Ont ∞ quasi-similarity of linear systems

Summary Conti defined t similarity of systems (1)x=tA(t)xand (2)y=B(t)y, and showed that it is an equivalence relation which preserves uniform and strict stability. Here the definition is weakened by imposing less stringent integrability conditions, some in terms of perhaps conditionally convergent improper integrals, on the matrix function relating A and B. The extended relation, t quasi-similarity, is not symmetric or transitive; however, it is shown that if (2) is t quasi-similar to (1)and (1)is uniformly, uniformly asymptotically, or strictly stable, then so is (2).Results are also given concerning linear asymptotic equilibrium of (2)in the case where (1)is stricly stable or has linear asymptotic equilibrium.     

Enclosings of λ-fold 4-cycle systems

In this paper we solve the problem of enclosing a λ-fold 4-cycle system of order v into a (λ + m)-fold 4-cycle system of order v + u for all m > 0 and u ≥ 1. An ingredient is constructed that is of interest on its own right, namely the problem of finding equitable partial 4-cycle systems of λ K _v . This supplementary solution builds on a result of Raines and Staniszlo.     

Relaxation Phenomenon of the P—systems

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Sampled-data control of Lur’e systems

The main purpose of this paper is to fill a gap in the literature concerning the problem of designing a sampled-data control law for continuous-time Lur’e systems. The goal is to design a state feedback sampled-data control for this class of nonlinear systems preserving global asymptotic stability and minimizing a guaranteed quadratic cost. The main challenge towards the solution of the proposed problem is to handle this class of nonlinear system in order to propose less conservative design conditions expressed through differential linear matrix inequalities - (DLMIs). Bellman’s Principle of Optimality applied together with the Popov–Lyapunov function that emerges from the celebrated Popov Stability Criterion is the key issue to obtain the reported results. Two examples are solved for illustration.     

Dynamics of Hodgkin–Huxley systems revisited

We consider the singularly perturbed Hodgkin–Huxley system subject to Neumann boundary conditions. We construct a family of exponential attractors {?_ε} which is continuous at ε = 0, ε being the parameter of perturbation. Moreover, this continuity result is obtained with respect to a metric independent of ε, compared with all previous results where the metric always depends on ε. In the latter case, one needs to consider more regular function spaces and more smoother absorbing sets. Our results show that we can construct and analyse the stability of exponential attractors in a natural phase-space as it is known for the global attractor. Also, a new proof of the upper semicontinuity of the global attractor 𝒜_ε at ε = 0 is given.     

Small embeddings of partial directed triple systems and partial triple systems with even λ

In 1976, Lindner and Rosa (Ars Combin. 1 (1976), 159–166) showed that a partial triple system with λ > 1 can be embedded in a finite triple system with the same λ. This result is improved in the case when λ is even by embedding a partial triple system on υ symbols in a triple system on t symbols, t ≡ 0,1 (mod 3), for all t >/ 3(λυ² + υ(2 ? λ) + 1). In the process, it is shown that for any λ >/ 1, a partial directed triple system on υ symbols can be embedded in a directed triple system on t symbols, t ≡ 0, 1 (mod 3), for all t ? 6λv² + 6v(1 ? λ) + 3, thus generalizing a result of Hamm (Proceedings, 14th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1983).     

Global solvability of massless Dirac–Maxwell systems

We consider the Cauchy problem for massless Dirac–Maxwell equations on an asymptotically flat background and give a global existence and uniqueness theorem for initial values small in an appropriate weighted Sobolev space. The result can be extended via analogous methods to Dirac–Higgs–Yang–Mills theories.     

Arithmetics of the numbers of orbits of the Fermat–Euler dynamical systems

Given positive integers a and n with (a,n)=1, we consider the Fermat–Euler dynamical system defined by the multiplication by a acting on the set of residues modulo n relatively prime to n. Given an integer M>1, the integers n for which the number of orbits of this dynamical system is a multiple of M form an ideal in the multiplicative semigroup of odd integers. We provide new results on the arithmetical properties of these ideals by using the topological properties of some directed graphs (the monads).      

Transcendental solutions of systems of complex differential equations北大核心CSCD

Using maximum modulus principle, we investigate the problem of the existence of the transcendental meromorphic solutions of system of complex algebraic differential equations, and obtain a result which the system only the algebraic solution under certain conditions. Examples show that our results are sharp. ©, 2015, Chinese Academy of Sciences. All right reserved.     

Stationary solutions of non-autonomous Maxwell–Dirac systems

The Maxwell–Dirac system describes the interaction of a particle with its self-generated electromagnetic field. In this paper, we study the existence of least energy stationary solutions for non-autonomous Maxwell–Dirac system with a nonlinear term in (3+1)$(3+1)$-Minkowski space–time via variational methods. This problem is strongly indefinite and presents a lack of compactness. To overcome these difficulties, we will use the linking and concentration compactness arguments.     

Stabilization of fractional bilinear systems with multiple inputs

In this paper, we study in a constructive way the stabilization problem of fractional bilinear systems with multiple inputs. Using the quadratic Lyapunov functions and some additional hypotheses on the unit sphere, we construct stabilizing feedback laws for the considered fractional bilinear system. A numerical example is given to illustrate the efficiency of the obtained result.     

Finite groups with systems of Σ-embedded subgroups

In recent years,a series of papers about cover-avoiding property of subgroups appeared and all the studies were connected with chief factors of a finite group.However,about the cover-avoiding property of subgroups for non-chief factor,there is no study up to now.The purpose of this paper is to build the theory.Let A be a subgroup of a finite group G and Σ:G0≤G1≤…≤Gn some subgroup series of G.Suppose that for each pair(K,H) such that K is a maximal subgroup of H and G i 1 K < H G i for some i,either A ∩ H = ...     

Stability of mass action reaction–diffusion systems

Results on stability of two types of chemical reactions, one represented by an acyclic graph and the other as a reversible reaction have been extended to the case of reaction–diffusion systems. Lyapunov functions are used as the major method for showing asymptotic stability of spatially homogeneous equilibria. Some examples are considered for illustration.     

Hölder continuity of three types of generalized synchronization manifolds of non-autonomous systems

This paper studies the existence of Hölder continuity of the generalized synchronization (GS) manifold. When the modified response system has an asymptotically stable equilibrium, periodic or quasi-periodic orbit, and chaotic attractor, GS is classified into four types accordingly. The first three types of GS are considered, and based on the Schauder fixed point theorem, sufficient conditions for Hölder continuous GS in the coupled non-autonomous systems are derived and theoretically proved.     

Some properties of Malgrange isomonodromic deformations of linear 2 × 2 systems

We study movable singularities of the Malgrange isomonodromic deformation of a linear differential 2 × 2 system with two irregular singularities of Poincaré rank 1 and with an arbitrary number of Fuchsian singular points.     

On the practical μ-stability of solutions of standard systems with delay

We study the problem of μ-stability of a dynamical system with delay. Conditions of the practical μ-stability are established for the general case and for a quasilinear system. The conditions suggested are illustrated by an example. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 204–213, February, 1999.