We present a study of the multiwavelength (MW) variability of the blazar AO 0235$+$164 based on the radio-to-$\gamma$-ray data covering a long time period from 1997 to 2023. The radio data are represented by the 1–22 GHz measurements from the SAO RAS RATAN-600 radio telescope, the 5 and 8 GHz data from the IAA RAS RT-32 telescopes, and the 37 GHz data from the RT-22 telescope of CrAO RAS. The optical measurements in the R-band were collected with the SAO RAS 1-m Zeiss-1000 and 0.5-m AS-500/2 telescopes. Additionally, we used the archive data at 230 GHz from the Submillimetre Array and the $\gamma$-ray data in the 0.1–100 GeV band from the Fermi-LAT point source 4FGL-DR2 catalogue. The variability properties during four epochs containing major flares and one epoch of relatively low activity were analysed using the fractional variability indices, discrete correlation functions, Lomb–Scargle periodograms, and structure functions. A significant correlation ($\ge \!2\sigma$) between the radio, optical, and $\gamma$-ray bands is found for all these periods with time delays from 0 to 1.7 yr. The relation between time delay and frequency is described by a linear law with a negative slope of $-10$ d GHz$^{-1}$. The discovered properties of MW variability for the low-activity period and for flaring states suggest that the mechanisms dominating the radio–$\gamma$-ray variations are not substantially different. The detected quasi-periodic oscillations of about 6 and 2 yr are tentative, as the time span of the observations includes fewer than four full cycles for the radio and optical data and only about three cycles for the Fermi-LAT data. These results should be interpreted with caution, given the limited number of observed cycles and the influence of red noise. We used cluster analysis to reliably separate the high and low-activity states and determined statistical differences in the main properties of AO 0235$+$164 non-thermal emission. The physical parameters of the radio jet were obtained using the Hedgehog model applied to the average radio spectrum of AO 0235$+$164 in the range 0.1–300 GHz. The effectiveness of replacing electrons with protons in the synchrotron radio emission of relativistic jets is shown for describing the nature of blazars and the generation of high-energy neutrinos.

During the last 40–50 yr, the TeV blazar Ton 599 has systematically experienced major outbursts detected in a wide wavelength range from radio to $\gamma$-rays. In this work, we present an analysis of Ton 599 quasi-periodic variability across multiple wavelengths using an observing baseline from 1983 to 2025. The $\gamma$-ray, optical, and radio emissions are found to be highly correlated with time lags $\sim 0$–360 d, which indicates that they are triggered by the same population of particles. Using the Lomb–Scargle periodogram and the Weighted Wavelet Z-transform, we revealed several periodic components with characteristic periods of 1.4, 1.7, 2.3, 6.5, and 7.5 yr. The result is consistent with the detection of periodic components in the 1997–2011 light curves, which means that we observe the same mechanism causing long-term periodic variability. A model of a binary supermassive black hole (SMBH) with a precessing jet, applied to the radio light curves of Ton 599, yields frequency-dependent best-fitting parameters with orbital periods ranging from $\sim$1.2 to 1.7 yr and precession periods from $\sim$5.8 to 7.7 yr. This result implies the existence of an SMBH system modulating emission through both the orbiting and jet precession effects, with differing observed periods possibly reflecting frequency-dependent emission regions along a structured stratified jet. None the less, the short-term periodicity and exceptionally strong flares likely arise from internal jet shocks, which aligns with typical blazar behaviour. We suggest that the multiband quasi-periodicity of Ton 599 is reasonably described by a combination of geometric effects (orbital motion and precession) and stochastic processes.

