包含Feynman-Sierpinski谐振器的单管振荡器中的高维动力学：分形深度和不规则性的影响
High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpinski resonators: effect of fractal depth and irregularity
摘要：
分形结构在自然界中普遍存在，并且在宽带谐振器和天线的实现方面受到越来越多的工程关注。我们表明，分形谐振器可以支持高维混沌动力学的出现，即使在一个基本的，单晶体管振荡器电路的上下文中。Sierpienski垫片的可变深度构造使用离散的电容器和电感器，其值根据一个简单的序列缩放。它被发现，在这种定期分形每次迭代有效地增加了共轭极点/零点对，逐渐产生更复杂和更广泛的频率响应，这也可以实现为更小的福斯特等效网络。谐振器在电路中作为单端口器件，取代了振荡器初始版本中的电感器。通过一个高度简化的数值模型，它表明，增加分形深度提高了混沌动力学的维数，导致高阶超混沌。这一结果得到了SPICE模拟和实验的全面证实，然而，这也表明，物理组件的非理想行为阻碍了获得高维动力学。这个问题实际上可以通过建立福斯特等效网络而不是逐字分形来缓解。此外，它表明，相当复杂的谐振，因此更丰富的动态，可以通过呈现分形谐振器不规则通过重新洗牌的电感器，甚至通过插入有限数量的焦点缺陷。目前的研究结果提请注意分形谐振器产生高维混沌动力学的潜在用途，并强调不规则性和组件的非理想性的重要性。
英文摘要：
Fractal structures pervade nature and are receiving increasing engineering attention towards the realization of broadband resonators and antennas. We show that fractal resonators can support the emergence of high-dimensional chaotic dynamics even in the context of an elementary, single-transistor oscillator circuit. Sierpiński gaskets of variable depth are constructed using discrete capacitors and inductors, whose values are scaled according to a simple sequence. It is found that in regular fractals of this kind each iteration effectively adds a conjugate pole/zero pair, yielding gradually more complex and broader frequency responses, which can also be implemented as much smaller Foster equivalent networks. The resonators are instanced in the circuit as one-port devices, replacing the inductors found in the initial version of the oscillator. By means of a highly simplified numerical model, it is shown that increasing the fractal depth elevates the dimension of the chaotic dynamics, leading to high-order hyperchaos. This result is overall confirmed by SPICE simulations and experiments, which however also reveal that the non-ideal behavior of physical components hinders obtaining high-dimensional dynamics. The issue could be practically mitigated by building the Foster equivalent networks rather than the verbatim fractals. Furthermore, it is shown that considerably more complex resonances, and consequently richer dynamics, can be obtained by rendering the fractal resonators irregular through reshuffling the inductors, or even by inserting a limited number of focal imperfections. The present results draw attention to the potential usefulness of fractal resonators for generating high-dimensional chaotic dynamics, and underline the importance of irregularities and component non-idealities.
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