Local first integrals of differential systems and diffeomorphisms

In this paper using theory of linear operators and normal forms we generalize a result of Poincaré [11] about the non-existence of local first integrals for systems of differential equations in a neighbourhood of a singular point. As an application of the generalized result, and under more weak conditions we obtain a result of Furta [8] about local first integrals of semi-quasi-homogeneous systems. Moreover, for diffeomorphisms and periodic differential systems we give definitions of their first integrals, and generalize the previous results about systems of differential equations to diffeomorphisms in a neighbourhood of a fixed point and to periodic differential systems in a neighbourhood of a constant solution.     

Formal first integrals for periodic systems

In the present paper, we give an elementary proof for the result of Li et al. (2003) [6] about nonexistence of formal first integrals for periodic systems in a neighborhood of a constant solution. Moreover, we present a criterion about partial existence of formal first integrals for the periodic system, by using the Floquet's theory.     

First integrals of local analytic differential systems

We investigate formal and analytic first integrals of local analytic ordinary differential equations near a stationary point. A natural approach is via the Poincaré–Dulac normal forms: If there exists a formal first integral for a system in normal form then it is also a first integral for the semisimple part of the linearization, which may be seen as “conserved” by the normal form. We discuss the maximal setting in which all such first integrals are conserved, and show that all first integrals are conserved for certain classes of reversible systems. Moreover we investigate the case of linearization with zero eigenvalues, and we consider a three-dimensional generalization of the quadratic Dulac–Frommer center problem.     

Non-commutative Hopf algebra of formal diffeomorphisms

This paper deals with two Hopf algebras which are the non-commutative analogues of two different groups of formal power series. The first group is the set of invertible series with the group law being multiplication of series, while the second is the set of formal diffeomorphisms with the group law being a composition of series. The motivation to introduce these Hopf algebras comes from the study of formal series with non-commutative coefficients. Invertible series with non-commutative coefficients still form a group, and we interpret the corresponding new non-commutative Hopf algebra as an alternative to the natural Hopf algebra given by the co-ordinate ring of the group, which has the advantage of being functorial in the algebra of coefficients. For the formal diffeomorphisms with non-commutative coefficients, this interpretation fails, because in this case the composition is not associative anymore. However, we show that for the dual non-commutative algebra there exists a natural co-associative co-product defining a non-commutative Hopf algebra. Moreover, we give an explicit formula for the antipode, which represents a non-commutative version of the Lagrange inversion formula, and we show that its coefficients are related to planar binary trees. Then we extend these results to the semi-direct co-product of the previous Hopf algebras, and to series in several variables. Finally, we show how the non-commutative Hopf algebras of formal series are related to some renormalization Hopf algebras, which are combinatorial Hopf algebras motivated by the renormalization procedure in quantum field theory, and to the renormalization functor given by the double-tensor algebra on a bi-algebra.     

Embedding smooth and formal diffeomorphisms through the Jordan-Chevalley decomposition

In [Xiang Zhang, The embedding flows of C^∞ hyperbolic diffeomorphisms, J. Differential Equations 250 (5) (2011) 2283-2298] Zhang proved that any local smooth hyperbolic diffeomorphism whose eigenvalues are weakly nonresonant is embedded in the flow of a smooth vector field. We present a new and more conceptual proof of such result using the Jordan-Chevalley decomposition in algebraic groups and the properties of the exponential operator.We characterize the hyperbolic smooth (resp. formal) diffeomorphisms that are embedded in a smooth (resp. formal) flow. We introduce a criterion showing that the presence of weak resonances for a diffeomorphism plus two natural conditions imply that it is not embeddable. This solves a conjecture of Zhang. The criterion is optimal, we provide a method to construct embeddable diffeomorphisms with weak resonances if we remove any of the conditions.     

Orbifolds as stratified diffeologies

We discuss general properties of stratified spaces in diffeology. This leads to a formal framework for the theory of stratifications. In particular, we consider the Klein stratification of diffeological orbifolds, defined by the action of local diffeomorphisms. We show that it is a standard stratification in the sense that the partition of the space into orbits of local diffeomorphisms is locally finite (for orbifolds with locally finite atlases), it satisfies the frontier condition and the orbits are locally closed manifolds.     

