Volume-preserving normal forms of Hopf-zero singularity

A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a non-zero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any non-degenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified R\"{o}ssler and generalized Kuramoto--Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple

Generalized Vector Variational-Like Inequalities

In this paper, we consider different types of generalized vector variational-like inequalities and study the relationships between their solutions. We study the general forms of Stampacchia and Minty type vector variational inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector variational inequalities for the Clarke’s subdifferential of non-differentiable locally Lipschitz functions and prove the existence of their solutions. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector variational inequalities

Nonsmooth Pseudolinearity

In this paper, we introduce the notion of generalized pseudolinearity for nondifferentiable and nonconvex but locally Lipschitz functions defined on a Banach space. We present some characterizations of generalized pseudolinear functions. The characterizations of the solution set of a convex and nondifferentiable but generalized pseudolinear program are obtained. The results of this paper extend various results for pseudolinear functions, pseudoinvex functions and pseudolinear functions, and also for pseudoinvex programs, pseudolinear programs and pseudolinear program