An Iterative Ab Initio Non-Hermitian Floquet Method for Photoionization Resonances

We present an efficient ab initio non-Hermitian Floquet method for computing the photoionization resonances of an electronic system interacting with linearly polarized<br>monochromatic laser light. Unlike the direct "brute force" diagonalization method, which has been used for huge Floquet matrix eigenvalue problems, the new method follows a simple iterative process. The computational advantages of<br>the iterative method are very remarkable as it avoids computation, storage, and diagonalization of the huge Floquet matrix. The new method can also be used in<br>conjunction with the ab initio computational techniques that were originally developed for the field-free bound state calculations. The method is best illustrated<br>with the photoionization resonance of the hydrogen atom.<br>

An Iterative Ab Initio Non-Hermitian Floquet Method for Photoionization Resonances

We present an efficient ab initio non-Hermitian Floquet method for computing the photoionization resonances of an electronic system interacting with linearly polarized<br>monochromatic laser light. Unlike the direct "brute force" diagonalization method, which has been used for huge Floquet matrix eigenvalue problems, the new method follows a simple iterative process. The computational advantages of<br>the iterative method are very remarkable as it avoids computation, storage, and diagonalization of the huge Floquet matrix. The new method can also be used in<br>conjunction with the ab initio computational techniques that were originally developed for the field-free bound state calculations. The method is best illustrated<br>with the photoionization resonance of the hydrogen atom.<br>

Διακλαδώσεις και ευστάθεια περιοδικών λύσεων μη γραμμικών πλεγμάτων με αναλυτικές μεθόδους

Σκοπός της διατριβής ήταν η μελέτη διακλαδώσεων και ευστάθειας περιοδικών λύσεων σε μη γραμμικά πλέγματα. Αρχικά μελετήσαμε ένα μοντέλο μίας διάστασης μαγνητικού μεταυλικού που αποτελείται από μία διακριτή σειρά από μη γραμμικούς συντονισμούς. Μελετήσαμε περιοδικά καθώς και στάσιμα οδεύοντα κύματα του μοντέλου. Χρησιμοποιώντας ανάλυση Melnikov μελετήσαμε την ύπαρξη καθώς και την παραμονή τέτοιων κυμάτων όπως επίσης και την γραμμική τους ευστάθεια. Επίσης βρήκαμε συνθήκες κάτω από τις οποίες μπορούμε να έχουμε ευστάθεια ή αστάθεια των λύσεων. Τα αναλυτικά μας αποτελέσματα συμφωνούν απόλυτα με τα αριθμητικά αποτελέσματα όπου χρησιμοποίησαμε την μέθοδο Runge-Kutta και την Floquet Method. Στην συνέχεια μελετήσαμε την δυναμική ενός ζεύγος από SQUIDS σε σειρά. Χρησιμοποιώντας πολυ-βαθμωτή ανάλυση υπολογίσε τις εξισώσεις πλάτους, οι οποίες περιγράφουν την αργή δυναμική του συστήματος. Απ' όπου βρήκαμε την ύπαρξη ομοκλινικών τροχιών στην δυναμική του ολοκληρώσιμου μέρους των αργών εξισώσεων τις κίνησης. Χρησιμοποιώντας ανάλυση Melnikov για μεγάλες διαστάσεις βρήκαμε τις τιμές των παραμέτρων για τις οποίες αυτές οι τροχιές εξακολουθούν να υπάρχουν και σε ολόκληρο το σύστημα, αποτελούμενες από Χαμιλτονιανές και μη Χαμιλτονιανές διαταραχές, σχηματίζοντας Silnikov τροχιές, οι οποίες υποδεικνύουν την έλλειψη ολοκληρωσιμότητας και την ύπαρξη χάους. Τέλος μελέτησαμε οδεύοντα κύματα σε πλέγματα μεταυλικών με μη γραμμική σύζευξη αλλά με γραμμικό δυναμικό και συζευγμένες εξισώσεις πλεγμάτων με ισορροπημένο κέρδος και χάσιμο χρησιμοποιώντας ανάλυση Melnikov.

THE FLOQUET METHOD FOR PT-SYMMETRIC PERIODIC POTENTIALS

By the general theory of PT-symmetric quantum systems, their energy levels are either real or occur in complex-conjugate pairs, which implies that the secular equation must be real. However, for periodic potentials it is by no means clear that the secular equation arising in the Floquet method is indeed real, since it involves two linearly independent solutions of the Schrödinger equation. In this brief note we elucidate how that reality can be established.

Observations Regarding Numerical Results Obtained by the Floquet-Method

Stability studies of parametrically excited systems are frequently carried out by numerical methods. Especially for LTP-systems, several such methods are known and in practical use. This study investigates and compares two methods that are both based on Floquet’s theorem. As an introductary benchmark problem a 1-dof system is employed, which is basically a mechanical representation of the damped Mathieu-equation. The second problem to be studied in this contribution is a time-periodic 2-dof vibrational system. The system equations are transformed into a modal representation to facilitate the application and interpretation of the results obtained by different methods. Both numerical methods are similar in the sense that a monodromy matrix for the LTP-system is calculated numerically. However, one method uses the period of the parametric excitation as the interval for establishing that matrix. The other method is based on the period of the solution, which is not known exactly. Numerical results are computed by both methods and compared in order to work out how they can be applied efficiently.

Dynamic modelling and stability of hingeless helicopter blades with a smart spring

The aeroelastic stability of a uniform, untwisted hingeless ‘smart’ helicopter rotor blade in hover has been analysed. The concept of a ‘smart’ blade is achieved by implementing a piezoelectric stack at an appropriate location along a host blade such that upon actuation it enters the load path becoming an integral part of the host structure. Thus, the stiffness characteristics of the rotor are altered causing modal damping augmentation of the blade. The perturbation equations of motion for the ‘smart’ blade that describe the unsteady blade motion about the equilibrium operating condition are obtained using Galerkin’s method. These differential equations with periodic time coefficients are analysed for stability utilising the Floquet method. Six different regimes of actuation are investigated, and a parametric study is carried out by considering six different design cases. It is shown that, compared to a ‘host’ blade the stability characteristics of the ‘smart’ blade are not affected adversely. In fact, a judicious design and actuation of the ‘smart’ spring has the potential of improving the stability boundaries of individual blades.