In two dimensional system of ordinary differential equations that provides a qualitative description of the thermohaline circulations locally in the upper layer of water in ocean [1], [2], [3], [4] we analyze loss of stability of its steady state for a generic finite parametric family of discontinuous transfer functions. It is shown that the problem is equivalent to the appearance of common fixed points of composition of involution for a generic family of pairs the involutions of real line with one common fixed point. The respective bifurcation diagrams in the space of parameters are also described. The study was financially supported by the Russian Science Foundation, project no. 19-11-00223.
References
[1] Davydov A.A., Melnikov N.B., Soft Loss of Stability in an Ocean Circulation Box Model with Turbulent Fluxes, Proc. Steklov Inst. Math., 259 (2007), 615.
[2] Davydov A.A., Melnikov N.B., AndronovHopf bifurcation in simple double diffusion models, Russian Math. Surveys, 62:2 (2007), 382384.
[3] Welander P.A., A simple heatsalt oscillator // Dyn. Atmos. and Oceans. 1982. V. 6, N 4. P. 233242.
[4] Davydov A.A., Zosimov S. O., Typical generation of self-oscillations in a oceanic circulation block model with turbulent flxes, Optimal control and dynamical systems, Collected papers. To the 95th anniversary of Academician Revaz Valerianovich Gamkrelidze, Proc. Steklov Inst. Math., 2023 (in print).