THIS WORK ESTABLISHES THE EXISTENCE OF ADDITION THEOREMS AND DOUBLE-ANGLE FORMULAS FOR CK REAL SCALAR FUNCTIONS. MOREOVER, WE DETERMINE NECESSARY AND SUFFICIENT CONDITIONS FOR A BIVARIATE FUNCTION TO BE AN ADDITION FORMULA FOR A CK REAL FUNCTION. THE DOUBLE-ANGLE FORMULAS ALLOW US TO GENERATE A DUPLICATION ALGORITHM, WHICH CAN BE USED AS AN ALTERNATIVE TO THE CLASSICAL NUMERICAL METHODS TO OBTAIN AN APPROXIMATION FOR THE SOLUTION OF AN ORDINARY DIFFERENTIAL EQUATION. WE DEMONSTRATE THAT THIS ALGORITHM CONVERGES UNIFORMLY IN ANY COMPACT DOMAIN CONTAINED IN THE MAXIMAL DOMAIN OF THAT SOLUTION. FINALLY, WE CARRY OUT SOME NUMERICAL SIMULATIONS SHOWING A GOOD PERFORMANCE OF THE DUPLICATION ALGORITHM WHEN COMPARED WITH STANDARD NUMERICAL METHODS

WE CONSIDER AN INTEGRABLE NON-HAMILTONIAN SYSTEM, WHICH BELONGS TO THE QUADRATIC KUKLES DIFFERENTIAL SYSTEMS. IT HAS A CENTER SURROUNDED BY A BOUNDED PERIOD ANNULUS. WE STUDY POLYNOMIAL PERTURBATIONS OF SUCH A KUKLES SYSTEM INSIDE THE KUKLES FAMILY. WE APPLY AVERAGING THEORY TO STUDY THE LIMIT CYCLES THAT BIFURCATE FROM THE PERIOD ANNULUS AND FROM THE CENTER OF THE UNPERTURBED SYSTEM. FIRST, WE SHOW THAT THE PERIODIC ORBITS OF THE PERIOD ANNULUS CAN BE PARAMETRIZED EXPLICITLY THROUGH THE LAMBERT FUNCTION. LATER, WE PROVE THAT AT MOST ONE LIMIT CYCLE BIFURCATES FROM THE PERIOD ANNULUS, UNDER QUADRATIC PERTURBATIONS. MOREOVER, WE GIVE CONDITIONS FOR THE NON-EXISTENCE, EXISTENCE, AND STABILITY OF THE BIFURCATED LIMIT CYCLES. FINALLY, BY USING AVERAGING THEORY OF SEVENTH ORDER, WE PROVE THAT THERE ARE CUBIC SYSTEMS, CLOSE TO THE UNPERTURBED SYSTEM, WITH 1 AND 2 SMALL LIMIT CYCLES.

WE CONSIDER POLYNOMIAL GENERALIZED LIÉNARD SYSTEMS THAT COME FROM SECOND ORDER POLYNOMIAL PERTURBATIONS OF A LINEAR CENTER. FOR THESE SYSTEMS WE FOUND A GENERIC (OPEN AND DENSE) SUBSET IN THE SPACE OF PERTURBATIONS OF DEGREE 2L, WITH , SUCH THAT EACH ASSOCIATED PERTURBED GENERALIZED LIÉNARD SYSTEM HAS AT MOST MEDIUM LIMIT CYCLES. MOREOVER, WE SHOW THAT THIS UPPER BOUND IS REACHED. IN ADDITION, SOME CONCRETE EXAMPLES ARE GIVEN.

WE PROVIDE EXPLICIT LOWER AND UPPER BOUNDS FOR THE MAXIMUM NUMBER OF ISOLATED ZEROS OF THE COMPLETE ABELIAN INTEGRAL ASSOCIATED WITH A RATIONAL POLYNOMIAL, WITH NON-TRIVIAL GLOBAL MONODROMY, AND A POLYNOMIAL 1-FORM OF DEGREE N. MOREOVER, WE OBTAIN THE EXPLICIT FORM OF THE RELATIVE COHOMOLOGY OF THE POLYNOMIAL 1-FORMS WITH RESPECT TO THE RATIONAL POLYNOMIAL.

IN THIS PAPER WE CONSIDER POLYNOMIAL PERTURBATIONS OF A FAMILY OF POLYNOMIAL HAMILTONIAN EQUATIONS WHOSE ASSOCIATED HAMILTONIAN IS NOT TRANSVERSAL TO INFINITY, AND ITS COMPLEXIFICATION IS NOT A MORSE POLYNOMIAL. WE LOOK FOR AN ALGORITHM TO COMPUTE THE FIRST NON-VANISHING POINCARÉ?PONTRYAGIN?MELNIKOV FUNCTION OF THE DISPLACEMENT FUNCTION ASSOCIATED WITH THE PERTURBED EQUATION. WE SHOW THAT THE ALGORITHM OF THE CASE WHEN THE HAMILTONIAN IS TRANSVERSAL TO INFINITY AND ITS COMPLEXIFICATION IS A MORSE POLYNOMIAL CAN BE EXTENDED TO OUR FAMILY OF PERTURBED EQUATIONS. WE APPLY THE RESULT TO STUDY THE MAXIMUM NUMBER OF ZEROS OF THE FIRST NON-VANISHING POINCARÉ?PONTRYAGIN?MELNIKOV FUNCTION ASSOCIATED WITH SOME PERTURBED HAMILTONIAN EQUATIONS.

EXPLICARÉ UN MODELO MATEMÁTICO SENCILLO QUE SE AJUSTA Y PREDICE EL COMPORTAMIENTO DE LOS DATOS OFICIALES EN ESPAÑA. ESTAS NOTAS PRETENDEN CONTRIBUIR A LA EDUCACIÓN CIENTÍFICA DE TODOS PARA CONCIENCIARNOS RESPECTO A LA SITUACIÓN TAN GRAVE DEL COVID-19 Y MOSTRAR, A TRAVÉS DE GRÁFICAS, LA IMPORTANCIA DEL CONFINAMIENTO Y AISLAMIENTO SOCIAL EN LA LUCHA CONTRA ESTA PANDEMIA.