Subharmonic oscillations in the Floquet circuit with the frequency-synthesis dimension

The period-doubling oscillation emerges with the coexistence between zero and {\pi} modes in Floquet topological insulator. Here, utilized the flexibility of the circuit, we construct the Floquet circuit with frequency-synthetic dimension and find the topological-protected deeply-subharmonic oscillations with the period extensively exceeding the doubling-driven period. In the construction framework, the periodically-driven mechanism is attained by implementing the circuit-oscillator hierarchy with the stepping-variation resonances in frequency domain. The zero and {\pi} modes that arise at the Floquet band in the circuit indicate the anomalous boundary-bulk correspondence. The coexistence of zero and {\pi} modes, results in a subharmonic oscillation with the extremely-low frequency on the edge of the Floquet circuit. Furthermore, we explore the Floquet band with the enhanced periodically-driven strength tailored by the component flexibility of the circuit. Our method provides a flexible scheme to study Floquet topological phases, and open a new path for realizing the deeply subwavelength system.