* Correspondence: tc_zhaodianli@139.com Department of mathematics, Shanghai Jiaotong University, Shanghai, 200240, China Full list of author information is available at the end of the article Abstract In this article, some inequalities on convolution equations are presented firstly. The mean square stability of the zero solution of the impulsive stochastic Volterra equation is studied by using ...

Stability of relative equilibria for Hamiltonian systems is generally equated with Liapunov stability of the corresponding fixed point of the flow on the reduced phase space. Under mild assumptions, a sharp interpretation of this stability is given in terms of concepts on the unreduced space. Suppose that (P, ω) is a symplectic manifold on which a Lie group G acts symplectically, and let H : P ...

In this paper we deal with the problem of stability and asymptotic stability of critical points of dynamical polysystems. We obtain results concerning polysystems with and without constraints, by means of uniform families of Liapunov functions. Dynamical polysystems may be interpreted as the topological counterpart of switched systems: our results are compared to those previously obtained in th...

The paper studies the exponential and non–exponential convergence rate to zero of solutions of scalar linear convolution Itô-Volterra equations in which the noise intensity depends linearly on the current state. By exploiting the positivity of the solution, various upper and lower bounds in first mean and almost sure sense are obtained, including Liapunov exponents .

We consider quasi-variational ordinary differential systems, which may be considered as the motion law for holonomic mechanical systems. Even when the potential energy of the system is not bounded from below, by constructing appropriate Liapunov functions and using the comparison method, we obtain sufficient conditions for global existence of solutions in the future and for their partial bounde...

The purpose of this work is to survey some possible deenitions of external stability for nonlinear systems and to provide the basic tools for investigating their relationship and dependence on internal stability properties. In particular, we review some early and more recent results about converse Liapunov theorems. To this respect, some improvements are pointed out without proof.

References [1] A. Adrover, S. Cerbelli & M. Giona. Exterior algebra-based algorithms to estimate Liapunov spectra and stretching statistics in high-dimensional and distributed systems, Int. [4] L. Allen & T.J. Bridges. Flow past a swept wing with a compliant surface: stabilizing the attachment line boundary layer, Stud. Hydrodynamic stability of the Ekman boundary layer including interaction wi...

It is well known that monomial average Liapunov functions of the form R(x1, x2, . . . , xn) = r0 ∏n i=1 x ri i (ri > 0, i = 0, 1, 2, . . . , n) play an eminent role in the permanence theory of ecological (or Kolmogorov) differential equations. A functional equation characterizing the above class of functions is presented.

In this paper, several dynamical systems for computing canonical correlations and canonical variates are proposed. These systems are shown to converge to the actual components rather than to a subspace spanned by these components. Using Liapunov stability theory, qualitative properties of the proposed systems are analyzed in detail including the limit of solutions as time approaches infinity.