Asymptotics of nonlinear dynamical systems with high degree of nonlinearity (Q1420653)

The nonlinear dynamical system described by second order ordinary differential equations of the form \[ \ddot x +\gamma \dot x + {\omega}^2 x +\varepsilon x^n = 0, \quad n=3, 5, 7 , \dots \quad \tag{1} \] is considered. In the case of small values of \(n\) \((n = 3, 5)\) and for \(\varepsilon\leq 1 \) a solution of (1) can be found by a standard Lindstedt-Poincaré procedure or by the averaging method. For large \(n\) the standard technique gives unsatisfactory results. In the article another form of the solution, i.e. \[ x= \frac{x_0}{n^{\frac {1}{n}}} \Biggl(1+ \varepsilon \frac{x_1}{x_0}+ {\varepsilon}^2 \frac{x_2}{x_0}+\dots \Biggr)^{\frac{1}{n}}, \tag{2} \] is proposed. It is noted that the solution (2) reflects essential properties of the problem and gives a sufficiently correct and smooth solution of (1) for \(n= \infty\).