Monotonicity of the Period Function for Hamiltonian Systems with Potentials

In this thesis, we study a new criterion that determines the monotonic behaviour of the period function for Hamiltonian systems with potentials, we denote as oscillators. Departing from traditional methods, we partition each orbit into two segments and analyse the period restricted to these segments. Analogously to the period function, these restrictions can be extended as functions of energy. By examining how the potential deviates from a suitable potential of a harmonic oscillator, we derive a monotonicity criterion for these restrictions. Comparing these restrictions then allows us to establish a new monotonicity criterion for the period function. The strength of this technique lies in modifying a potential, while influencing and controlling the monotonic behaviour of the respective period function. In addition, we establish a link between a nonresonance condition - important for the theory of open book spaces - and a specific local monotonic behaviour of the period function. This relationship prompts us to modify potentials so that the newly obtained potential satisfies the prerequisites needed to apply the theory of open book spaces. Finally, we explore the dynamics of N oscillators, each sharing a common parameter-dependent double-well potential, and implement an all-to-all velocity coupling with a scalar coupling parameter. To adhere to the constraints of the open book spaces, we alter this double-well potential according to modifications we established. After replacing the original potential with its modification, we explore the dynamics of the resulting limiting process as the coupling parameter approaches zero. With a careful choice of initial conditions, we discover an intriguing movement through the pages of the open book.

Level thesis:degree_level

thesis.doctoral

Grantor dc:publisher

Universität Bielefeld

Year

2024

Author dc:creator

• Krause, Timo

Repository record source_url

https://pub.uni-bielefeld.de/record/2992894

OAI identifier oai:identifier

oai:pub.uni-bielefeld.de:2992894