Stability and Boundedness of Solutions to Some Non-autonomous Multidimensional Nonlinear Systems

Assessment of degree of boundedness and stability of multidimensional nonlinear systems with time-dependent and especially nonperiodic coefficients is an important applied problem which has no adequate resolution yet. Most of the known techniques mostly provide computationally intensive and conservative stability criteria in this area which frequently fail to gage the degrees of stability and especially boundedness of solutions to the corresponding systems. Recently, we outline a new approach to this task resting on analysis of solutions to a scalar auxiliary equation bounding from above time-histories of the norms of solutions to the original systems. This paper develops a new technique casting the auxiliary equation in a simplified form which, in turn, amplifies its application domain and reduces the computational hamper of our prior approach. Consequently, we develop novel boundedness and stability criteria and estimated the trapping and stability regions for some multidimensional nonlinear systems with time - dependent coefficients. This let us to assess in target simulations the degree of boundedness and stability of multidimensional nonlinear and non-autonomous systems which were intractable to our prior methodolog

Estimation of non-symmetric and unbounded region of attraction using shifted shape function and R-composition

A general numerical method using sum of squares programming is proposed to address the problem of estimating the region of attraction (ROA) of an asymptotically stable equilibrium point of a nonlinear polynomial system. The method is based on Lyapunov theory, and a shape function is defined to enlarge the provable subset of a local Lyapunov function. In contrast with existing methods with a shape function centered at the equilibrium point, the proposed method utilizes a shifted shape function (SSF) with its center shifted iteratively towards the boundary of the newly obtained invariant subset to improve ROA estimation. A set of shifting centers with corresponding SSFs is generated to produce proven subsets of the exact ROA and then a composition method, namely R-composition, is employed to express these independent sets in a compact form by just a single but richer-shaped level set. The proposed method denoted as RcomSSF brings a significant improvement for general ROA estimation problems, especially for non-symmetric or unbounded ROA, while keeping the computational burden at a reasonable level. Its effectiveness and advantages are demonstrated by several benchmark examples from literature.Comment: 40 pages, 9 figure