IN THIS WORK WE EXAMINE TRAPEZOIDAL CENTRAL CONFIGURATIONS IN THE PLANAR FOUR-VORTEX PROBLEM. MORE SPECIFICALLY, WE CONSIDER THE CONVEX CENTRAL CONFIGURATIONS IN WHICH THE CONVEX QUADRILATERAL HAS TWO PARALLEL SIDES. WITH ANALYTICAL ARGUMENTS WE CLASSIFY ALL POSSIBLE ARRANGEMENTS. ADDITIONALLY, WE PROVE THE UNIQUENESS OF THE TRAPEZOIDAL CENTRAL CONFIGURATION FOR GIVING FOUR VORTICITIES WITH THE SAME SIGN.

IN THE PRESENT PAPER, WE CONSIDER THE RESTRICTED PLANAR(N +1)-BODY PROBLEM WHERE N = 3 BODIES (CALLED PRIMARIES) INTERACTING WITH ONE ANOTHER ACCORDING TO NEWTONIAN LAW WHICH ARE IN THE VERTICES OF A REGULAR N-GON WITH THE ORIGIN AT THE CENTER OF MASSES OF THE COORDINATE SYSTEM. WE PROVE THAT THE SIMULTANEOUS BINARY COLLISION BETWEEN THE INFINITESIMAL MASS AND ANY PRIMARY ARE REGULARIZABLE, THROUGH THE IMPLEMENTATION OF BIRKHOFF-TYPE TRANSFORMATION

WE PROVIDE CANONICAL FORMS FOR THE HOMOGENEOUS POLYNOMIALS OF DEGREE FIVE. THEN WE CHARACTERIZE ALL THE PHASE PORTRAITS IN THE POINCARÉ DISK FOR ALL QUARTIC HOMOGENEOUS POLYNOMIAL DIFFERENTIAL SYSTEMS. MORE PRECISELY, THERE ARE EXACTLY 23 DIFFERENT TOPOLOGICAL PHASE PORTRAITS FOR THE QUARTIC HOMOGENEOUS POLYNOMIAL DIFFERENTIAL SYSTEMS.

WE ESTABLISH THE EXISTENCE OF BOUNDED, ALMOST PERIODIC SOLUTIONS AND ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS FOR FUNCTIONAL DIFFERENCE EQUATIONS WITH INFINITE DELAY ON ABSTRACT PHASE SPACE. AN APPLICATION TO VOLTERRA DIFFERENCE SYSTEMS WITH INFINITE DELAY IS GIVEN.

