Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

We consider natural complex Hamiltonian systems with n degrees of freedom given by a Hamiltonian function which is a sum of the standard kinetic energy and a homogeneous polynomial potential V of degree k > 2. The well known Morales-Ramis theorem gives the strongest known necessary conditions for the Liouville integrability of such systems. It states that for each k there exists an explicitly known infinite set ⊂ ℚ such that if the system is integrable, then all eigenvalues of the Hessian matrix V″(d) calculated at a non-zero d ∈ ℂ ⁿ satisfying V′(d) = d, belong to . The aim of this paper is, among others, to sharpen this result. Under certain genericity assumption concerning V we prove the following fact. For each k and n there exists a finite set such that if the system is integrable, then all eigenvalues of the Hessian matrix V″(d) belong to . We give an algorithm which allows to find sets . We applied this results for the case n = k = 3 and we found all integrable potentials satisfying the genericity assumption. Among them several are new and they are integrable in a highly non-trivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta.      

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases

In this paper the problem of classification of integrable natural Hamiltonian systems with n degrees of freedom given by a Hamilton function, which is the sum of the standard kinetic energy and a homogeneous polynomial potential V of degree k > 2, is investigated. It is assumed that the potential is not generic. Except for some particular cases a potential V is not generic if it admits a nonzero solution of equation V′(d) = 0. The existence of such a solution gives very strong integrability obstructions obtained in the frame of the Morales-Ramis theory. This theory also gives additional integrability obstructions which have the form of restrictions imposed on the eigenvalues (λ ₁, …, λ _n ) of the Hessian matrix V″(d) calculated at a nonzero d ∈ ℂ ⁿ satisfying V′(d) = d. In our previous work we showed that for generic potentials some universal relations between (λ ₁, …, λ _n ) calculated at various solutions of V′ (d) = d exist. These relations allow one to prove that the number of potentials satisfying the necessary conditions for the integrability is finite. The main aim of this paper was to show that relations of such forms also exist for nongeneric potentials. We show their existence and derive them for the case n = k = 3 applying the multivariable residue calculus. We demonstrate the strength of the results analyzing in details the nongeneric cases for n = k = 3. Our analysis covers all the possibilities and we distinguish those cases where known methods are too weak to decide if the potential is integrable or not. Moreover, for n = k = 3, thanks to this analysis, a three-parameter family of potentials integrable or superintegrable with additional polynomial first integrals which seemingly can be of an arbitrarily high degree with respect to the momenta was distinguished.      

Limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 4 via the averaging method

By using the averaging method, we study the limit cycles for a class of quartic polynomial differential systems as well as their global shape in the plane. More specifically, we analyze the global shape of limit cycles bifurcating from a Hopf bifurcation and also from periodic orbits with linear center , . The perturbation of these systems is made inside the class of quartic polynomial differential systems without quadratic and cubic terms.     

Strong isochronism of polynomial reversible dynamical systems on the plane with homogeneous nonlinearities of degree 4

We find the maximum order and initial polar angle of strongly isochronous two-dimensional polynomial reversible systems with homogeneous nonlinearities of the fourth degree.     

Algebraic limit cycles of degree 4 for quadratic systems

Yablonskii (Differential Equations 2 (1996) 335) and Filipstov (Differential Equations 9 (1973) 983) proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems. It was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems. Chavarriga (A new example of a quartic algebraic limit cycle for quadratic sytems, Universitat de Lleida, Preprint 1999) found a third family of this kind of algebraic limit cycles. Here, we prove that quadratic systems have exactly four different families of algebraic limit cycles. The proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields.     

Characterization of isolated homogeneous hypersurface singularities in C4

Let V be a hypersurface with an isolated singularity at the origin in Cn+1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial.For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bi.er than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.     

Origin and infinity manifolds for mechanical systems with homogeneous potentials

We study manifolds describing the behavior of motions close to the origin and at infinity of configuration space, for mechanical systems with homogeneous potentials. We find an inversion between these behaviors when the sign of the degree of homogeneity is changed. In some cases, the blow up equations can be written in canonical form, by first reducing to a contact structure. A motivation for the use of blow-up techniques is given, and some examples are studied in detail.Research partially supported by CONACyT (Mexico), under grants PCCBNAL 790178 and PCCBBNA 022553.Member of CIFMA (Mexico). On sabbatical leave at the University of Barcelona during the year 1987–88.     

