Geoscientific systems are typically characterized by inherently nonlinear processes determining their structure and dynamics. Therefore, it is usually not possible to infer sufficiently holistic information on their temporal behavior from time series data when considering only classical linear concepts of statistics. As an alternative, a variety of dynamical systems based approaches for analyzing nonlinear time series have been developed, the foundations of which originate in the theory of nonlinear dynamical systems. In this short course, we will specifically introduce some prominent examples of contemporary analysis frameworks highlighting the methodological variety of nonlinear time series analysis.

The preliminary agenda of this short course will be as follows:

1. Why do we need nonlinear time series analysis methods in the geosciences?

2. Some fundamental notions
2.1. Dynamical systems and phase space
2.2. Chaos and predictability
2.3. Fractal scaling and dynamical persistence
2.4. A brief overview on existing nonlinear time series analysis methods

3. Recurrence analysis
3.1. Attractor reconstruction from time series: Embedding
3.2. Recurrences and recurrence plots
3.3. Dynamical invariants from recurrence plots
3.4. Measures of complexity: Recurrence quantification analysis
3.5. Recurrences and networks

4. Synchronization analysis
4.1. What is synchronization, and what not?
4.2. Types of synchronization
4.3. Phase synchronization: Hilbert transform approach
4.4. Event synchronization
4.5. Generalized synchronization: Detection by means of recurrences

All mentioned techniques will be illustrated by applications to real-world geoscientific data. Wherever possible, the presentation will also address possible solutions to treating typical problems of geoscientific data, such as non-stationarities, missing values, and uncertainties. Course materials (including example codes for platform-independent open-source software such as Python and R) will be made available to the participants upon request after the course.