Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

Reduction of partially invariant submodels of rank 3 defect 1 to invariant submodels

2018 ◽  
Vol 13 (3) ◽  
pp. 59-63 ◽  
Author(s):  
D.T. Siraeva
Keyword(s):  
Equation Of State ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Gas Dynamics ◽  
Hydrodynamic Type ◽  
System Of Equations ◽  
Two Dimensional ◽  
Equations Of Gas Dynamics ◽  
Entropy Functions

Equations of hydrodynamic type with the equation of state in the form of pressure separated into a sum of density and entropy functions are considered. Such a system of equations admits a twelve-dimensional Lie algebra. In the case of the equation of state of the general form, the equations of gas dynamics admit an eleven-dimensional Lie algebra. For both Lie algebras the optimal systems of non-similar subalgebras are constructed. In this paper two partially invariant submodels of rank 3 defect 1 are constructed for two-dimensional subalgebras of the twelve-dimensional Lie algebra. The reduction of the constructed submodels to invariant submodels of eleven-dimensional and twelve-dimensional Lie algebras is proved.

Zero-dimensional covers of finite dimensional dynamical systems

1995 ◽  
Vol 15 (5) ◽  
pp. 939-950 ◽  
Author(s):  
John Kulesza
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Points ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

AbstractIf (X, f) is a compact metric, finite-dimensional dynamical system with a zero-dimensional set of periodic points, then there is a zero-dimensional compact metric dynamical system (C, g) and a finite-to-one (in fact, at most (n + l)n-to-one) surjection h: C → X such that h o g = f o h. An example shows that the requirement on the set of periodic points is necessary.

scholarly journals MAD SUBALGEBRAS AND LIE SUBALGEBRAS OF AN ENVELOPING ALGEBRA

2010 ◽  
Vol 82 (3) ◽  
pp. 401-423
Author(s):  
XIN TANG
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional ◽  
Base Field ◽  
Inner Derivation ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Characterization Theorems

AbstractLet 𝒰(𝔯(1)) denote the enveloping algebra of the two-dimensional nonabelian Lie algebra 𝔯(1) over a base field 𝕂. We study the maximal abelian ad-nilpotent (mad) associative subalgebras and finite-dimensional Lie subalgebras of 𝒰(𝔯(1)). We first prove that the set of noncentral elements of 𝒰(𝔯(1)) admits the Dixmier partition, 𝒰(𝔯(1))−𝕂=⋃ 5i=1Δi, and establish characterization theorems for elements in Δi, i=1,3,4. Then we determine the elements in Δi, i=1,3 , and describe the eigenvalues for the inner derivation ad Bx,x∈Δi, i=3,4 . We also derive other useful results for elements in Δi, i=2,3,4,5 . As an application, we find all framed mad subalgebras of 𝒰(𝔯(1)) and determine all finite-dimensional nonabelian Lie algebras that can be realized as Lie subalgebras of 𝒰(𝔯(1)) . We also study the realizations of the Lie algebra 𝔯(1) in 𝒰(𝔯(1)) in detail.

scholarly journals STABILITY ANALYSIS WITH APPLICATIONS OF A TWO-DIMENSIONAL DYNAMICAL SYSTEM ARISING FROM A STOCHASTIC MODEL FOR AN ASSET MARKET

2011 ◽  
Vol 11 (04) ◽  
pp. 715-752
Author(s):  
VLADIMIR BELITSKY ◽  
ANTONIO LUIZ PEREIRA ◽  
FERNANDO PIGEARD DE ALMEIDA PRADO
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Asset Price ◽  
Excess Demand ◽  
Market Model ◽  
Asset Market ◽  
Two Dimensional ◽  
Equilibrium Solutions ◽  
Dimensional Dynamical System ◽  
Demand Fluctuations

