AI 中文总结
研究具有多个小参数的基因调控网络平面常微分方程系统，通过参数空间爆破识别奇异极限，进行分岔分析与几何爆破确定动力学，揭示不同分岔情况，表明系统动力学和敏感性取决于参数相对大小，提供奇异摄动问题分析框架。
AI 中文摘要
我们考虑一类重要的基因调控动力学模型中的平面常微分方程系统。该系统奇异依赖于陡度参数\(0 < \varepsilon_1,\varepsilon_2 \ll 1\)，当这些参数趋于零时收敛到一个分段光滑系统。与先前研究不同，我们不假设\(\varepsilon_1 = \varepsilon_2\)。因此，当\((\varepsilon_1, \varepsilon_2) \to (0,0)\)时的动力学取决于极限的取法。通过在参数空间进行初步爆破，我们识别出三个不同的奇异极限。我们在每种情况下进行双参数分岔分析，并在变量和参数空间应用多次几何爆破来确定分岔结构和相关的全局动力学。在三种情况中的两种揭示了 Bogdanov - Takens 分岔，特别是在一种情况下，正则化的可见 - 不可见二重奇点被证明在鸭轨道附近组织奇异分岔的展开。我们的结果表明系统的定性动力学和对参数变化的整体敏感性取决于陡度参数的相对大小。更一般地，本文开发的分析框架为具有多个独立小参数的奇异摄动问题提供了一种系统方法，该方法应广泛适用于基因调控网络模型之外。
英文摘要
We consider a planar ODE system from an important class of models for gene regulatory dynamics. The system depends singularly on the steepness parameters $0<\varepsilon_1,\varepsilon_2 \ll 1$ and converges to a piecewise-smooth system as these parameters tend to zero. Unlike previous studies, we do not assume that $\varepsilon_1 = \varepsilon_2$. As a consequence, the dynamics when $(\varepsilon_1, \varepsilon_2) \to (0,0)$ depends upon how the limit is taken. Using a preliminary blow-up in parameter space, we identify three distinct singular limits. We perform a two-parameter bifurcation analysis in each case, and apply multiple geometric blow-ups in variable and parameter space to determine the bifurcation structure and the associated global dynamics. Bogdanov-Takens bifurcations are revealed in two of three cases, and in one case in particular, a regularised visible-invisible two-fold singularity is shown to organise the unfolding of singular bifurcations in the vicinity of canards. Our results show that the qualitative dynamics and overall sensitivity of the system to parameter variation depends on the relative size of the steepness parameters. More generally, the analytical framework developed herein provides a systematic approach to singular perturbation problems with multiple independent small parameters that should apply well beyond gene regulatory network models.
Comments58 pages, 21 figures
