Minkowski维数与无穷远附近的快慢多项式Liénard方程
Minkowski dimension and slow-fast polynomial Liénard equations near infinity
AI总结：
该研究将Minkowski维数定义推广至无界序列，结合Poincaré–Lyapunov紧化分析无穷远附近的快慢广义Liénard方程，以揭示含无穷远部分的快慢环的分形本质。
摘要：
在平面快慢系统中，对实数域$\u211d$内（有界）序列的分形分析已被证明具有重要作用，可用于检测奇异Hopf分岔中的首个非零Lyapunov量、确定有限平面内定义的快慢环产生的极限环最大数目等。研究者采用由慢关系函数生成序列的Minkowski维数概念。本文采用类似方法，结合Poincaré–Lyapunov紧化，聚焦于快慢广义Liénard方程$\u1e21 x=y-\u2211_{k=0}^{n+1} B_kx^k,\u1e21 y=-ε\u2211_{k=0}^{m}A_kx^k$在无穷远附近的分形分析。我们将Minkowski维数的定义推广到无界序列，这有助于我们更深入地理解快慢Liénard方程中检测到的、包含无穷远部分的快慢环的分形本质。
英文摘要：
In planar slow-fast systems, fractal analysis of (bounded) sequences in $\mathbb R$ has proved important for detection of the first non-zero Lyapunov quantity in singular Hopf bifurcations, determination of the maximum number of limit cycles produced by slow-fast cycles, defined in the finite plane, etc. One uses the notion of Minkowski dimension of sequences generated by slow relation function. Following a similar approach, together with Poincaré--Lyapunov compactification, in this paper we focus on a fractal analysis near infinity of the slow-fast generalized Liénard equations $\dot x=y-\sum_{k=0}^{n+1} B_kx^k,\ \dot y=-ε\sum_{k=0}^{m}A_kx^k$. We extend the definition of the Minkowski dimension to unbounded sequences. This helps us better understand the fractal nature of slow-fast cycles that are detected inside the slow-fast Liénard equations and contain a part at infinity.
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