WE CONSIDER THE PLANAR RESTRICTED -BODY PROBLEM, WHERE THE PRIMARIES ARE MOVING IN A CENTRAL CONFIGURATION. IT IS VERIFIED THAT, WHEN THE ENERGY APPROACHES MINUS INFINITY, THE INFINITESIMAL MASS IS ARBITRARILY CLOSE TO A PRIMARY. WE USE LEVI-CIVITA AND MCGEHEE COORDINATES TO REGULARIZE THE BINARY COLLISION IN THIS SETTING. A CANONICAL TRANSFORMATION IS CONSTRUCTED IN SUCH A WAY THAT IT TRANSFORMS THE EQUATIONS INTO THE FORM OF A PERTURBED RESONANT PAIR OF HARMONIC OSCILLATORS WHERE THE PERTURBATION PARAMETER IS THE RECIPROCAL OF THE ENERGY. WE FIRST PROVE THE EXISTENCE OF FOUR TRANSVERSAL EJECTION?COLLISION ORBITS. AFTER THAT, WE CARRY OUT THE CONSTRUCTION OF THE ANNULUS MAPPING AND VERIFY THE CONDITIONS OF THE MOSER INVARIANT CURVE THEOREM; WE ARE ABLE TO SHOW THE EXISTENCE OF LONG PERIODIC SOLUTIONS FOR THE RESTRICTED ()-BODY PROBLEM. WE ALSO PROVE THE EXISTENCE OF QUASI-PERIODIC SOLUTIONS CLOSE TO THE BINARY COLLISION. THE FIRST RESULT IMPLIES, VIA THE KAM THEOREM, THE EXISTENCE OF AN UNCOUNTABLE NUMBER OF INVARIANT PUNCTURED TORI IN THE CORRESPONDING ENERGY SURFACE FOR CERTAIN INTERVALS OF VALUES OF THE JACOBI CONSTANT. THIS WORK GREW FROM AN ATTEMPT TO CARRY OVER THE METHODS USED TO STUDY THE RESTRICTED THREE-BODY PROBLEM FOR HIGH VALUES OF THE JACOBIAN CONSTANT BY CONLEY (1963, 1968) [3], [18]. CHENCINER [4] AND CHENCINER AND LLIBRE (1988) [5] APPLIED THEIR TECHNIQUES TO A MORE GENERAL RESTRICTED PROBLEM. OUR GOAL IN THIS PAPER IS TO GIVE A GENERALIZATION OF CONLEY?S RESULTS (CONLEY, 1968 [18]). IN ADDITION, WE SHOW THAT THE HILL TERMS (THE TERMS OF SIXTH ORDER) THAT APPEAR IN THIS STUDY HAVE THE SAME NATURE BUT WITH DIFFERENT COEFFICIENTS THAN THOSE IN THE MENTIONED PAPERS. THIS FACT ALLOWS US TO PRESENT SOME DIFFERENCES WITH RESPECT TO KNOWN RESULTS. THUS, WE POINT OUT CONDITIONS ON THE RELATIVE EQUILIBRIUM OF THE -BODY PROBLEM IN ORDER TO OVERCOME THE APPARENT DIFFICULTIES.

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.

USING THE EXISTENCE OF SUMMABLE DICHOTOMIES IN LINEAR FUNCTIONAL DIFFERENCE EQUATIONS, THE CONTRACTION PRINCIPLE AND THE SCHAUDER FIXED POINT THEOREM, WE OBTAIN THE EXISTENCE OF BOUNDED AND PERIODIC SOLUTIONS UNDER QUITE GENERAL HYPOTHESES FOR NONLINEAR FUNCTIONAL DIFFERENCE EQUATIONS ON PHASE SPACES. APPLICATIONS OF OUR MAIN RESULTS TO VOLTERRA EQUATIONS ARE GIVEN. MOREOVER, EXAMPLES ARE ALSO GIVEN TO ILLUSTRATE OBTAINED RESULTS.

USING THE NOTION OF SUMMABLE DICHOTOMY IN ORDINARY DIFFERENCE EQUATIONS AND THE CONTRACTION AND SCHAUDER FIXED POINT THEOREM, WE OBTAIN THE EXISTENCE OF BOUNDED AND PERIODIC SOLUTIONS ON ? UNDER QUITE GENERAL HYPOTHESES FOR NON HOMOGENOUS RETARDED DIFFERENCE EQUATIONS.

THE CENTRAL CONFIGURATIONS GIVEN BY AN EQUILATERAL TRIANGLE AND A REGULAR TETRAHEDRON WITH EQUAL MASSES AT THE VERTICES AND A BODY AT THE BARYCENTER HAVE BEEN WIDELY STUDIED IN [9] AND [14] DUE TO THE PHENOMENA OF BIFURCATION OCCURRING WHEN THE CENTRAL MASS HAS A DETERMINED VALUE ? . WE PROPOSE A VARIATION OF THIS PROBLEM SETTING THE CENTRAL MASS AS THE CRITICAL VALUE ? AND LETTING A MASS AT A VERTEX TO BE THE PARAMETER OF BIFURCATION. IN BOTH CASES, 2D AND 3D, WE VERIFY THE EXISTENCE OF BIFURCATION, THAT IS, FOR A SAME SET OF MASSES WE DETERMINE TWO NEW CENTRAL CONFIGURATIONS. THE COMPUTATION OF THE BIFURCATIONS, AS WELL AS THEIR PICTURES HAVE BEEN PERFORMED CONSIDERING HOMOGENEOUS FORCE LAWS WITH EXPONENT .

