A three-dimensional set of ordinary differential equations that constitutes a simple abstract model of Darcy convection is investigated. The model reproduces a number of effects that are typical for dynamic systems with nontrivial cosymmetry. Nontrivial cosymmetry can give rise to a continuous family of equilibria where, in this case, the equilibrium stability spectrum varies along the family. The family of equilibria and its stability are examined analytically, and special bifurcations that occur in the system are investigated. It is shown that discrete and continual symmetries, called “flash symmetries,” can be present in the system for certain parameter values. Computer experiments on the selection of equilibria in the symmetric and cosymmetric cases have been carried out. They showed that, for initial points that are far enough from a cycle of equilibria, the neighborhood of a single equilibrium is established in the case of cosymmetry, but all the equilibria are equivalent in the case of symmetry. The authors hope that these results, as well as the formulation of the problems and the approach to their solution, will serve as a sample in the investigation of more complex systems in mathematical physics.

The paper addresses the nonlinear dynamics of planar inviscid incompressible flows in the straight channel of a finite length. Our attention is focused on the effects of boundary conditions on vorticity dynamics. The renowned Yudovich's boundary conditions (YBC) are the normal component of velocity given at all boundaries, while vorticity is prescribed at an inlet only. The YBC are fully justified mathematically: the well posedness of the problem is proven. In this paper we study general nonlinear properties of channel flows with YBC. There are 10 main results in this paper: (i) the trapping phenomenon of a point vortex has been discovered, explained and generalized to continuously distributed vorticity such as vortex patches and harmonic perturbations; (ii) the conditions sufficient for decreasing Arnold's and enstrophy functionals have been found, these conditions lead us to the washout property of channel flows; (iii) we have shown that only YBC provide the decrease of Arnold's functional; (iv) three criteria of nonlinear stability of steady channel flows have been formulated and proven; (v) the counterbalance between the washout and trapping has been recognized as the main factor in the dynamics of vorticity; (vi) a physical analogy between the properties of inviscid channel flows with YBC, viscous flows and dissipative dynamical systems has been proposed; (vii) this analogy allows us to formulate two major conjectures (C1 and C2) which are related to the relaxation of arbitrary initial data to C1: steady flows, and C2: steady, self-oscillating or chaotic flows; (viii) a sufficient condition for the complete washout of fluid particles has been established; (ix) the nonlinear asymptotic stability of selected steady flows is proven and the related thresholds have been evaluated; (x) computational solutions that clarify C1 and C2 and discover three qualitatively different scenarios of flow relaxation have been obtained.