Characterization of isolated homogeneous hypersurface singularities in ?4

Let V be a hypersurface with an isolated singularity at the origin in ℂ ⁿ⁺¹. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface signularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau’s theorem remains true for singularities with geometric genus equal to zero. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

关于广义齐性Cochrane和的加权均值

研究了广义齐性Cochrane和的一些加权均值,并利用不完全区间上特征和的性质,Dirichlet函数的均值估计以及周期Bernoulli多项式的性质,得到一些较强的渐近公式．     

The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree

One of the considerable discussions in data interpolation is to find the optimal number of data which minimizes the error of the interpolation polynomial. In this paper, first the theorems corresponding to the equidistant nodes and the roots of the Chebyshev polynomials are proved in order to estimate the accuracy of the interpolation polynomial, when the number of data increases. Based on these theorems, then we show that by using a perturbation method based on the CESTAC method, it is possible to find the optimal degree of the interpolation polynomial. The results of numerical experiments are presented.     

On the Detection of Limit Cycles by the Variational Velocity Method

This paper summarizes, clarifies, and corrects some aspects of the variational velocity methodfor the detection of limit cycles. After definitions and statements of the most important theoremsassociated with this method, some aspects of the proof of the main theorem are corrected andreworked. An example from the original paper in Acta Appl. Math. 48 (1997),13–32, is then discussed and criticized. Finally, the limitations of this method are discussed,especially as it applies to systems involving multiple limit cycles (and therefore as it applies toHilberts XVIth Problem).     

Polynomial first integrals for weight-homogeneous planar polynomial differential systems of weight degree 3

Our main result is the classification of all weight-homogeneous planar polynomial differential systems of weight degree 3 having a polynomial first integral.     

Characterization of isolated homogeneous hypersurface singularities in C~4 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Let V be a hypersurface with an isolated singularity at the origin in Cn 1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface singularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau's theorem remains true for singularities with geometric genus equal to zero.     

On stability of switched homogeneous nonlinear systems

In this paper, the problem of stability of switched homogeneous systems is addressed. First of all, if there is a quadratic Lyapunov function such that nonlinear homogeneous systems are asymptotically stable, a matrix Lyapunov-like equation is obtained for a stable nonlinear homogeneous system using semi-tensor product of matrices, and Lyapunov equation of linear system is just its particular case. Following the previous results, a sufficient condition is obtained for stability of switched nonlinear homogeneous systems, and a switching law is designed by partition of state space. In particular, a constructive approach is provided to avoid chattering phenomena which is caused by the switching rule. Then for planar switched homogeneous systems, an LMI approach to stability of planar switched homogeneous systems is presented. Similar to the condition for linear systems, the LMI-type condition is easily verifiable. An example is given to illustrate that candidate common Lyapunov function is a key point for design of switching law.     

On the dynamics of generic non-Abelian free actions

We investigate some global generic properties of the dynamics associated to non-Abelian free actions in certain special cases. The main properties considered in this paper are related to the existence of dense orbits, to ergodicity and to topological rigidity. We first deal with them in the case of conservative homeomorphisms of a manifold and C ¹-diffeomorphisms of a surface. Groups of analytic diffeomorphisms of a manifold which, in addition, contain a Morse-Smale element and possess a generating set close to the identity are considered as well. From our discussion we also derive the existence of a rigidity phenomenon for groups of skew-products which is opposed to the phenomenon present in Furstenbergs celebrated example of a minimal diffeomorphism that is not ergodic (cf. [Ma]).     

Classification of the centers, their cyclicity and isochronicity for the generalized quadratic polynomial differential systems

In this paper we classify the centers, the cyclicity of their Hopf bifurcation and the isochronicity of the polynomial differential systems in R² of degree d that in complex notation z=x+iy can be written as

     

Liouvillian first integrals of quadratic-linear polynomial differential systems

For a large class of quadratic-linear polynomial differential systems with a unique singular point at the origin having non-zero eigenvalues, we classify the ones which have a Liouvillian first integral, and we provide the explicit expression of them.