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Fractal structures pervade nature and are receiving increasing engineeringattention towards the realization of broadband resonators and antennas. We showthat fractal resonators can support the emergence of high-dimensional chaoticdynamics even in the context of an elementary, single-transistor oscillatorcircuit. Sierpi\'nski gaskets of variable depth are constructed using discretecapacitors and inductors, whose values are scaled according to a simplesequence. It is found that in regular fractals of this kind each iterationeffectively adds a conjugate pole/zero pair, yielding gradually more complexand broader frequency responses, which can also be implemented as much smallerFoster equivalent networks. The resonators are instanced in the circuit asone-port devices, replacing the inductors found in the initial version of theoscillator. By means of a highly simplified numerical model, it is shown thatincreasing the fractal depth elevates the dimension of the chaotic dynamics,leading to high-order hyperchaos. This result is overall confirmed by SPICEsimulations and experiments, which however also reveal that the non-idealbehavior of physical components hinders obtaining high-dimensional dynamics.The issue could be practically mitigated by building the Foster equivalentnetworks rather than the verbatim fractals. Furthermore, it is shown thatconsiderably more complex resonances, and consequently richer dynamics, can beobtained by rendering the fractal resonators irregular through reshuffling theinductors, or even by inserting a limited number of focal imperfections. Thepresent results draw attention to the potential usefulness of fractalresonators for generating high-dimensional chaotic dynamics, and underline theimportance of irregularities and component non-idealities.

High-dimensional dynamics in a single-transistor oscillator containing Feynman-Sierpinski resonators: effect of fractal depth and irregularity