We analyze the stability properties of equilibrium solutions and periodicity of orbits in a two-dimensional dynamical system whose orbits mimic the evolution of the price of an asset and the excess demand for that asset. The construction of the system is grounded upon a heterogeneous interacting agent model for a single risky asset market. An advantage of this construction procedure is that the resulting dynamical system becomes a macroscopic market model which mirrors the market quantities and qualities that would typically be taken into account solely at the microscopic level of modeling. The system's parameters correspond to: (a) the proportion of speculators in a market; (b) the traders' speculative trend; (c) the degree of heterogeneity of idiosyncratic evaluations of the market agents with respect to the asset's fundamental value; and (d) the strength of the feedback of the population excess demand on the asset price update increment. This correspondence allows us to employ our results in order to infer plausible causes for the emergence of price and demand fluctuations in a real asset market. The employment of dynamical systems for studying evolution of stochastic models of socio-economic phenomena is quite usual in the area of heterogeneous interacting agent models. However, in the vast majority of the cases present in the literature, these dynamical systems are one-dimensional. Our work is among the few in the area that construct and study analytically a two-dimensional dynamical system and apply it for explanation of socio-economic phenomena.

BIANCHI COSMOLOGIES AS DYNAMICAL SYSTEMS WITHOUT THE CONSTRAINT

2000 ◽  
Vol 15 (17) ◽  
pp. 2771-2791
Author(s):  
MAREK SZYDŁOWSKI ◽  
ADAM KRAWIEC
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Dynamical System ◽  
Phase Portraits ◽  
Finite Domain ◽  
Hamiltonian Constraint ◽  
Two Dimensional ◽  
Class A ◽  
Dimensional Dynamical System ◽  
Analysis Of Dynamics

The Bianchi class A cosmology is treated as a nonlinear dynamical system. In the new variables in which Hamiltonian constraint is solved algebraically, the Bianchi class A model assumes the form of autonomous dynamical system in ℝ4 with polynomial form of vector field. It is proposed that the dimension of minimum reduced phase spaces of unconstrained autonomous systems be treated as a measure of generality of solution. The behavior of these models is studied in terms of qualitative analysis of differential equations. It is shown that the more general Bianchi IX and Bianchi VIII models (called Mixmaster models) can be presented as four-dimensional. We argue that the reduced Mixmaster dynamical systems are chaotic in the same sense as the original ones. The Bianchi I and Bianchi II world models are described by one-dimensional and two-dimensional systems, respectively. We also study dynamics of Bianchi VI0 and Bianchi VII0 models as a three-dimensional dynamical system. For two-dimensional dynamical system, the phase portraits are constructed with the Poincaré sphere which allows the analysis of dynamics both in finite domain and at infinity. For the last class of models we find an invariant submanifold on which systems are analyzed in details.

scholarly journals Structure of n-Lie Algebras with Involutive Derivations

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Ruipu Bai ◽  
Shuai Hou ◽  
Yansha Gao
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Two Dimensional

We study the structure of n-Lie algebras with involutive derivations for n≥2. We obtain that a 3-Lie algebra A is a two-dimensional extension of Lie algebras if and only if there is an involutive derivation D on A=A1  ∔  A-1 such that dim A1=2 or dim A-1=2, where A1 and A-1 are subspaces of A with eigenvalues 1 and -1, respectively. We show that there does not exist involutive derivations on nonabelian n-Lie algebras with n=2s for s≥1. We also prove that if A is a (2s+2)-dimensional (2s+1)-Lie algebra with dim A1=r, then there are involutive derivations on A if and only if r is even, or r satisfies 1≤r≤s+2. We discuss also the existence of involutive derivations on (2s+3)-dimensional (2s+1)-Lie algebras.

scholarly journals Vector bundles associated to Lie algebras

2016 ◽  
Vol 2016 (716) ◽  
Author(s):  
Jon F. Carlson ◽  
Eric M. Friedlander ◽  
Julia Pevtsova
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Vector Bundles ◽  
Coherent Sheaves ◽  
Restricted Lie Algebra ◽  
Finite Dimensional

AbstractWe introduce and investigate a functorial construction which associates coherent sheaves to finite dimensional (restricted) representations of a restricted Lie algebra

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