Let a given set be endowed with a preference preordering, and consider the problem of finding a solution to the differential inclusion A(OeS(*(O) which remains in the given set and evolves monotonically with respect to the preordering. We give sufficient conditions for the existence of such a trajectory, couched in terms of a notion of tangency developed by Clarke. No smoothness or convexity is...

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G f , or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated over-determined ‘defining system’ of differential equations. The usual computer classification...

We develop a theory of mean-square random invariant manifolds for dynamical systems generated by stochastic differential equations. This is applicable to partial equations driven nonlinear noise. The existence unstable proved the Lyapunov-Perron method based on backward equation involving conditional expectation with respect filtration. stable sets also established but remains open.