Jaume Llibre

Combinatorial dynamics and entropy in dimension one

Preliminaries: general notation graphs, loops and cycles. Interval maps: the Sharkovskii Theorem maps with the prescribed set of periods forcing relation patterns for interval maps antisymmetry of the forcing relation P-monotone maps and oriented patterns consequences of Theorem 2.6.13 stability of patterns and periods primary patterns extensions characterization of primary oriented patterns more about primary oriented patterns. Circle maps: liftings and degree of circle maps lifted cycles cycles and lifted cycles periods for maps of degree different from -1, 0 and 1 periods for maps of degree 0 periods for maps of degree -1 rotation numbers and twist lifted cycles estimate of a rotation interval periods for maps of degree 1 maps of degree 1 with the prescribed set of periods other results. Appendix: lifted patterns. Entropy: definitions entropy for interval maps horseshoes entropy of cycles continuity properties of the entropy semiconjugacy to a map of a constant slope entropy for circle maps proof of Theorem 4.7.3.

On the nonexistence, existence and uniqueness of limit cycles

We present two new criteria for studying the nonexistence, existence and uniqueness of limit cycles of planar vector fields. We apply these criteria to some families of quadratic and cubic polynomial vector fields, and to compute an explicit formula for the number of limit cycles which bifurcate out of the linear centre , when we deal with the system . Moreover, by using the second criterion we present a method to derive the shape of the bifurcated limit cycles from a centre.

Tranversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem

Abstract The restricted three-body problem is considered for values of the Jacobi constant C near the value C 2 associated to the Euler critical point L 2 . A Lyapunov family of periodic orbits near L 2 , the so-called family ( c ), is born for C = C 2 and exists for values of C less than C 2 . These periodic orbits are hyperbolic. The corresponding invariant manifolds meet transversally along homoclinic orbits. In this paper the variation of the transversality is analyzed as a function of the Jacobi constant C and of the mass parameter μ. Asymptotical expressions of the invariant manifolds for C ≲ C 2 and μ ≳ 0 are found. Several numerical experiments provide accurate information for the manifolds and a good agreement is found with the asymptotical expressions. Symbolic dynamic techniques are used to show the existence of a large class of motions. In particular the existence of orbits passing in a random way (in a given sense) from the region near one primary to the region near the other is proved.

Darboux integrability and the inverse integrating factor

Abstract We mainly study polynomial differential systems of the form dx / dt = P ( x , y ), dy / dt = Q ( x , y ), where P and Q are complex polynomials in the dependent complex variables x and y , and the independent variable t is either real or complex. We assume that the polynomials P and Q are relatively prime and that the differential system has a Darboux first integral of the form H=f 1 λ 1 ⋯f p λ p exp h 1 g 1 n 1 μ 1 ⋯ exp h q g q n q μ q , where the polynomials f i and g j are irreducible, the polynomials g j and h j are coprime, and the λ i and μ j are complex numbers, when i =1,…, p and j =1,…, q . Prelle and Singer proved that these systems have a rational integrating factor. We improve this result as follows. Assume that H is a rational function which is not polynomial. Following to Poincare we define the critical remarkable values of H . Then, we prove that the system has a polynomial inverse integrating factor if and only if H has at most two critical remarkable values. Under some assumptions over the Darboux first integral H we show, first that the system has a polynomial inverse integrating factor; and secondly that if the degree of the system is m , the homogeneous part of highest degree of H is a multi-valued function, and the functions exp( h j / g j ) are exponential factors for j =1,…, q , then the system has a polynomial inverse integrating factor of degree m +1. We also present versions of these results for real polynomial differential systems. Finally, we apply these results to real polynomial differential systems having a Darboux first integral and limit cycles or foci.

Local analytic integrability for nilpotent centers

Let X(x,y) and Y(x,y) be real analytic functions without constant and linear terms defined in a neighborhood of the origin. Assume that the analytic differential system \dot{x}=y+ X(x,y) , \dot{y}=Y(x,y) has a nilpotent center at the origin. The first integrals, formal or analytic, will be real except if we say explicitly the converse. We prove the following. If X= y f(x,y^2) and Y= g(x,y^2) , then the systemhas a local analytic first integral of the form H=y^2+F(x,y) ,where F starts with terms of order higher than two. If the system has a formal first integral, then it hasa formal first integral of the form H=y^2+F(x,y) , where F starts with terms of order higher than two. In particular, if thesystem has a local analytic first integral defined at the origin,then it has a local analytic first integral of the form H=y^2+F(x,y) , where F starts with terms of order higher than two. As an application we characterize the nilpotent centersfor the differential systems \dot{x}=y+P_3(x,y) , \dot{y}=Q_3(x,y) , which have a local analytic first integral,where P_3 and Q_3 are homogeneous polynomials of degree three.

The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems

Abstract In this work we study the centers of planar analytic vector fields which are limit of linear type centers. It is proved that all the nilpotent centers are limit of linear type centers and consequently the Poincare–Liapunov method to find linear type centers can be also used to find the nilpotent centers. Moreover, we show that the degenerate centers which are limit of linear type centers are also detectable with the Poincare–Liapunov method.

CHAPTER 5 - Integrability of Polynomial Differential Systems

This chapter presents the planar polynomial differential systems. The notion of first integral is introduced; and the definition of integrating factor is discussed. The chapter introduces the notion of an exponential factor due to Christopher. An exponential factor appears when an invariant algebraic curve has in some sense multiplicity larger than 1.

Algebraic aspects of integrability for polynomial systems

We present an introductory survey to the Darboux integrability theory of planar complex and real polynomial differential systems. Our presentation contains some improvements to the classical theory.

On the optimal station keeping control of halo orbits

Abstract Techniques for computing and controlling a halo orbit are considered in this paper. It presents a semi-analytical theory for the halo orbits in the Restricted Three Bodies Problem (RTBP), that is valid and amenable to computation to any order. Results are presented up to order 11. The Floquet modes of the monodromy matrix are used to define a local optimal control procedure through the concepts of projection and gain functions.

Limit cycles of the generalized polynomial Liénard differential equations

We apply the averaging theory of first, second and third order to the class of generalized polynomial Lienard differential equations. Our main result shows that for any n , m ≥ 1 there are differential equations of the form ẍ + f ( x ) ẋ + g ( x ) = 0, with f and g polynomials of degree n and m respectively, having at least [( n + m − 1)/2] limit cycles, where [·] denotes the integer part function.