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We use atmospheric neutrino data and MINOS data to constrain the MaVaN (MassVarying Neutrinos) mechanism. The MaVaN model was largely studied in cosmologyscenarios and comes from the coupling of the neutrinos with a neutral scalardepending on the local matter density. For atmospheric neutrinos, this newinteraction affects the neutrino propagation inside the Earth, and asconsequence, induces modifications in their oscillation pattern. To performsuch test for a non-standard oscillation mechanism with a non-diagonal neutrinocoupling in the mass basis, we analyze the angular distribution of atmosphericneutrino events as seen by the Super-Kamiokande experiment for the events inthe Sub-GeV and multi-GeV range and muon neutrinos (anti-neutrinos) in MINOSexperiment. From the combined analysis of these two sets of data we obtain thebest fit for $\Delta m^{2}_{32}=2.45 \times 10^{-3}$ eV$^{2}$,$sin^{2}(\theta_{23})=0.42 $ and MaVaN parameter $\alpha_{32}=0.28$ with modestimprovement, $\Delta \chi^{2}= 1.8$, over the standard oscillation scenario.The combination of MINOS data and Super-Kamiokande data prefers small values ofMaVaN parameter $\alpha_{32} < 0.31 $ at $90 \%$ C. L..

Searching for non-diagonal Mass varying mechanism in the $\nu_{\mu}$-$\nu_{\tau}$ system