Extension of formal conjugations between diffeomorphisms

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ? ⁿ . More precisely, we are interested in the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms ??, ?? such that every formal conjugation $\hat \sigma$ (i.e. $\eta \circ \hat \sigma = \hat \sigma \circ \phi$ ) does not extend to the fixed points set Fix(??) of ??, meaning that it is not transversally formal (or semi-convergent) along Fix(??). We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.     

Local Darboux first integrals of analytic differential systems

We discuss local and formal Darboux first integrals of analytic differential systems, using the theory of Poincaré–Dulac normal forms, and we study the effect of local Darboux integrability on analytic normalization. Moreover we determine local restrictions on classical Darboux integrability of polynomial systems.     

New partially hyperbolic dynamical systems I

We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms in the past forty years.     

Commutators of contactomorphisms

The group of volume preserving diffeomorphisms, the group of symplectomorphisms and the group of contactomorphisms constitute the classical groups of diffeomorphisms. The first homology groups of the compactly supported identity components of the first two groups have been computed by Thurston and Banyaga, respectively. In this paper we solve the long standing problem on the algebraic structure of the third classical diffeomorphism group, i.e. the contactomorphism group. Namely we show that the compactly supported identity component of the group of contactomorphisms is perfect and simple (if the underlying manifold is connected). The result could be applied in various ways.     

On the nonexistence of rational first integrals for nonlinear systems and semiquasihomogeneous systems

In this paper we first give a necessary condition for general nonlinear systems to have rational first integrals. Then by using the so-called Kowalevsky exponents we present a criterion for nonexistence of rational first integrals for semiquasihomogeneous systems.     

第一积分中值函数

通过上下确界,给出了"第一积分中值函数"的定义,对"第一积分中值函数"的分析性质进行了系统的讨论,证明了"第一积分中值函数"的单调性、可积性、连续性、可导性等分析性质.     

一类具有1:-4共振奇点的复三次Lotka-Volterra系统的可积性条件

对于一类具有1:-4共振奇点的复三次Lotka-Volterra系统,通过前12阶广义奇点量的计算,给出系统可积的充分条件.这些条件通过构造积分因子或形式积分得以证明.     

Asymptotics of unitary and orthogonal matrix integrals

In this paper, we prove that in small parameter regions, arbitrary unitary matrix integrals converge in the large N limit and match their formal expansion. Secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the HCIZ integral. Our convergence result also leads us to new results of smoothness of microstates. We finally generalize our approach to integrals over the orthogonal group.     

Flow equivalence,hyperbolic systems and a new zeta function for flows

We analyze the dynamics of diffeomorphisms in terms of their suspension flows. For many Axion A diffeomorphisms we find simplest representatives in their flow equivalence class and so reduce flow equivalence to conjugacy. The zeta functions of maps in a flow equivalence class are correlated with a zeta function ζ _H for their suspended flow. This zeta function is defined for any flow with only finitely many closed orbits in each homology class, and is proven rational for Axiom A flows. The flow equivalence of Anosov diffeomorphisms is used to relate the spectrum of the induced map on first homology to the existence of fixed points. For Morse-Smale maps, we extend a result of Asimov on the geometric index. Partially supported by MCS 76-08795.     

A representation of solution of stochastic differential equations

We prove that the logarithm of the formal power series, obtained from a stochastic differential equation, is an element in the closure of the Lie algebra generated by vector fields being coefficients of equations. By using this result, we obtain a representation of the solution of stochastic differential equations in terms of Lie brackets and iterated Stratonovich integrals in the algebra of formal power series.     

Integrals of Bessel functions

We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique that we propose is based on the formal reduction of functions in this family to Gaussians.     

Analytical integrability of the Rikitake system

In this paper, we study the first integrals of the Rikitake system $$\dot{x} = -\mu x + yz, \quad \dot{y} = -\mu y + x (z-a), \quad \dot{z} = 1-xy,$$ that can be described by formal power series when the system has singular points.