Newtonian and Pseudo-Newtonian Hill Problem

A pseudo-Newtonian Hill problem based on the Paczynski-Wiita pseudo-Newtonian potential that reproduces general relativistic effects is presented and compared with the usual Newtonian Hill problem. Poincare maps, Lyapunov exponents and fractal escape techniques are employed to study bounded and unbounded orbits. In particular we consider the systems composed by Sun, Earth and Moon and composed by the Milky Way, the M2 cluster and a star. We find that some pseudo-Newtonian systems - including the M2 system - are more stable than their Newtonian equivalent.Comment: 12 pages, 4 figures, 1 tabl

NON-INTEGRABILI TY CRITERIA IN HAMILTONIAN DYNAMICAL SYSTEMS

ΣΚΟΠΟΣ ΤΗΣ ΠΑΡΟΥΣΗΣ ΔΙΔΑΚΤΟΡΙΚΗΣ ΔΙΑΤΡΙΒΗΣ ΕΙΝΑΙ Η ΕΥΡΕΣΗ ΚΡΙΤΗΡΙΩΝ ΓΙΑ ΤΗ ΜΗ ΥΠΑΡΞΗ ΟΛΟΚΛΗΡΩΜΑΤΩΝ ΤΗΣ ΚΙΝΗΣΗΣ ΣΕ ΧΑΜΙΛΤΟΝΙΑΝΑ ΔΥΝΑΜΙΚΑ ΣΥΣΤΗΜΑΤΑ. ΑΝΑΠΤΥΣΣΟΝΤΑΣ ΣΕ ΣΕΙΡΑ ΩΣ ΠΡΟΣ ΤΗ ΔΙΑΤΑΡΑΚΤΙΚΗ ΠΑΡΑΜΕΤΡΟ Ε ΤΗΝ ΑΝΑΓΚΑΙΑ ΣΥΝΘΗΚΗ ΓΙΑ ΤΗΝ ΥΠΑΡΞΗ ΟΛΟΚΛΗΡΩΜΑΤΟΣ ΤΗΣ ΚΙΝΗΣΗΣ ΣΕ ΜΙΑ ΔΙΑΤΑΡΑΓΜΕΝΗ ΧΑΜΙΛΤΟΝΙΑΝΗ ΚΑΙ ΠΑΡΑΜΕΤΡΟΠΟΩΝΤΑΣ ΤΗΝ ΤΑΥΤΟΤΗΤΑ ΠΟΥ ΠΡΟΚΥΠΤΕΙ ΣΕ ΟΡΟΥΣ Ε ΠΑΝΩ ΣΤΙΣ ΠΕΡΙΟΔΙΚΕΣ ΤΡΟΧΙΕΣ ΤΟΥ ΑΔΙΑΤΑΡΑΚΤΟΥ ΣΥΣΤΗΜΑΤΟΣ, ΟΔΗΓΟΥΜΑΣΤΕ ΣΕ ΤΡΙΑ ΘΕΩΡΗΜΑΤΑ, ΤΑ ΟΠΟΙΑ ΠΑΡΕΧΟΥΝ ΚΡΙΤΗΡΙΑ ΜΗ ΟΛΟΚΛΗΡΩΣΙΜΟΤΗΤΑΣ. ΤΟ ΠΡΩΤΟ ΘΕΩΡΗΜΑ ΔΙΝΕΙ ΜΙΑ ΙΚΑΝΗ ΣΥΝΘΗΚΗ ΜΗ ΟΛΟΚΛΗΡΩΣΙΜΟΤΗΤΑΣ ΓΙΑ ΣΥΝΑΡΤΗΣΕΙΣ HAMILTON ΔΥΟ ΒΑΘΜΩΝ ΕΛΕΥΘΕΡΙΑΣ, ΕΝΩ ΤΟ ΔΕΥΤΕΡΟ ΔΙΝΕΙ ΜΙΑ ΑΝΑΛΟΓΗ ΣΥΝΘΗΚΗ ΓΙΑ ΣΥΝΑΡΤΗΣΕΙΣ N ΒΑΘΜΩΝ ΕΛΕΥΘΕΡΙΑΣ ΚΑΙ ΕΠΙΠΛΕΟΝ ΕΝΑ ΚΡΙΤΗΡΙΟ ΓΙ ΤΟΝ ΜΕΓΙΣΤΟ ΕΠΙΤΡΕΠΤΟ ΑΡΙΘΜΟ ΟΛΟΚΛΗΡΩΜΑΤΩΝ. ΤΟ ΤΡΙΤΟ ΘΕΩΡΗΜΑ ΑΝΑΦΕΡΕΤΑΙ ΣΕ ΔΙΑΤΑΡΑΓΜΕΝΑ ΣΥΣΤΗΜΑΤΑ ΕΝΟΣ ΒΑΘΜΟΥ ΕΛΕΥΘΕΡΙΑΣ, ΜΕ ΔΙΑΤΑΡΑΧΗ ΠΟΥ ΕΞΑΡΤΑΤΑΙ ΠΕΡΙΟΔΙΚΑ ΑΠΟ ΤΟ ΧΡΟΝΟ . ΕΠΙΠΛΕΟΝ, ΤΟ ΠΡΩΤΟ ΑΠΟ ΤΑ ΠΑΡΑΠΑΝΩ ΘΕΩΡΗΜΑΤΑ ΣΥΝΔΕΕΤΑΙ ΜΕ ΤΟ ΓΝΩΣΤΟ ΘΕΩΡΗΜΑ ΜΗ ΟΛΟΚΛΗΡΩΣΙΜΟΤΗΤΑΣ ΤΟΥ POINCARE, ΕΝΩ ΚΑΙ ΤΑ ΤΡΙΑ ΘΕΩΡΗΜΑΤΑ ΣΥΝΔΕΟΝΤΑΙ ΜΕ ΤΗ ΣΥΝΕΧΙΣΗ ΤΩΝ ΠΕΡΙΟΔΙΚΩΝ ΤΡΟΧΙΩΝ ΚΑΤΩ ΑΠΟ ΤΗ ΔΙΑΤΑΡΑΧΗ, ΜΕΣΩ ΤΗΣ ΘΕΩΡΙΑΣ POINCARE-MELNIKOV. ΤΕΛΟΣ ΓΙΝΕΤΑΙ ΕΦΑΡΜΟΓΗ ΚΑΙ ΤΩΝ ΤΡΙΩΝΠΑΡΑΠΑΝΩ ΚΡΙΤΗΡΙΩΝ ΓΙΑ ΤΗΝ ΑΠΟΔΕΙΞΗ ΜΗ ΟΛΟΚΛΗΡΩΣΙΜΟΤΗΤΑΣ ΣΥΓΚΕΚΡΙΜΕΝΩΝ ΧΑΜΙΛΤΟΝΙΑΝΩΝ ΣΥΣΤΗΜΑΤΩΝ ΜΕ ΦΥΣΙΚΗ ΣΗΜΑΣΙΑ.THE AIM OF THIS THESIS IS TO PROVIDE CRITERIA FOR THE NON-EXISTENCE OF INTEGRALS OF MOTION IN HAMILTONIAN DYNAMICAL SYSTEMS. BY EXPANDING THE NECESSARY CONDITION FOR THE EXISTENCE OF AN INTEGRAL OF MOTION IN A PERTURBED HAMILTONIAN WITH RESPECT TO THE PERTURBATIVE PARAMETER Ε AND PARAMETRIZING THE RESULTING IDENTITY IN ORDER Ε ALONG THE PERIODIC ORBITS OF THE UNPERTURBED SYSTEM, WE DERIVE THREE THEOREMS WHICH PROVIDE NONINTEGRABILITY CRITERIA. THE FIRST THEOREMSUPPLIES A SUFFICIENT CONDITION FOR NON-INTEGRABILITY OF TWO DEGREE OF FREEDOM HAMILTONIANS, WHILE THE SECOND ONE SUPPLIES A SIMILAR CONDITION FOR HAMILTONIANS OF N DEGREES OF FREEDOM AND, IN ADDITION, A CRITERION WHICH RESTRICTS THE MAXIMUM ALLOWED NUMBER OF INTEGRALS OF MOTION. THE THIRD THEOREM REFERS TO PERTURBED ONE DEGREE OF FREEDOM SYSTEMS WITH A PERTURBATION DEPENDING PERIODICALLY ON TIME. MOREOVER, THE FIRST OF THE ABOVE THEOREMS IS RELATED TO THE WELLKNOWH NON-INTEGRABILITY THEOREM OF POINCARE, WHILE ALL THEOREMS ARE RELATED TO THE CONTINUATION OF THE PERIODIC ORBITS WITH RESPECT TO THE PERTURBATION, THROUGH THE POINCARE-MELNIKOV THEORY. FINALLY, THE ABOVE THREE CRITERIA ARE APPLIED IN PROVING NON-INTEGRABILITY OF SPECIFIC HAMILTONIANS WITH PHYSICAL INTEREST

