Abstract
A deep understanding of the dynamical behavior in the parameter-state space plays a vital role in both the optimal design and motion control of mechanical systems. By combining the GPU parallel computing technique with two determinate indicators, namely the Lyapunov exponents and Poincaré section, this paper presents a detailed study on the two-parameter dynamics of a mechanical governor system with different time delays. By identifying different responses in the two-parameter plane, the effect of time delay on the complexity of the evolutionary process is fully revealed. The path-following calculation scheme and time domain collocation method are used to explore the detailed bifurcation mechanisms. An interesting phenomenon that the number of intersection points of some periodic responses on the specified Poincaré section differs from the actual period characteristics is found in classifying the dynamic behavior. For example, the commonly exhibited period-one orbit may have two or more intersection points on the Poincaré section rather than one point. The variations of the basins of attraction are also discussed in the plane of initial history conditions to demonstrate the multistability phenomena and chaotic transitions.
Highlights
Time-delay effect is widely encountered in many dynamical systems including biological systems [1,2,3], social and economic systems [4,5], neural networks [6,7], chemical reactions [8,9], and mechanical systems [10,11], due to the essential dependency of present states on past states
To develop an in-depth insight into the open questions unaddressed in Ref. [15], this paper aims to reveal the global dynamics of the autonomous governor system with and without time delay, e.g., the two-parameter classification of system response and the related multistability phenomena, which will develop a deep understanding of time-delay induced global dynamical behavior of the present governor system
The methodologies used for computation and bifurcation identification of two-parameter dynamics of the mechanical governor system are presented in Sec. 3, including the numerical integration strategy of time-delayed system, determinate indicators of dynamical response, GPU parallel computing technique and bifurcation identification method of periodic response
Summary
Time-delay effect is widely encountered in many dynamical systems including biological systems [1,2,3], social and economic systems [4,5], neural networks [6,7], chemical reactions [8,9], and mechanical systems [10,11], due to the essential dependency of present states on past states. It is expected that this method will be effective in solving the high-dimensional dynamical system, especially for the delayed dynamical system with complex nonlinearity Based on these two reasons, the TDCM is chosen in this study to obtain the approximate analytical solution of periodic response and identify corresponding bifurcations. The methodologies used for computation and bifurcation identification of two-parameter dynamics of the mechanical governor system are presented in Sec. 3, including the numerical integration strategy of time-delayed system, determinate indicators of dynamical response, GPU parallel computing technique and bifurcation identification method of periodic response. It might be inaccurate to determine the period property of periodic response by only counting the points on the Poincaré section for the autonomous system with time delay, since the self-intersection phenomenon of phase trajectory [49] may mislead the classification To overcome this confusion, the other methods will be applied and described in subsequent anlaysis. The Poincaré section is mainly used for the classification of different periodic responses in two-parameter diagram and the presentation of single-parameter bifurcation diagram
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