To ensure the stability of a young seedling of a tree and proper rooting, tree stems are usually staked with one, two or three stakes. In this paper, we present a mathematical model and its approximations of a nonlinear biodynamical system in the form of a complex discrete structure on a cantilever. The nonlinear biodynamical system corresponds to a staked tree with one stake. The geometric nonlinearity of the system is introduced by a spring with cubic nonlinear properties that oscillates in the vertical plane. The influence coefficients of deflection of the cantilever were used for describing single-frequency forced oscillations of a complex biodynamical system under an external single-frequency force. This external single-frequency force has a circular frequency that is close to frequencies from a set of eigen circular frequencies of a corresponding linearized system. The system oscillates in two orthogonal planes - the horizontal and the vertical plane. Forced oscillations, in the vertical direction, of this discrete complex system, are described by subsystems of nonlinear differential equations that are solved in the first approximation by using two methods: the method of variation of constants in combination with the method of averaging and the other extended asymptotic method of nonlinear mechanics Krylov- Bogoliubov- Mitropolsky. The results of the qualitative analysis of the solution of the system of nonlinear differential equations by amplitudes and phases of four nonlinear modes are presented and discussed. In both cases, the frequencies of external force are within the range of resonant frequencies, but constant in time in the case when the generalized method of variation of constants with the method of averaging was used. In the case when the Krylov-Bogoliubov- Mitropolsky asymptotic method of nonlinear mechanics of approximation was used, the frequencies of external force have changeable values within time- frequency changes with different speeds.

This system is simpler than a system of governing differential equations along amplitudes and phases of generalized coordinates, and permits a qualitative analysis of nonlinear phenomena in systems with nonlinear dynamics. The method described in this paper is suitable for studying the stability and instability of amplitudes and of phases of nonlinear modes in the first approximation.

The asymptotic method of nonlinear mechanics Krylov-Bogoliubov- Mitropolsky allows the analysis of stationary discrete modes of nonlinear small oscillations in the resonant range as well as of non-stationary nonlinear modes. This method can be applied to studying nonlinear dynamics of more complex systems based on cantilevers with complex structure.

The most valuable and important elements of the work are emphasized.

为保证树木幼苗的稳定性和适当的生根，通常在树干上打上一根、两根或三根木桩。在本文中，我们以悬臂上的复杂离散结构的形式提出了非线性生物动力学系统的数学模型及其近似值。非线性生物动力系统对应于一棵有一根桩的桩树。系统的几何非线性是由一个在垂直平面上振动的具有三次非线性特性的弹簧引入的。悬臂挠度影响系数用于描述复杂生物动力系统在单频外力作用下的单频强迫振荡。这种外部单频力的圆频率接近于相应线性化系统的一组特征圆频率中的频率。该系统在两个正交平面中振荡 - 水平平面和垂直平面。这个离散复杂系统在垂直方向上的强制振荡由非线性微分方程的子系统描述，这些子系统通过使用两种方法在一次近似中求解：常数变化法结合平均法和另一种方法非线性力学 Krylov- Bogoliubov- Mitropolsky 的扩展渐近方法。给出并讨论了非线性微分方程组通过四种非线性模式的幅值和相位的解的定性分析结果。在这两种情况下，外力的频率在共振频率的范围内，但在使用广义的常数变化法和平均法的情况下，在时间上是恒定的。在使用非线性逼近力学的Krylov-Bogoliubov-Mitropolsky渐近方法的情况下，外力的频率随不同速度的时频变化而变化。

该系统比沿广义坐标的幅度和相位控制微分方程的系统更简单，并且允许对具有非线性动力学的系统中的非线性现象进行定性分析。本文描述的方法适用于研究非线性模式的振幅和相位的稳定性和不稳定性。

非线性力学 Krylov-Bogoliubov- Mitropolsky 的渐近方法允许分析谐振范围内的非线性小振荡的平稳离散模式以及非平稳非线性模式。该方法可用于研究基于复杂结构悬臂梁的更复杂系统的非线性动力学。

强调了工作中最有价值和最重要的元素。