50,558 research outputs found
Painleve property and the first integrals of nonlinear differential equations
Link between the Painleve property and the first integrals of nonlinear
ordinary differential equations in polynomial form is discussed. The form of
the first integrals of the nonlinear differential equations is shown to
determine by the values of the Fuchs indices. Taking this idea into
consideration we present the algorithm to look for the first integrals of the
nonlinear differential equations in the polynomial form. The first integrals of
five nonlinear ordinary differential equations are found. The general solution
of one of the fourth ordinary differential equations is given.Comment: 22 page
Explicit Construction of First Integrals with Quasi-monomial Terms from the Painlev\'{e} Series
The Painlev\'{e} and weak Painlev\'{e} conjectures have been used widely to
identify new integrable nonlinear dynamical systems. For a system which passes
the Painlev\'{e} test, the calculation of the integrals relies on a variety of
methods which are independent from Painlev\'{e} analysis. The present paper
proposes an explicit algorithm to build first integrals of a dynamical system,
expressed as `quasi-polynomial' functions, from the information provided solely
by the Painlev\'{e} - Laurent series solutions of a system of ODEs.
Restrictions on the number and form of quasi-monomial terms appearing in a
quasi-polynomial integral are obtained by an application of a theorem by
Yoshida (1983). The integrals are obtained by a proper balancing of the
coefficients in a quasi-polynomial function selected as initial ansatz for the
integral, so that all dependence on powers of the time is
eliminated. Both right and left Painlev\'{e} series are useful in the method.
Alternatively, the method can be used to show the non-existence of a
quasi-polynomial first integral. Examples from specific dynamical systems are
given.Comment: 16 pages, 0 figure
A class of finite dimensional optimal nonlinear estimators
Finite dimensional optimal nonlinear state estimators are derived for bilinear systems evolving on nilpotent and solvable Lie groups. These results are extended to other classes of systems involving polynomial nonlinearities. The concepts of exact differentials and path-independent integrals are used to derive optimal finite dimensional estimators for a further class of nonlinear systems
Non-existence criteria for Laurent polynomial first integrals
In this paper we derived some simple criteria for non-existence and partial non-existence Laurent polynomial first integrals for a general nonlinear systems of ordinary differential equations , with . We show that if the eigenvalues of the Jacobi matrix of the vector field are -independent, then the system has no nontrivial Laurent polynomial integrals
The impact of the diagonals of polynomial forms on limit theorems with long memory
We start with an i.i.d. sequence and consider the product of two
polynomial-forms moving averages based on that sequence. The coefficients of
the polynomial forms are asymptotically slowly decaying homogeneous functions
so that these processes have long memory. The product of these two polynomial
forms is a stationary nonlinear process. Our goal is to obtain limit theorems
for the normalized sums of this product process in three cases: exclusion of
the diagonal terms of the polynomial form, inclusion, or the mixed case (one
polynomial form excludes the diagonals while the other one includes them). In
any one of these cases, if the product has long memory, then the limits are
given by Wiener chaos. But the limits in each of the cases are quite different.
If the diagonals are excluded, then the limit is expressed as in the product
formula of two Wiener-It\^{o} integrals. When the diagonals are included, the
limit stochastic integrals are typically due to a single factor of the product,
namely the one with the strongest memory. In the mixed case, the limit
stochastic integral is due to the polynomial form without the diagonals
irrespective of the strength of the memory.Comment: Published at http://dx.doi.org/10.3150/15-BEJ697 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Manifold constrained non-uniformly elliptic problems
We consider the problem of minimizing variational integrals defined on
\cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The
main novelty is that the underlying energy features a non-uniformly elliptic
integrand exhibiting different polynomial growth conditions and no homogeneity.
We develop a few intrinsic methods aimed at proving partial regularity of
minima and providing techniques for treating larger classes of similar
constrained non-uniformly elliptic variational problems. In order to give
estimates for the singular sets we use a general family of Hausdorff type
measures following the local geometry of the integrand. A suitable comparison
is provided with respect to the naturally associated capacities.Comment: 50 page
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