Abstract
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.
Highlights
Functional differential systems with multiple delays are important mathematic models to describe all kinds of natural and society phenomena
With more general case, the incommensurate, we mainly study critical delays when the system appears purely imaginary eigenvalue
The asymptotic stability of differential systems with multiple delays can be established from the rightmost part of the spectrum
Summary
Functional differential systems with multiple delays are important mathematic models to describe all kinds of natural and society phenomena. The numerical methods mainly contain linear multistep methods, RungeKutta methods, Newton methods, θ-methods, and so on Those methods are main keys to solve problems of stability on functional differential equations with delays all the time. We mainly discuss neutral delayed differential systems by algebraic methods. By algebraic methods, such as matrix pencil, linear operators, and Kronecker product, delayed margin and stability of the neutral linear differential system (1) are derived. For both types of above systems, we derive the criteria of stability and the distribution of eigenvalues or generalized eigenvalues of constant matrix pencil.
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