relationship enables one to present several more or less closed expressions for the oscillator energy. One of such expressions is given in the form of simple recurrence relations derived by means of the method of orthogonal polynomials which is one of the basic tools in the theory of random matrices.
Nodal Sets of Random Eigenfunctions for the Isotropic Harmonic Oscillator  [PDF]
Boris Hanin,Steve Zelditch,Peng Zhou
Mathematics , 2013,
Abstract: We consider Gaussian random eigenfunctions (Hermite functions) of fixed energy level of the isotropic semi-classical Harmonic Oscillator on ${\bf R}^n$. We calculate the expected density of zeros of a random eigenfunction in the semi-classical limit $h \to 0.$ In the allowed region the density is of order $h^{-1},$ while in the forbidden region the density is of order $h^{-\frac{1}{2}}$. The computer graphics due to E.J. Heller illustrate this difference in "frequency" between the allowed and forbidden nodal sets.
Coherent state of the effective mass harmonic oscillator  [PDF]
Atreyee Biswas,Barnana Roy
Physics , 2009, DOI: 10.1142/S0217732309028977
Abstract: We construct coherent state of the effective mass harmonic oscillator and examine some of its properties. In particular closed form expressions of coherent states for different choices of the mass function are obtained and it is shown that such states are not in general x-p uncertainty states. We also compute the associated Wigner functions.
Stability Analysis of the Helmholtz Oscillator with Time Varying Mass  [PDF]
Richard A. Morrison
Mathematics , 2007,
Abstract: We study the effects of periodically time varying mass on the stability of the Helmholtz oscillator, which, when linearised, takes the form of Ince's equation and exhibits parametric resonance. The resonance regions in the parameter space are mapped and we use the Melnikov function to demonstrate that the parametric instability does not affect the qualitative dynamics of the nonlinear system.
Frequency synchronization in random oscillator network  [PDF]
Takashi Ichinomiya
Physics , 2004, DOI: 10.1103/PhysRevE.70.026116
Abstract: We study the frequency-synchronization of randomly coupled oscillators. By analyzing the continuum limit, we obtain the sufficient condition for the mean-field type synchronization. We especially find that the critical coupling constant $K$ becomes 0 in the random scale free network, $P(k)\propto k^{-\gamma}$, if $2 < \gamma \le 3$. Numerical simulations in finite networks are consistent with these analysis.
On Quantized Lienard Oscillator and Momentum Dependent Mass  [PDF]
B. Bagchi,A. Ghose Choudhury,P. Guha
Physics , 2013,
Abstract: We examine the analytical structure of the nonlinear Lienard oscillator and show that it is a bi-Hamiltonian system depending upon the choice of the coupling parameters. While one has been recently studied in the context of a quantized momentum- dependent mass system, the other Hamiltonian also reflects a similar feature in the mass function and also depicts an isotonic character. We solve for such a Hamiltonian and give the complete solution in terms of a confluent hypergeometric function.
Mass Dependence of Instabilities of an Oscillator with Multiplicative and Additive Noise  [PDF]
Moshe Gitterman,David A. Kessler
Physics , 2012, DOI: 10.1103/PhysRevE.87.022137
Abstract: We study the instabilities of a harmonic oscillator subject to additive and dichotomous multiplicative noise, focussing on the dependance of the instability threshold on the mass. For multiplicative noise in the damping, the instability threshold is crossed as the mass is decreased, as long as the smaller damping is in fact negative. For multiplicative noise in the stiffness, the situation is more complicated and in fact the transition is reentrant for intermediate noise strength and damping. For multiplicative noise in the mass, the results depend on the implementation of the noise. One can take the velocity or the momentum to be conserved as the mass is changed. In these cases increasing the mass destabilizes the system. Alternatively, if the change in mass is caused by the accretion/loss of particles to the Brownian particle, these processes are asymmetric with momentum conserved upon accretion and velocity upon loss. In this case, there is no instability, as opposed to the other two implementations. We also study the distribution of the energy, finding a power-law cutoff at a value which increases with time.
Generalized Harmonic Oscillator and the Schr?dinger Equation with Position-Dependent Mass  [PDF]
Ju Guo-Xing,Cai Chang-Ying,Ren Zhong-Zhou
Physics , 2007,
Abstract: We study the generalized harmonic oscillator which has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and the its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.
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