Discussion: In this example, after a sinusoidal curve-fit of the (O-C) vs. Time data showed (Figure 1. below) a periodicity (P_ttv) of 489.48 ± 24.13 days (488.81 days was observed in the Lomb-Scargle Periodogram (LSP) of the same data), a plot (Figure 2.) of the Residuals also gave a good sinusoidal curve-fit with a periodicity of 1122.23 ± 163.38 days (LSP: 1174.81 days). The added combination of both sinusoidal curves is arrayed in Figure 3. and reasonably produces a complex overall curvature consistent with the initial data. While it is certainly possible (see recent work of Lithwick and others) that some of this unusual curvature obtains from eccentric orbits of planetary objects in this system, it is also possible that at least 3 planets in near-circular orbits are mutually-interacting to give the TTV distribution observed.
TTVs are derived from Q1-Q16 Kepler data. x-axes: “Observed Tc” (Mid-Transit Time): EXOFAST’s best-fits from Normalized PDCSAP_FLUX Kepler light flux vs. time (BJD_tdb - 2454900) data. y-axes: “(O – C)”: difference between Observed Tc and the Calculated Tc from the graphically obtained linear ephemeris.