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An interdisciplinary analysis is presented in this paper on the relationship between the bifurcation behavior of chaotic systems, specifically Chua’s circuit, and the quantized energy level of atoms. We show that it is possible to associate special capacitance values in Chua’s circuits with the electronic energy levels En of atoms calculated from Bohr’s Law. In particular, these particular capacitance values correspond to the bifurcation points αn of Chua’s circuits. We found a functional relation associating a specific Chua’s circuit to the Hydrogen atom and obtained a map of “Atom” to “Chua’s Circuit”. This map can be extended to other elements in the Periodic Table, thereby demonstrating an almost universal relation between the two different physical systems. With this map we can calculate the energy levels of different atoms from bifurcation diagrams (or vice-versa) and express the analytical relations in terms of the two variables αn and En