Abstract. The authors consider actual problems of the modern theory of dynamical systems on manifolds, which are being actively developed. A brief review of the fields of the theory of dynamical systems is given. The results of the algebra of dual numbers, quaternionic algebra, biquaternions (dual quaternion) and their application to the study of infinitesimal neighborhoods and infinitesimal deformations of varieties (schemes) are presented. Summarized the theory of differential-algebraic equations over the field of real numbers and their dynamics, as well as relevant elements of trajectory optimization of dynamic systems. On the basis of the connection in the bundles, an extension of the theory of differential-algebraic equations to algebraic manifolds and schemes over, arbitrary fields and schemes, respectively, is given.

Keywords: dual number, dual quaternion, motor, quaternionic algebra, algebraic manifold, scheme, deformation, differential-algebraic equation, mathematical model, dynamical system, differential equation on algebraic manifold.