The slide sets collected here are marginally reworked versions of the sets used for the lecture *Chaotic, Complex, and Evolving Environmental Systems* during the winter term 2020/21 in the Department of Physics and Astronomy at Heidelberg University.

This course consists of 4 parts that are covered in 27 two-hour lectures. There is a slide set for each lecture.

#### part I – embedding

A glimpse at the operation of our environment, as a motivation and a source of examples, followed by an introduction to nonlinear dynamical systems.

##### 1 introduction

Why study chaos, complexity, and evolution? What is in this course?

As background: (i) Spaceship Earth and our attempts to operate it as both the essential motivation for studying chaotic, complex, and evolving systems and prime example for them. (ii) What is different between these systems and what is more commonly studied in physics? (iii) Some thoughts about representations of real systems.

Some thoughts on reading, more precisely on building knowledge from random access information.

##### 2 nonlinear dynamical systems I

Definition and illustration of dynamical systems, graphical and numerical solution of 1d-systems, linear stability of their fixpoints, Lipschitz continuity, example of logistic equation.

##### 3 nonlinear dynamical systems II

Two-dimensional dynamical systems, linear stability of their fixpoints, invariant sets and manifolds. Glycolysis model for illustration.

##### 4 nonlinear dynamical systems III

What can a differentiable dynamical system do, depending on its dimension? Topological limitations and system transitions with bifurcations.

#### part II – chaos

##### 5 discrete chaotic systems I

Deterministic chaos in dynamical systems that are discrete in time – discrete is different! – with the logistic map as workhorse. Period-doubling cascade and its universality.

##### 6 discrete chaotic systems II

Chaotic regime of logistic map with supertracks and supercycles. Universality also in the chaotic regime: Sharkovskii’s theorem.

##### 7 discrete chaotic systems III

The logistic map simpler (horseshoe map) and more complicated (fluid dynamics). The prevalence of chaotic regimes in discrete and continuous dynamical systems. Two-dimensional discrete systems.

##### 8 continuous chaotic systems I

Deterministic chaos in dynamical systems that are continuous in time: the periodically driven and viscously damped physical pendulum (I).

##### 9 continuous chaotic systems II

The periodically driven and viscously damped physical pendulum (II): conservative vs dissipative, the nature of strange attractors, temporal analysis (instead of our common spatial analysis), and transitions in parameter space.

##### 10 continuous chaotic systems III

Fluid convection (Rayleigh-Bénard) and its abstraction into the L63 system. Analysis as a dynamical system (fixpoints and non-chaotic regimes).

##### 11 continuous chaotic systems IV

The chaotic regime of the L63 system with a focus on the transition from stable fixpoints into chaos with no intervening period-doubling cascade, but with an interval where stable fixpoints and the chaotic regime coexist stably, in separate domains of the state space. Deeper understanding of the nature of strange attractors.

#### part III – complexity

##### 12 complexity – fundamentals I

Complexity vs chaos. Gain some inspiration from the phenomenology of landscapes (I): critical slopes. More general geometries: fractal sets and iterated affine maps.

##### 13 discrete complex systems I

Cellular automata, the workhorse for discrete complex systems and beyond. First applications: (i) BTW sand pile model and (ii) some basics of contact processes including percolation and identification of clusters.

##### 14 discrete complex systems II

Further applications of cellular automata: (i) forest fire models and (ii) contagious disease model.

##### 15 complexity – fundamentals II

Power law distributions for describing scale-free situations & an attempt to characterize complexity.

##### 16 complexity – patterns I

Patterned landscapes – sand ripples, crack networks, and vegetation patterns – and some physics behind them & the representation and classification of patterns and their dynamics.

##### 17 complexity – patterns II

The Swift-Hohenberg model, its construction and phenomenology.

##### 18 complexity – patterns III

The reaction-diffusion model, a large class of process-based representations exemplified by Turing instability and Gray-Scott model.

##### 19 population dynamics I

Some fundamentals of population dynamics. Urn model for well-mixed stochastic systems: Master equation and transition to deterministic regime.

##### 20 population dynamics II

Deterministic two-species interactions in well-mixed (non-spatial) domains, corresponding to two large populations in a common domain, with each population easily capable to traverse the entire domain. The interaction between them are either competition-cooperation (p-process) or predator-prey (s-process).

##### 21 population dynamics III

Spatial domain with two and three species a linear food chain (s-process).

#### part IV – evolution

##### 22 evolution I

Introduction to evolution with some definitions. The phases of physical and chemical self-organization. Necessities for and consequences of life. Principles of (traditional) Darwinian evolution.

##### 23 evolution II

From the origin of the biomolecular machinery to the origin of life.

##### 24 evolution III

Oxygenation of system Earth with the resulting major reorganization of global biogeochemical regimes and cycles. The eventual emergence and radiation of higher life (animals). An overview of the major evolutionary transitions in individuality.

##### 25 evolution IV

Elements of an evolution mechanics for the unfolding’s front with some thoughts on evolutions in different fields. The key element, the self-replicator, with its fundamental limitations. The evolutionary spiral as a conceptual sketch for evolution’s progression through major transitions in individuality.

##### 26 evolution V

A more comprehensive grasp on current life: the hierarchical structure of all beings, parasites as an evolutionary strategy and driving force, individuals as ecosystems (holobionts), and culture as the current front of evolutionary unfolding.

##### 27 evolution VI

Humankind’s cultural evolution, its emergence and unfolding, and the situation in transition today.

last modified May 11, 2021