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首页> 外文期刊>Journal of Differential Equations >Global Hopf bifurcation of differential equations with threshold type state-dependent delay
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Global Hopf bifurcation of differential equations with threshold type state-dependent delay

机译：阈值类型依赖状态的时滞微分方程的全局Hopf分支

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摘要

We develop global Hopf bifurcation theory of differential equations with state-dependent delay using the S~1-equivariant degree and investigate a two-degree-of-freedom mechanical model of turning processes. For the model of turning processes we show that the extreme points of each vibration component of the non-constant periodic solutions can be embedded into a manifold with explicit algebraic expression. This observation enables us to establish analytical upper and lower bounds of the amplitudes of the peri-odic solutions in terms of the system parameters and to exclude certain periods. Using the achieved global bifurcation theory we reveal that if the relative frequency between the natural frequency and the turning fre-quency varies in a certain interval, then generically every bifurcated continuum of periodic solutions must terminate at a bifurcation point. This termination means that the underlying system with parameters in the stability region near the vertical asymptotes of the stability lobes is less subject to chatter. In the process, several sufficient conditions for the non-existence of non-constant periodic solutions are also obtained.
机译：我们使用S〜1-等变度数发展了具有状态依赖时滞的微分方程的全局Hopf分岔理论，并研究了车削过程的两自由度力学模型。对于车削过程模型，我们表明，非恒定周期解的每个振动分量的极点都可以嵌入具有显式代数表达式的流形中。这种观察使我们能够根据系统参数确定周期解的振幅的分析上限和下限，并排除某些时段。使用已实现的全局分叉理论，我们揭示出，如果固有频率与转折频率之间的相对频率在一定间隔内变化，那么一般而言，周期解的每个分叉连续体都必须在分叉点处终止。此终止意味着在稳定波瓣的垂直渐近线附近的稳定区域中具有参数的基础系统较少受到震颤的影响。在该过程中，还获得了不存在非恒定周期解的几个充分条件。

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