Vaskin V. V., Vaskina A. V., Mamaev I. S. Problems of stability and asymptotic behavior of vortex patches on the plane 2010, Vol. 6, No. 2, pp.  327-343 Abstract With the help of mathematical modelling, we study the dynamics of many point vortices system on the plane. For this system, we consider the following cases: — vortex rings with outer radius $r = 1$ and variable inner radius $r_0$, — vortex ellipses with semiaxes $a$, $b$. The emphasis is on the analysis of the asymptotic $(t → ∞)$ behavior of the system and on the verification of the stability criteria for vorticity continuous distributions. Keywords: vortex dynamics, point vortex, hydrodynamics, asymptotic behavior Citation: Vaskin V. V., Vaskina A. V., Mamaev I. S.,  Problems of stability and asymptotic behavior of vortex patches on the plane, Rus. J. Nonlin. Dyn., 2010, Vol. 6, No. 2, pp.  327-343 DOI:10.20537/nd1002007
 Vaskin V. V., Erdakova N. N., Mamaev I. S. Statistical mechanics of nonlinear dynamical systems 2009, Vol. 5, No. 3, pp.  385-402 Abstract With the help of mathematical modeling, we study the behavior of a gas ($\sim10^6$ particles) in a one-dimensional tube. For this dynamical system, we consider the following cases: — collisionless gas (with and without gravity) in a tube with both ends closed, the particles of the gas bounce elastically between the ends, — collisionless gas in a tube with its left end vibrating harmonically in a prescribed manner, — collisionless gas in a tube with a moving piston, the piston’s mass is comparable to the mass of a particle. The emphasis is on the analysis of the asymptotic ($t→∞$)) behavior of the system and specifically on the transition to the state of statistical or thermal equilibrium. This analysis allows preliminary conclusions on the nature of relaxation processes. At the end of the paper the numerical and theoretical results obtained are discussed. It should be noted that not all the results fit well the generally accepted theories and conjectures from the standard texts and modern works on the subject. Keywords: one-dimensional collisionless gas, statistical equilibrium, thermodynamical equilibrium, weak limit Citation: Vaskin V. V., Erdakova N. N., Mamaev I. S.,  Statistical mechanics of nonlinear dynamical systems, Rus. J. Nonlin. Dyn., 2009, Vol. 5, No. 3, pp.  385-402 DOI:10.20537/nd0903006