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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., Comment: 31 pages, 10 figures

Quadrupole topological insulators are a new class of topological insulators with quantized quadrupole moments, which support protected gapless corner states. The experimental demonstrations of quadrupole-topological insulators were reported in a series of artificial materials, such as photonic crystals, acoustic crystals, and electrical circuits. In all these cases, the underlying structures have discrete translational symmetry and thus are periodic. Here we experimentally realize two-dimensional aperiodic-quasicrystalline quadrupole-topological insulators by constructing them in electrical circuits, and observe the spectrally and spatially localized corner modes. In measurement, the modes appear as topological boundary resonances in the corner impedance spectra. Additionally, we demonstrate the robustness of corner modes on the circuit. Our circuit design may be extended to study topological phases in higher-dimensional aperiodic structures.

Weyl points (WPs), as nodal degenerate points in three-dimensional (3D) momentum space, are ideal if they are symmetry-related, well-separated, residing at the same energy and far from the nontopological bands. Although type-II WPs show some unique features compared with type-I counterparts, ideal type-II WPs have not yet been reported due to the lack of an ideal Weyl system with enough flexibility to tilt the dispersion bands. By stacking two-dimensional (2D) layers of inductor-capacitor (LC) resonator dimers with the breaking of parity inversion symmetry, here we experimentally realize the topological circuits with only the topological bands and observe a minimal number of four ideal type-II WPs. Two hallmark features of type-II WPs: a strongly tilted band structure with two group velocities having the same sign near type-II WPs and the topological surface states in an incomplete bandgap have been demonstrated. Our results establish an ideal system to the further study of Weyl physics and provide a new perspective to the access of topological matters.