1,043 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
Analytic integrable systems: Analytic normalization and embedding flows
In this paper we mainly study the existence of analytic normalization and the
normal form of finite dimensional complete analytic integrable dynamical
systems. More details, we will prove that any complete analytic integrable
diffeomorphism in with having eigenvalues
not modulus and is locally analytically conjugate to its
normal form. Meanwhile, we also prove that any complete analytic integrable
differential system in with having
nonzero eigenvalues and is locally analytically conjugate to
its normal form. Furthermore we will prove that any complete analytic
integrable diffeomorphism defined on an analytic manifold can be embedded in a
complete analytic integrable flow. We note that parts of our results are the
improvement of Moser's one in {\it Comm. Pure Appl. Math.} 91956,
673--692 and of Poincar\'e's one in {\it Rendiconti del circolo matematico di
Palermo} 51897, 193--239. These results also improve the ones in {\it J.
Diff. Eqns.} 2442008, 1080--1092 in the sense that the linear part of the
systems can be nonhyperbolic, and the one in {\it Math. Res. Lett.}
92002, 217--228 in the way that our paper presents the concrete
expression of the normal form in a restricted case.Comment: 32. Journal of Differential Equations, 201
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