We present a study of the multiwavelength (MW) variability of the blazar AO 0235+164 based on the radio-to-$\gamma$-ray data covering a period from 1997 to 2023. The radio data are represented by the measurements from the SAO RAS, IAA RAS, and CrAO RAS telescopes. The optical measurements were collected with the SAO RAS 1-m and 0.5-m reflectors. The archive data at 230 GHz from the Submillimeter Array and the $\gamma$-ray data from the Fermi-LAT mission were used too. A significant correlation between different spectral bands is found with time delays up to 1.7 years. The relation between time delay and frequency is well described by a linear law. The revealed features of MW variability for the quiet period and for flaring states suggest that the mechanisms dominating the radio-$\gamma$-ray variations are not substantially different. AO 0235+164 shows a total variability period of $\sim$6 years for all wavelength bands, and 1.4-2.3 years during the low state, which may reflect its general properties.

We present different methods for analyzing the light curves of active galactic nuclei (AGNs). Based on the example of AGN multiwavelength measurements, features of the following methods are considered: (1) structure functions (SFs)—to search for variability timescales; (2) discrete correlation functions (DCFs)—to search for connections between processes; (3) the Lomb–Scargle (L–S) periodogram—to search for periodicity. We analyze advantages and disadvantages of the methods and discuss their constraints and interpretation.

We present a study of the multiwavelength variability of the blazar Ton 599 based on the radio-to--ray data covering the period of 1997-2024. The data are represented by the 1-22 GHz measurements from the SAO RAS RATAN-600 radio telescope, the 5 and 8 GHz data from the IAA RAS RT-32 telescopes, and the 37 GHz data from the RT-22 telescope of CrAO RAS. The optical measurements in the R-band were collected with the SAO RAS 1-m Zeiss-1000 and 0.5-m AS-500/2, SPbSU 0.4-m LX-200, and CrAO RAS 0.7-m AZT-8 telescopes. We also used the archive data at 230 GHz from the Submillimeter Array (SMA) and the -ray data in the 0.1-100 GeV band from the Fermi-LAT point source catalogue 4FGL-DR2. The blazar exhibits extreme variability properties with a variability index varying from 0.47 to 0.98. A characteristic feature of the light curves is the presence of double and triple flares with different spectral properties. A significant correlation ($\geq 2\sigma$) between the radio, optical, and -ray bands is found with time delays from 0 to 365 days. The L-S periodograms show the significant peaks ($>3\sigma$) related to periods of 2.4-3.9 years, which coincides with global oscillations obtained earlier on the time scale of 1987-2010.

We present a study of the multiwavelength (MW) variability of the blazar AO 0235+164 based on the radio-to-$γ$-ray data covering a long time period from 1997 to 2023. The radio data are represented by the 1-22 GHz measurements from the RATAN-600 radio telescope, the 5 and 8 GHz data from the RT-32 telescopes, and the 37 GHz data from the RT-22 telescope. The optical measurements in the $R$-band were collected with the 1-m Zeiss-1000 and 0.5-m AS-500/2 telescopes. Additionally we used the archive data at 230~GHz from the SMA and the $γ$-ray data in the 0.1-100 GeV band from the Fermi-LAT point source 4FGL-DR2 catalogue. The variability properties during four epochs containing major flares and one epoch of relatively low activity were analysed. A significant correlation ($\geq\!2σ$) between the radio, optical, and $γ$-ray bands is found for all these periods with time delays from 0 to 1.7 yrs. The relation between time delay and frequency is described by a linear law with a negative slope of -10 day/GHz. The discovered properties of MW variability for the low activity period and for flaring states suggest that the mechanisms dominating the radio-$γ$-ray variations are not substantially different. The detected quasi-periodic oscillations of about 6 and 2 years are tentative, as the time span of the observations includes fewer than 4 full cycles for the radio and optical data and only about 3 cycles for the Fermi-LAT data. The physical parameters of the radio jet were obtained using the Hedgehog model applied to the average radio spectrum of AO 0235+164 in the range 0.1-300 GHz. The effectiveness of replacing electrons with protons in the synchrotron radio emission of relativistic jets is shown for describing the nature of blazars and the generation of high energy neutrinos.