NL Dynamics w. Diffeomorphism
Explore nonlinear dynamics properties without tracking. Novel methods using diffeomorphism maps to understand beam dynamics in storage rings.
Nonlinear Dynamics Diffeomorphism Storage Rings
Overview
This project develops new analytical tools to study nonlinear beam dynamics in storage rings without relying on particle tracking simulations. By leveraging diffeomorphism maps — smooth, invertible coordinate transformations — we can characterize the topology of phase space and predict dynamic aperture from first principles.
Motivation
Particle tracking is a traditional method for determine the long-term stability of a ring accelerator. However, it is also computationally expensive and miss the physics interpretation of non-stable motions. Diffeomorphism-based approaches allow us to:
- Identify invariant tori and chaotic regions analytically
- Apply the KAM (Kolmogorov–Arnold–Moser) theorem to assess stability
- Faster optimizing lattice designs for maximum dynamic aperture
Methods
- Construction of square matrix near fixed points
- Computation of approximated action-angle variables
- Iterative method to get precise frequency and orbit diffeomorphism to a rigid rotation
- Infer topology using the iterative results
Extensions
- Expand the method from 4-D to 6-D or even higher dimensions
- Include self-field nonlinearity, such as beam-beam effect and space charge effects