44,770 research outputs found
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
Macaulay inverse systems and Cartan-Kahler theorem
During the last months or so we had the opportunity to read two papers trying
to relate the study of Macaulay (1916) inverse systems with the so-called
Riquier (1910)-Janet (1920) initial conditions for the integration of linear
analytic systems of partial differential equations. One paper has been written
by F. Piras (1998) and the other by U. Oberst (2013), both papers being written
in a rather algebraic style though using quite different techniques. It is
however evident that the respective authors, though knowing the computational
works of C. done during the first half of the last century in a way not
intrinsic at all, are not familiar with the formal theory of systems of
ordinary or partial differential equations developped by D.C. Spencer
(1912-2001) and coworkers around 1965 in an intrinsic way, in particular with
its application to the study of differential modules in the framework of
algebraic analysis. As a byproduct, the first purpose of this paper is to
establish a close link between the work done by F. S. Macaulay (1862-1937) on
inverse systems in 1916 and the well-known Cartan-K{\"a}hler theorem (1934).
The second purpose is also to extend the work of Macaulay to the study of
arbitrary linear systems with variable coefficients. The reader will notice how
powerful and elegant is the use of the Spencer operator acting on sections in
this general framework. However, we point out the fact that the literature on
differential modules mostly only refers to a complex analytic structure on
manifolds while the Spencer sequences have been created in order to study any
kind of structure on manifolds defined by a Lie pseudogroup of transformations,
not just only complex analytic ones. Many tricky explicit examples illustrate
the paper, including the ones provided by the two authors quoted but in a quite
different framework
A Geometric Index Reduction Method for Implicit Systems of Differential Algebraic Equations
This paper deals with the index reduction problem for the class of
quasi-regular DAE systems. It is shown that any of these systems can be
transformed to a generically equivalent first order DAE system consisting of a
single purely algebraic (polynomial) equation plus an under-determined ODE
(that is, a semi-explicit DAE system of differentiation index 1) in as many
variables as the order of the input system. This can be done by means of a
Kronecker-type algorithm with bounded complexity
Variable-delay feedback control of unstable steady states in retarded time-delayed systems
We study the stability of unstable steady states in scalar retarded
time-delayed systems subjected to a variable-delay feedback control. The
important aspect of such a control problem is that time-delayed systems are
already infinite-dimensional before the delayed feedback control is turned on.
When the frequency of the modulation is large compared to the system's
dynamics, the analytic approach consists of relating the stability properties
of the resulting variable-delay system with those of an analogous distributed
delay system. Otherwise, the stability domains are obtained by a numerical
integration of the linearized variable-delay system. The analysis shows that
the control domains are significantly larger than those in the usual
time-delayed feedback control, and that the complexity of the domain structure
depends on the form and the frequency of the delay modulation.Comment: 13 pages, 8 figures, RevTeX, accepted for publication in Physical
Review
Reduction Operators of Linear Second-Order Parabolic Equations
The reduction operators, i.e., the operators of nonclassical (conditional)
symmetry, of (1+1)-dimensional second order linear parabolic partial
differential equations and all the possible reductions of these equations to
ordinary differential ones are exhaustively described. This problem proves to
be equivalent, in some sense, to solving the initial equations. The ``no-go''
result is extended to the investigation of point transformations (admissible
transformations, equivalence transformations, Lie symmetries) and Lie
reductions of the determining equations for the nonclassical symmetries.
Transformations linearizing the determining equations are obtained in the
general case and under different additional constraints. A nontrivial example
illustrating applications of reduction operators to finding exact solutions of
equations from the class under consideration is presented. An observed
connection between reduction operators and Darboux transformations is
discussed.Comment: 31 pages, minor misprints are correcte
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