Weakly connected nonlinear systems : boundedness and stability of motion Preface The investigation of nonlinear systems with a small parameter is attributable by a lot of modern problems of mechanics physics hydrodynamics electrodynamics of charge particle beams space technology astrodynamics and many others. The key problem in solution of various applied problems is that of the stability of solutions of systems of equations in various senses. The methods of the classical stability theory if appropriately adapted may be applied to systems containing a small parameter. The progress in solving problems of the theory of stability and nonlinear perturbations is associated with finding way around significant difficulties connected with the growth of the number of variables characterizing the state of a system which may include critical variables. In addition the presence of critical variables may result in a situation when not only the first approximation cannot solve a stability problem but also the further nonlinear approximations below some order cannot solve it. New approaches recently developed for systems with a small parameter may include the following. A. The development of the direct Lyapunov method for the study of the boundedness and stability of systems with a finite number of degrees of freedom with respect to two different measures. B. The analysis of stability on the basis of the combination of the concepts of the direct Lyapunov method and the averaging method of nonlinear mechanics for some classes of linear and nonlinear systems. C. The generalization of the direct Lyapunov method on the basis of the concepts of the comparison principle and the averaging method of nonlinear mechanics. D. The development of the method of matrix valued Lyapunov functions and its application in the study of stability of