WE STUDY CONVEXITY AND SYMMETRY OF CENTRAL CONFIGURATIONS IN THE FIVE BODY PROBLEM WHEN THREE OF THE MASSES ARA LOCATED AT THE VERTICES OF AN EQUILATERAL TRIANGLE, THAT WE CALL LAGRANGE PLUS TWO CENTRAL CONFIGURATIONS. FIRST, WE PROVE THAT THE TWO BODIES OUT OF THE VERTICES OF THE TRIANGLE CANNOT BE PLACED ON CERTAIN LINES. NEXT, WE GIVE A GEOMETRICAL CHARACTERIZATION OF SUCH CONFIGURATIONS IN THE SENSE AS THAT OF DZIOBEK, AND WE DESCRIBE THE ADMISSIBLE REGIONS WHERE THE TWO REMAINING BODIES CAN BE PLACED. FURTHERMORE, WE PROVE THAT ANY LAGRANGE PLUS TWO CENTRAL CONFIGURATION IS CONCAVE. FINALLY, WE SHOW NUMERICALLY THE EXISTENCE OF NON-SYMMETRIC CENTRAL CONFIGURATIONS OF THE FIVE BODY PROBLEM.

IN THIS PAPER WE STUDY THE CONTINUOUS AND DISCONTINUOUS PLANAR PIECEWISE DIFFERENTIAL SYSTEMS FORMED BY FOUR LINEAR CENTERS SEPARATED BY THREE PARALLEL STRAIGHT LINES DENOTED BY ?={(X,Y)?R2:X=?P,X=0,X=Q, P,Q>0}. WE PROVE THAT WHEN THESE PIECEWISE DIFFERENTIAL SYSTEMS ARE CONTINUOUS THEY HAVE NO LIMIT CYCLES. WHILE FOR THE DISCONTINUOUS CASE WE SHOW THAT THEY CAN HAVE AT MOST FOUR LIMIT CYCLES AND WE ALSO PROVIDE EXAMPLES OF SUCH SYSTEMS WITH ZERO, ONE, AND TWO LIMIT CYCLES. IN PARTICULAR WE HAVE SOLVED THE EXTENSION OF THE 16TH HILBERT PROBLEM TO THIS CLASS OF PIECEWISE DIFFERENTIAL SYSTEMS.

WE CONSIDER THE DYNAMICS OF A PASSIVE TRACER, ADVECTED BY THE PRESENCE OF A LATITUDINAL RING OF IDENTICAL POINT VORTICES. THE CORRESPONDING INSTANTANEOUS MOTION IS MODELED BY A ONE DEGREE OF FREEDOM HAMILTONIAN SYSTEM. SUCH A DYNAMICS PRESENTS A RICH VARIETY OF BEHAVIORS WITH RESPECT TO THE NUMBER OF VORTICES, N, AND THE RING?S CO-LATITUDE, ?O?OR, EQUIVALENTLY, ITS VERTICAL POSITION QO = COS??O. WE CARRY OUT A COMPLETE DESCRIPTION OF THE GLOBAL PHASE PORTRAIT FOR THE CASES N = 2, 3, 4 BY DETERMINING EQUILIBRIUM POINTS, THEIR STABILITY, AND BIFURCATIONS WITH RESPECT TO THE PARAMETER ?O, AND BY CHARACTERIZING THE SEPARATRIX SKELETON. MOREOVER, FOR N ? 5, WE PROVE THE EXISTENCE OF A VALUE OF BIFURCATION ?ON SUCH THAT WHEN ?O = ?ON (?O = ? ? ?ON, RESPECTIVELY) THE SOUTH (NORTH, RESPECTIVELY) POLE BECOMES A N-BIFURCATION POINT, I.E., A SYMMETRIC WEB OF N CENTERS AND N SADDLES BIFURCATES FROM THE CORRESPONDING POLE.

