The Problem of Bicenter and Isochronicity for a Class of Quasi Symmetric Planar Systems

We study a class of quasi symmetric seventh degree systems and obtain the conditions that its two singular points can be two centers at the same step by careful computing and strict proof. In addition, the condition of an isochronous center is also given. In terms of quasi symmetric systems, our work is interesting and obtained conclusions about bicenters are new

Periodic oscillators, isochronous centers and resonance

An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context, resonance means that all solutions are unbounded. The theory of resonance is well known for the harmonic oscillator and we extend it to nonlinear isochronous oscillators.Comment: 28 page

Classical Mechanics

An overview of the foundations of Classical Mechanic

Simultaneity of centres in Zq-equivariant systems

We study the simultaneous existence of centres for two families of planar Zq-equivariant systems. First, we give a short review about Zq-equivariant systems. Next, we present the necessary and sufficient conditions for the simultaneous existence of centres for a Z2-equivariant cubic system and for a Z2- equivariant quintic system

The Problem of Bicenter and Isochronicity for a Class of Quasi Symmetric Planar Systems

We study a class of quasi symmetric seventh degree systems and obtain the conditions that its two singular points can be two centers at the same step by careful computing and strict proof. In addition, the condition of an isochronous center is also given. In terms of quasi symmetric systems, our work is interesting and obtained conclusions about bicenters are new

Some open problems in low dimensional dynamical systems

The aim of this paper is to share with the mathematical community a list of 33 problems that I have found along the years during my research. I believe that it is worth to think about them and, hopefully, it will be possible either to solve some of the problems or to make some substantial progress. Many of them are about planar differential equations but there are also questions about other mathematical aspects: Abel differential equations, difference equations, global asymptotic stability, geometrical questions, problems involving polynomials or some recreational problems with a dynamical component

Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems

We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeroes of a suitable function, called the Melnikov function - by analogy with the periodic case. We show that, if the Melnikov function has a zero of odd order and under some further condition on the sign of the perturbation parameter, then there exists at least one resonant solution which continues an unperturbed solution. If the Melnikov function is identically zero then one can push perturbation theory up to the order where a counterpart of Melnikov function appears and does not vanish identically: if such a function has a zero of odd order and a suitable positiveness condition is met, again the same persistence result is obtained. If the system is Hamiltonian, then the procedure can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the result follows that at least one resonant quasi-periodic solution always exists with no assumption on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.Comment: 60 pages, 16 figures. arXiv admin note: substantial text overlap with arXiv:1011.093

Spektralstatisitk abseits der Standard Universalitätsklassen

Random matrix theory (RMT) and semi-classical methods are both used to study the spectra of chaotic quantum systems. The former is usually applied to small distances within the spectrum residing on scales of the mean level spacing where the phenomenon of universality emerges. Through the quantum-classical correspondence provided by semi-classics this regime is, conversely, described by long time dynamical properties of the classical system. In the first half of this thesis we explore such a connection for the study of spectral properties of quantum graphs with dynamically broken time reversal invariance. In physical systems this invariance is often not fully broken and to emulate this we include a rank-1 perturbation on the quantum level resulting in a RMT model outside of the usual universality classes. As we further show, the outcome can depend on graph specific properties and the rank of the perturbation. For the second half of the thesis we consider an opposite limit: instead of long times we study short times, on which universality can not be expected, in a chain-like spin system. In these systems the number of spins takes on a similar role as time, therefore the short time behavior in long chains has remarkable similarities to long time dynamics in few-body system. For instance, the formulation of an "evolution" operator in spatial direction is possible. Exploring this spatial-time duality we address long range spectral statistic in many-body systems and, for the first time, resolve periodic orbits in a genuine many-body system from the traces of its quantum evolution.Zufallsmatrixtheorie (RMT) und Semiklassik stellen zwei Methoden zum Studium chaotischer Quantenspektren dar. Erstere beschreibt für gewöhnlich die Spektralstatistik auf Skalen des mittleren Niveauabstandes, für welche man universelle Eigenschaften findet. Ausgehend vom semiklassischen Ansatz, dass eine Korrespondenz zwischen dem quantenmechanischen Spektrum und der klassischen Dynamik eines Systems besteht, wird dieser Bereich gleichermaßen durch klassische Langzeiteigenschaften beschrieben. Im ersten Teil dieser Arbeit nutzen wir diesen Zusammenhang für Studien an einem Quantengraphen mit gebrochener Zeitumkehrinvarianz. In physikalsichen Systemen ist diese Invarianz oft jedoch nicht vollständig gebrochen, was wir durch eine Rang-1 Störung im Quantensystem abbilden. Dies führt zu einem RMT Model abseits der standard Universalitätsklassen. Weiterhin zeigt sich, dass das Ergebnis sowohl von spezifischen Eigenschaften des Graphen als auch vom Rang der Störung abhängt. Der zweite Teil der Arbeit widmet sich einem diametralen Grenzfall: anstelle langer Zeiten untersuchen wir, innerhalb einer Spinkette, kurze Zeitskalen, auf denen Univeralität nicht zu erwarten ist. In diesen Systemen spielt die Teilchenzahl eine ähnliche Rolle wie die Zeit, entsprechend weist das Kurzzeitverhalten bemerkenswerte Übereinstimmungen zum Langzeitverhalten von Systemen mit wenigen Freiheitsgeraden auf. Zum Beispiel ist es möglich einen "Zeitentwicklungsoperator" in räumlicher Richtung aufzustellen. Dieser Zugang ermöglicht es uns Spektralstatistik auf langreichweitigen Energieskalen zu studieren und, zum ersten Mal, periodische Bahnen im Quantenspektrum eines Vielteilchensystems zu identifizieren

The Dynamics of Coupled Oscillators

The subject is introduced by considering the treatment of oscillators in Mathematics from the simple Poincar´e oscillator, a single variable dynamical process defined on a circle, to the oscillatory dynamics of systems of differential equations. Some models of real oscillator systems are considered. Noise processes are included in the dynamics of the system. Coupling between oscillators is investigated both in terms of analytical systems and as coupled oscillator models. It is seen that driven oscillators can be used as a model of 2 coupled oscillators in 2 and 3 dimensions due to the dependence of the dynamics on the phase difference of the oscillators. This means that the dynamics are easily able to be modelled by a 1D or 2D map. The analysis of N coupled oscillator systems is also described. The human cardiovascular system is studied as an example of a coupled oscillator system. The heart oscillator system is described by a system of delay differential equations and the dynamics characterised. The mechanics of the coupling with the respiration is described. In particular the model of the heart oscillator includes the baroreceptor reflex with time delay whereby the aortic fluid pressure influences the heart rate and the peripheral resistance. Respiration is modelled as forcing the heart oscillator system. Locking zones caused by respiratory sinus arrhythmia (RSA), the synchronisation of the heart with respiration, are found by plotting the rotation number against respiration frequency. These are seen to be relatively narrow for typical physiological parameters and only occur for low ratios of heart rate to respiration frequency. Plots of the diastolic pressure and heart interval in terms of respiration phase parameterised by respiration frequency illustrate the dynamics of synchronisation in the human cardiovascular system