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12EXTENDED MATHIEU EQUATION IN DUSTY PLASMA
Semyonov V.P.,*1 Timofeev A.V.2
1 Moscow State Inst. of Electronics and Mathematics-Higher School of Economics,
Russia
Joint Institute for High Temperatures RAS, Moscow, Russia
Phenomenon and mechanisms of an energy transfer between degrees of freedom of a dusty plasma system are of great interest in the field of dusty plasma. Since fluctuations of dust particles in the gas-discharge plasma on vertical and horizontal directions are largely independent, motion in these directions can be separated. So the dust particles motion and some of the energy transfer mechanisms can be
described by the extended Mathieu equation: x + 2xx + c0 (1 + h cos copt) x = n (t), where n( t) is a stochastic force.
The approximate analytical solution for the Mathieu equation x + 2xx + c0 (1 + h cos cpt) x = 0 in the approximation of small h, s = cp!c0 - 2/n
and X is the basis for the study. Acting by an analogy with standard approach an expression for the growth rate of the amplitude s can be obtained. The resonance areas boundaries obtained analytically and the ones obtained numerically are close only for nonfriction system with h < 1.
The classical approach leads to serious differences with the numerical solution of the equation in the presence of friction (X ^ 0 ). It can be explained by the fact that this approach takes into account friction terms only of zero-order of smallness.The results of more accurate solution are closer to the data obtained numerically. This solution also explains such phenomenon as the shift of the c0jcp value wherein
the resonance occurs with a minimum value of h.
The extended Mathieu equation is studied for a wide range of parameters values. Using analytical and numerical approaches boundaries of the resonance area, the time of onset and the growth rate of the amplitude at various system parameters are derived. The results of the calculation are compared with analytical solution and the theory is specified in the places of disagreements.
The work is supported by Russian Science Foundation (14-19-01295).