\Psi-series and obstructions to integrability of periodically perturbed one degree of freedom Hamiltonians

A connection between the \Psi-series local expansion of the solution of a perturbed system of O.D.E.'s and the evaluation of the Mel'nikov vector with the method of residues has recently been found by Goriely and Tabor. By following an analogous procedure, we find a straightforward relation between the failure of the compatibility condition of the Painlev'e test and the absence of an analytic integral for periodically perturbed Hamiltonians whose unperturbed part does not necessarily possess a homoclinic loop. We apply these results to a periodically perturbed unharmonic oscillator. MSC codes: 70H05, 58F05. PACS codes: 03.20.+i Key words: Hamiltonian systems, integrability, Painlev'e property, \Psi-series 1 Introduction This work deals with the connection between the \Psi-series expression of the solution in complex time and a recently developed [1] real-time non-integrability criterion for one degree of freedom periodically perturbed Hamiltonian systems, whose integrable part is als..

Convergence of Birkhoff Normal form for Essentially Isochronous Systems

We reconsider the problem of the convergence of Birkhoff's normal form for a system of perturbed harmonic oscillators, under the condition that the system is essentially isochronous. In contrast with previous proofs based on the so called quadratically convergent method, the present proof uses only classical expansions in a parameter. This allows us to bring into light some mechanisms of accumulation of small divisors, which can be useful in more complicated and interesting cases. These same mechanisms allows us to prove the theorem with the Bruno condition on the frequencies in a very natural way

Τhe Study of Square Periodic Perturbations as an Immunotherapy Process on a Tumor Growth Chaotic Model

In the present study, the simulation of an immunotherapy effect for a known dynamical system, that describes the process for avascular, vascular, and metastasis tumor growth based on a chemical network model, has been presented. To this end, square signals of various amplitudes have been used, to model the effect of external therapy control, in order to affect the population of immune cells. The results of the simulations show that for certain values of the amplitude of the square signal, the populations of the proliferating tumor cells in the vascular and metastasis stages have been reduced