**Dynamic System Analytics in the Earth System Sciences**

Analysing complex and dynamic earth and environmental systems often reaches technical and theoretical limits of statistical standard tools. This is the case as most classical statistical approaches fundamentally require assumptions about statistical independence and stationarity, which are especially problematic as soon as we deal with non-linearity and coevolution. Many promising approaches founded in scientific fields ranging from theoretical physics to information theory have been developed. Still, their application to applied climate and environmental sciences remains challenging.

This 6 ECTS course introduces a set of modern tools for rigorous analyses of dynamical systems. Starting off with examples from geophysics and fluid mechanics the course will guide you during an excursion through stochastic physics, information theory, phase-state analyses and applications with big data and scaling. We especially invite you to bring own data and application questions to provide hands-on utilisation of the concepts and tools within your field of research.

**Programme**

**1 – From fluid mechanics to dynamical systems**

The first module provides a common basis for all participants. With fundamentals from classical fluid dynamics, thermodynamics, stability and scaling laws the foundation is laid. The theoretical lectures are complemented with practical analytical and numerical examples across the earth sciences.

**2 – Coevolutionary dynamical systems**

The second module extends the classical fluid dynamics with stochastic physics and information theory. With this, complexity is rigorously treated in a simple and coherent framework providing the physical background to coevolutionary dynamics and organisation. The module is halved into theoretical derivations and real-world applications from multiscale geophysical fluid dynamics.

**3 – Mastering conjugated dimensions**

The third module will redefine the state-space system analyses with physical principles and thermodynamic limits as enhanced phase-state analyses. This steers the participants towards a thorough dynamic system understanding without the requirement for single attractors, finite phases and fixed scales. It also includes conjugated pairs and a new breed of non-paired conjugates. The theoretical derivations are followed by analytical and numerical exercises from simple synthetic cases to real-world applications.

**4 – Implementation and frontier topics**

The fourth module will start with the application of the prior modules to systems chosen by the participants, who are encouraged to bring associated resources (e.g. datasets). Subsequently frontier topics of coevolutionary scaling and big data applications will be motivated, by presenting ongoing methodological development projects where participants are encouraged to engage.