98,355 research outputs found
On Algebraic Approach in Quadratic Systems
When considering friction or resistance, many physical processes are mathematically simulated by quadratic systems of ODEs or discrete quadratic dynamical systems. Probably the most important problem when such systems are applied in engineering is the stability of critical points and (non)chaotic dynamics. In this paper we consider homogeneous quadratic systems via the so-called Markus approach. We use the one-to-one correspondence between homogeneous quadratic dynamical systems and algebra which was originally introduced by Markus in (1960). We resume some connections between the dynamics of the quadratic systems and (algebraic) properties of the corresponding algebras. We consider some general connections and the influence of power-associativity in the corresponding quadratic system
A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions
Combinig the harmonic balance method (HBM) and a continuation method is a
well-known technique to follow the periodic solutions of dynamical systems when
a control parameter is varied. However, since deriving the algebraic system
containing the Fourier coefficients can be a highly cumbersome procedure, the
classical HBM is often limited to polynomial (quadratic and cubic)
nonlinearities and/or a few harmonics. Several variations on the classical HBM,
such as the incremental HBM or the alternating frequency/time domain HBM, have
been presented in the literature to overcome this shortcoming. Here, we present
an alternative approach that can be applied to a very large class of dynamical
systems (autonomous or forced) with smooth equations. The main idea is to
systematically recast the dynamical system in quadratic polynomial form before
applying the HBM. Once the equations have been rendered quadratic, it becomes
obvious to derive the algebraic system and solve it by the so-called ANM
continuation technique. Several classical examples are presented to illustrate
the use of this numerical approach.Comment: PACS numbers: 02.30.Mv, 02.30.Nw, 02.30.Px, 02.60.-x, 02.70.-
Algebraic structures and Hamiltonians from the equivalence classes of 2D conformal algebras
The construction of superintegrable systems based on Lie algebras and their
universal enveloping algebras has been widely studied over the past decades.
However, most constructions rely on explicit differential operator realisations
and Marsden-Weinstein reductions. In this paper, we develop an algebraic
approach based on the subalgebras of the 2D conformal algebra
. This allows us to classify the centralisers of the
enveloping algebra of the conformal algebra and construct the corresponding
Hamiltonians with integrals in algebraic form. It is found that the symmetry
algebras underlying these algebraic Hamiltonians are six-dimensional quadratic
algebras. The Berezin brackets and commutation relations of the quadratic
algebraic structures are closed without relying on explicit realisations or
representations. We also give the Casimir invariants of the symmetry algebras.
Our approach provides algebraic perspectives for the recent work by Fordy and
Huang on the construction of superintegrable systems in the Darboux spaces
Constructing Involutive Tableaux with Guillemin Normal Form
Involutivity is the algebraic property that guarantees solutions to an
analytic and torsion-free exterior differential system or partial differential
equation via the Cartan-K\"ahler theorem. Guillemin normal form establishes
that the prolonged symbol of an involutive system admits a commutativity
property on certain subspaces of the prolonged tableau. This article examines
Guillemin normal form in detail, aiming at a more systematic approach to
classifying involutive systems. The main result is an explicit quadratic
condition for involutivity of the type suggested but not completed in Chapter
IV, \S 5 of the book Exterior Differential Systems by Bryant, Chern, Gardner,
Goldschmidt, and Griffiths. This condition enhances Guillemin normal form and
characterizes involutive tableaux.Comment: This article co-evolved with "Degeneracy of the Characteristic
Variety," arXiv:1410.6947 and most notation is shared. However, be aware that
the meaning of the indices i,j,k,l and the space Y is not the same between
these article
Feedback Nash Equilibria for Descriptor Differential Games Using Matrix Projectors
In this article we address the problem of finding feedback Nash equilibria for linear quadratic differential games defined on descriptor systems. First, we decouple the dynamic and algebraic parts of a descriptor system using canonical projectors. We discuss the effects of feedback on the behavior of the descriptor system. We derive necessary and sufficient conditions for the existence of the feedback Nash equilibria for index 1 descriptor systems and show that there exist many informationally non-unique equilibria corresponding to a single solution of the game. Further, for descriptor systems with index greater than 1, we give a regularization based approach and discuss the associated drawbacks.
Long-range effects on superdiffusive solitons in anharmonic chains
Studies on thermal diffusion of lattice solitons in Fermi-Pasta-Ulam
(FPU)-like lattices were recently generalized to the case of dispersive
long-range interactions (LRI) of the Kac-Baker form. The position variance of
the soliton shows a stronger than linear time-dependence (superdiffusion) as
found earlier for lattice solitons on FPU chains with nearest neighbour
interactions (NNI). In contrast to the NNI case where the position variance at
moderate soliton velocities has a considerable linear time-dependence (normal
diffusion), the solitons with LRI are dominated by a superdiffusive mechanism
where the position variance mainly depends quadratic and cubic on time. Since
the superdiffusion seems to be generic for nontopological solitons, we want to
illuminate the role of the soliton shape on the superdiffusive mechanism.
Therefore, we concentrate on a FPU-like lattice with a certain class of
power-law long-range interactions where the solitons have algebraic tails
instead of exponential tails in the case of FPU-type interactions (with or
without Kac-Baker LRI). A collective variable (CV) approach in the continuum
approximation of the system leads to stochastic integro-differential equations
which can be reduced to Langevin-type equations for the CV position and width.
We are able to derive an analytical result for the soliton diffusion which
agrees well with the simulations of the discrete system. Despite of
structurally similar Langevin systems for the two soliton types, the algebraic
solitons reach the superdiffusive long-time limit with a characteristic
time-dependence much faster than exponential solitons. The soliton
shape determines the diffusion constant in the long-time limit that is
approximately a factor of smaller for algebraic solitons.Comment: 7 figure
One some algebraic formulations within enveloping algebras related to superintegrability
We report on some recent purely algebraic approaches to superintegrable systems from the perspective of subspaces of commuting polynomials in the enveloping algebras of Lie algebras that generate quadratic (and eventually higher-order) algebras. In this context, two algebraic formulations are possible; a first one strongly dependent on representation theory, as well as a second formal approach that focuses on the explicit construction within commutants of algebraic integrals for appropriate algebraic Hamiltonians defined in terms of suitable subalgebras. The potential use in this context of the notion of virtual copies of Lie algebras is briefly commented
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