Algebraic properties of some quadratic dynamical systems

We study the interrelation between the algebraic properties and the dynamical behavior of elementary cellular automata and other deterministic quadratic dynamical systems. The examples presented indicate that complex behavior, often observed in these systems, can be connected to the nonassociativity of algebraic structures on which systems evolve.     

Algebraic properties of quadratic quaternary rings

A quadratic quaternary ring (abbreviated QQR) was defined for generalized quadrangles [1] as the analogue of a planar ternary ring (abbreviated PTR) for projective planes. To be useful, this algebraic structure should have nice properties whenever the generalized quadrangle has a large automorphism group. This interaction is described in this paper. Both authors are supported by the National Fund of Scientific Research Belgium (NFWO).     

Relationships among some chaotic properties of non-autonomous discrete dynamical systems

This paper is concerned with relationships among some chaotic properties of non-autonomous discrete dynamical systems. Some relationships among weak mixing, topologically weak mixing, generic chaos, dense chaos, and sensitivity are investigated. In addition, some equivalent conditions of sensitivity are given and the relationships between sensitivity and Li–Yorke sensitivity are obtained. These results generalize some existing results of autonomous discrete systems, some of which relax the corresponding conditions.     

Finite-dimensional time-invariant linear dynamical systems: Algebraic theory

Willems' approach to dynamical systems without a priori distinguishing between inputs and outputs is accepted, and with this as a starting point, new linear dynamical systems are introduced and studied. It is proved in particular that (in the complex case) the set of isomorphism classes of completely observable (or completely reachable) linear systems with given input and output numbers and McMillan degree, has a natural structure of a compact algebraic variety. This variety is closely connected to the one constructed by Hazewinkel using the Rosenbrock linear systems % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]=Ax+Bu, v=Cx+D(·)u, where D is a polynomial matrix, and may be regarded as the most natural compactification of it. (The connection is very similar to that of Grass_m,mx+p() and Mat_m.p(). Input/output linear systems, i.e. linear systems equipped with an extra structure which distinguishes input and output signals, are also considered. It is shown that each of them may be represented by the equations K% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]+L% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeyDayaaca% aaaa!35D8!\[{\rm{\dot u}}\]=Fx+Gu, v=Hx+Ju (det(K–sF)0). Such systems clearly contain the so-called generalized linear systems. They also contain the Rosenbrock linear systems mentioned above and essentially coincide with them.     

Algebraic limit cycles of degree 4 for quadratic systems

Yablonskii (Differential Equations 2 (1996) 335) and Filipstov (Differential Equations 9 (1973) 983) proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems. It was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems. Chavarriga (A new example of a quartic algebraic limit cycle for quadratic sytems, Universitat de Lleida, Preprint 1999) found a third family of this kind of algebraic limit cycles. Here, we prove that quadratic systems have exactly four different families of algebraic limit cycles. The proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields.     

Algebraic approximation of attractors of dynamical systems on manifolds

We consider an algebraic approximation of attractors of dynamical systems defined on a Euclidean space, a flat cylinder, and a projective space. We present the Foias-Temam method for the approximation of attractors of systems with continuous time and apply it to the investigation of Lorenz and Rössler systems. A modification of this method for systems with discrete time is also described. We consider elements of the generalization of the method to the case of an arbitrary Riemannian analytic manifold.     

On some qualitative properties of monotone linear extensions of dynamical systems

We study monotone linear extensions of dynamical systems. The problem of the existence of invariant manifolds and exponential separation is investigated for linear extensions that preserve the order structure. We also study the relationship between the monotonicity of linear extensions and the existence (weak regularity, quasiregularity) of bounded solutions of inhomogeneous linear extensions.     

On some variants of dynamical systems

A dynamical system motivated by discrete physics is studied. Fuzzy dynamical systems are used to study fuzzy discrete replicator dynamics of hawk–dove (HD) and prisoner's dilemma (PD) games. New solutions are obtained. Finally a preliminary study for fuzzy predator–prey model is studied and a new equilibrium is found.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

Algebraic K-theory of quadratic forms

The survey is devoted to the algebraic K-theory of quadratic forms over rings of general nature and over some special rings. Applications of this theory to algebraic topology are also considered.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 24, pp. 121–194, 1986.     

Algebraic and geometric properties of quadratic Hamiltonians determined by sectional operators

Following the terminology introduced by V. V. Trofimov and A. T. Fomenko, we say that a self-adjoint operator $\varphi :\mathfrak{g}* \to \mathfrak{g}$ is sectional if it satisfies the identity ad _?x ^* a = ad _β ^* x, $x \in \mathfrak{g}*$ , where $\mathfrak{g}$ is a finite-dimensional Lie algebra and $a \in \mathfrak{g}*$ and $\beta \in \mathfrak{g}$ are fixed elements. In the case of a semisimple Lie algebra $\mathfrak{g}$ , the above identity takes the form [?x, a] = [β, x] and naturally arises in the theory of integrable systems and differential geometry (namely, in the dynamics of n-dimensional rigid bodies, the argument shift method, and the classification of projectively equivalent Riemannian metrics). This paper studies general properties of sectional operators, in particular, integrability and the bi-Hamiltonian property for the corresponding Euler equation $\dot x = ad_{\varphi x}^* x$ .     

Stochastic stability in some chaotic dynamical systems

For a certain class of piecewise monotonic transformations it is shown using a spectral decomposition of the Perron-Frobenius-operator ofT that invariant measures depend continuously on 3 types of perturbations: 1) deterministic perturbations, 2) stochastic perturbations, 3) randomly occuring deterministic perturbations. The topology on the space of perturbed transformations is derived from a metric on the space of Perron-Frobenius-operators.With 1 Figure     

Algebraic properties of some new vector-valued rational interpolants

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory, 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F(z), where , were proposed, and some of their properties were studied. In this work, after modifying their definition slightly, we continue the study of these interpolation procedures. We show that the interpolants produced via these procedures are unique in some sense and that they are symmetric functions of the points of interpolation. We also show that, under the conditions that guarantee uniqueness, they also reproduce F(z) in case F(z) is a rational function.     

Application of chaos degree to some dynamical systems

Chaos degree defined through two complexities in information dynamics is applied to some deterministic dynamical models. It is shown that this degree well describes the chaotic feature of the models.     

Finite-dimensional discrete linear stochastic accelerated-time systems and their application to quadratic stochastic dynamical systems

The limiting behavior of the trajectories {x ⁽ⁿ⁾ } of linear discrete stochastic systems of the form (K, P ^an+b ) _nN , whereK is the standard simplex in ^N ,P: ^{N N} is a linear operator,PK K,a ft,b ,a+b>0, is described. An application to a class of quadratic stochastic dynamical systems is considered.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 709–718, May, 1996.