Guo Yu, Luo Albert C J
McCoy School of Engineering, Midwestern State University, Wichita Falls, Texas 76308, USA.
Department of Mechanical and Industrial Engineering, Southern Illinois University Edwardsville, Edwardsville, Illinois 62026-1805, USA.
Chaos. 2022 Oct;32(10):103129. doi: 10.1063/5.0121990.
In this paper, the complete bifurcation dynamics of period-3 motions to chaos are obtained semi-analytically through the implicit mapping method. Such an implicit mapping method employs discrete implicit maps to construct mapping structures of periodic motions to determine complex periodic motions. Analytical bifurcation trees of period-3 motions to chaos are determined through nonlinear algebraic equations generated through the discrete implicit maps, and the corresponding stability and bifurcations of periodic motions are achieved through eigenvalue analysis. To study the periodic motion complexity, harmonic amplitudes varying with excitation amplitudes are presented. Once more, significant harmonic terms are involved in periodic motions, and such periodic motions will be more complex. To illustrate periodic motion complexity, numerical and analytical solutions of periodic motions are presented for comparison, and the corresponding harmonic amplitudes and phases are also presented for such periodic motions in the bifurcation trees of period-3 motions to chaos. Similarly, other higher-order periodic motions and bifurcation dynamics for the nonlinear spring pendulum can be determined. The methods and analysis presented herein can be applied for other nonlinear dynamical systems.
在本文中，通过隐映射方法半解析地得到了从周期3运动到混沌的完整分岔动力学。这种隐映射方法采用离散隐映射来构建周期运动的映射结构，以确定复杂的周期运动。通过离散隐映射生成的非线性代数方程确定从周期3运动到混沌的解析分岔树，并通过特征值分析实现周期运动的相应稳定性和分岔。为了研究周期运动的复杂性，给出了随激励幅度变化的谐波幅值。再一次，周期运动中涉及显著的谐波项，并且这样的周期运动将更加复杂。为了说明周期运动的复杂性，给出了周期运动的数值解和解析解进行比较，并且在从周期3运动到混沌的分岔树中也给出了此类周期运动相应的谐波幅值和相位。类似地，可以确定非线性弹簧摆的其他高阶周期运动和分岔动力学。本文提出的方法和分析可应用于其他非线性动力系统。