CHRONIC MYELOID LEUKEMIA IS A BLOOD CANCER FOR WHICH STANDARD THERAPY WITH TYROSINE-KINASE INHIBITORS IS SUCCESSFUL IN THE MAJORITY OF PATIENTS. AFTER DISCONTINUATION OF TREATMENT HALF OF THE WELL-RESPONDING PATIENTS EITHER PRESENT UNDETECTABLE LEVELS OF TUMOR CELLS FOR A LONG TIME OR EXHIBIT SUSTAINED FLUCTUATIONS OF TUMOR LOAD OSCILLATING AT VERY LOW LEVELS. MOTIVATED BY THE CONSEQUENT QUESTION OF WHETHER THE OBSERVED KINETICS REFLECT PERIODIC OSCILLATIONS EMERGING FROM TUMOR-IMMUNE INTERACTIONS, IN THIS WORK, WE ANALYZE A SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS DESCRIBING THE IMMUNE RESPONSE TO CML WHERE BOTH THE FUNCTIONAL RESPONSE AGAINST LEUKEMIA AND THE IMMUNE RECRUITMENT EXHIBIT OPTIMAL ACTIVATION WINDOWS. BESIDES INVESTIGATING THE STABILITY OF THE EQUILIBRIUM POINTS, WE PROVIDE RIGOROUS PROOFS THAT THE MODEL EXHIBITS AT LEAST TWO TYPES OF BIFURCATIONS: A TRANSCRITICAL BIFURCATION AROUND THE TUMOR-FREE EQUILIBRIUM POINT AND A HOPF BIFURCATION AROUND A BIOLOGICALLY PLAUSIBLE EQUILIBRIUM POINT, PROVIDING AN AFFIRMATIVE ANSWER TO OUR INITIAL QUESTION. FOCUSING OUR ATTENTION ON THE HOPF BIFURCATION, WE EXAMINE THE EMERGENCE OF LIMIT CYCLES AND ANALYZE THEIR STABILITY THROUGH THE CALCULATION OF LYAPUNOV COEFFICIENTS. THEN WE ILLUSTRATE OUR THEORETICAL RESULTS WITH NUMERICAL SIMULATIONS BASED ON CLINICALLY RELEVANT PARAMETERS. BESIDES THE MATHEMATICAL INTEREST, OUR RESULTS SUGGEST THAT THE FLUCTUATING LEVELS OF LOW TUMOR LOAD OBSERVED IN CML PATIENTS MAY BE A CONSEQUENCE OF PERIODIC ORBITS ARISING FROM PREDATOR?PREY-LIKE INTERACTIONS.

IN THIS PAPER, A PLANAR SYSTEM OF ORDINARY DIFFERENTIAL EQUATIONS IS CONSIDERED, WHICH IS A MODIFIED LESLIE-GOWER MODEL, CONSIDERING A BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE. IT GENERATES A COMPLEX DYNAMICS OF THE PREDATOR-PREY INTERACTIONS ACCORDING TO THE ASSOCIATED PARAMETERS. FROM THE SYSTEM OBTAINED, WE CHARACTERIZE ALL THE EQUILIBRIA AND ITS LOCAL BEHAVIOR, AND THE EXISTENCE OF A TRAPPING SET IS PROVED. WE DESCRIBE DIFFERENT TYPES OF BIFURCATIONS (SUCH AS HOPF, BOGDANOV-TAKENS, AND HOMOCLINIC BIFURCATION), AND THE EXISTENCE OF LIMIT CYCLES IS SHOWN. ANALYTIC PROOFS ARE PROVIDED FOR ALL RESULTS. ECOLOGICAL IMPLICATIONS AND A SET OF NUMERICAL SIMULATIONS SUPPORTING THE MATHEMATICAL RESULTS ARE ALSO PRESENTED.

