The Applications of Algebraic Methods on Stable Analysis for General Differential Dynamical Systems with Multidelays

The distribution of purely imaginary eigenvalues and stabilities of generally singular or neutral differential dynamical systems with multidelays are discussed. Choosing delays as parameters, firstly with commensurate case, we find new algebraic criteria to determine the distribution of purely imaginary eigenvalues by using matrix pencil, linear operator, matrix polynomial eigenvalues problem, and the Kronecker product. Additionally, we get practical checkable conditions to verdict the asymptotic stability and Hopf bifurcation of differential dynamical systems. At last, with more general case, the incommensurate, we mainly study critical delays when the system appears purely imaginary eigenvalue.

A centre theorem for two-dimensional complex holomorphic systems and its generalization

We consider a two-dimensional complex holomorphic system. In particular, we use the centre manifold theory together with the singular point theory of C. H. Briot & J. C. Bouquet ( J . Éc . imp . Polyt . 21, 133 (1856)) to establish a centre theorem concerning the behaviour of the phase paths of the system in the neighbourhood of an equilibrium point having a single purely imaginary eigenvalue. An extended centre theorem is established for the corresponding N -dimensional complex holomorphic system ( N ≥ 3).

Stability of water waves

We apply some general results for Hamiltonian systems, depending on the notion of signature of eigenvalues, to determine the circumstances under which collisions of imaginary eigenvalue for the linearized problem about a travelling water wave of permanent form are avoided or lead to loss of stability, up to non-degeneracy assumptions. A new superharmonic instability is predicted and verified.