WE CONSIDER THE HAMILTONIAN FUNCTION DEFINED BY THE CUBIC POLYNOMIAL H = 1/2(Y(1)(2) + Y(2)(2)) + V(X(1), X(2)) WHERE THE POTENTIAL V(X) = DELTA V-2(X(1), X(2)) + V-3(X(1), X(2)), WITH V-2(X(1), X(2)) = 1/2(X(1)(2) + X(2)(2)) AND V-3(X(1), X(2)) = 1/3X(1)(3) + F X(1)X(2)(2) + GX(2)(3), WITH F AND G ARE REAL PARAMETERS SUCH THAT F NOT EQUAL 0 AND DELTA IS 0 OR 1. OUR OBJECTIVE IS TO STUDY THE NUMBER AND BIFURCATIONS OF THE EQUILIBRIA AND ITS TYPE OF STABILITY. MOREOVER, WE OBTAIN THE EXISTENCE OF PERIODIC SOLUTIONS CLOSE TO SOME EQUILIBRIUM POINTS AND AN ISOLATED SYMMETRIC PERIODIC SOLUTION DISTANT OF THE EQUILIBRIA FOR SOME CONVENIENT REGION OF THE PARAMETERS. WE POINT OUT THE ROLE OF THE PARAMETERS AND THE DIFFERENCE BETWEEN THE HOMOGENEOUS POTENTIAL CASE (DELTA = 0) AND THE GENERAL CASE (DELTA = 1).

WE CONSIDER A THREE-DIMENSIONAL MATHEMATICAL MODEL THAT DESCRIBES THE INTERACTION BETWEEN THE EFFECTOR CELLS, TUMOR CELLS, AND THE CYTOKINE (IL-2) OF A PATIENT. THIS IS CALLED THE KIRSCHNER?PANETTA MODEL. OUR OBJECTIVE IS TO EXPLAIN THE TUMOR OSCILLATIONS IN TUMOR SIZES AS WELL AS LONG-TERM TUMOR RELAPSE. WE THEN EXPLORE THE EFFECTS OF ADOPTIVE CELLULAR IMMUNOTHERAPY ON THE MODEL AND DESCRIBE UNDER WHAT CIRCUMSTANCES THE TUMOR CAN BE ELIMINATED OR CAN REMAIN OVER TIME BUT IN A CONTROLLED MANNER. NONLINEAR DYNAMICS OF IMMUNOGENIC TUMORS ARE GIVEN, FOR EXAMPLE: WE PROVE THAT THE TRAJECTORIES OF THE ASSOCIATED SYSTEM ARE BOUNDED AND DEFINED FOR ALL POSITIVE TIME; THERE ARE SOME INVARIANT SUBSETS; THERE ARE OPEN SUBSETS OF PARAMETERS, SUCH THAT THE SYSTEM IN THE FIRST OCTANT HAS AT MOST FIVE EQUILIBRIUM SOLUTIONS, ONE OF THEM IS TUMOR-FREE AND THE OTHERS ARE OF CO-EXISTENCE. WE ARE ABLE TO PROVE THE EXISTENCE OF TRANSCRITICAL AND PITCHFORK BIFURCATIONS FROM THE TUMOR-FREE EQUILIBRIUM POINT. FIXING AN EQUILIBRIUM AND INTRODUCING A SMALL PERTURBATION, WE ARE ABLE TO SHOW THE EXISTENCE OF A HOPF PERIODIC ORBIT, SHOWING A CYCLIC BEHAVIOR AMONG THE POPULATION, WITH A STRONG DOMINANCE OF THE PARENTAL ANOMALOUS GROWTH CELL POPULATION. THE PREVIOUS INFORMATION REVEALS THE EFFECTS OF THE PARAMETERS. IN OUR STUDY, WE OBSERVE THAT OUR MATHEMATICAL MODEL EXHIBITS A VERY RICH DYNAMIC BEHAVIOR AND THE PARAMETER ?? (DEATH RATE OF THE EFFECTOR CELLS) AND ?P1 (PRODUCTION RATE OF THE EFFECTOR CELL STIMULATED BY THE CYTOKINE IL-2) PLAYS AN IMPORTANT ROLE. MORE PRECISELY, IN OUR APPROACH THE INEQUALITY ??2 > ?P1 IS VERY IMPORTANT, THAT IS, THE DEATH RATE OF THE EFFECTOR CELLS IS GREATER THAN THE PRODUCTION RATE OF THE EFFECTOR CELL STIMULATED BY THE CYTOKINE IL-2. FINALLY, MEDICAL IMPLICATIONS AND A SET OF NUMERICAL SIMULATIONS SUPPORTING THE MATHEMATICAL RESULTS ARE ALSO PRESENTED.