##### Kurt Roth

This page collects material for the lecture on *Chaotic, Complex, and Evolving Environmental Systems* in the winter term 2020/21. This course belongs to the master curriculum of the Department of Physics and Astronomy at the University of Heidelberg. As of now, this material is intended exclusively for students who are registered for that course.

I ask you to * not share any of this material*. If you feel that someone else should have access, please have that person drop me a note at kurt.roth@iup.uni-heidelberg.de.

**Legal**: This material is copyrighted – © Kurt Roth 2020 or by the copyright holders of cited material – and you are not allowed to distribute it by any means.

### lecture notes

These lecture notes are v0.6 of Summer 2020 (318 MB). They are more detailed than the slides and they are the reference whenever a discrepancy should exist. While the notes may appear like a completed work, they are not: there are abrupt ends, missing part, and chapter 9 is grossly incomplete. We’ll go with what there is.

Technically, this is one large pdf file with equations, figures, and references fully linked. The figures are typically high-resolution and you can zoom in to reveal greater detail. Finally, there is ample space for your annotations.

### lectures

Each of the following units covers an approximately two-hour lecture, with some of them a bit more packed, others lighter. With each comes a single slide presentation (pdf for your annotations) accompanied by 2…4 videos of the actual presentation.

**unit 1 – introduction & embedding**

Motivation for the course, larger thematic setting, and introduction of some language.

**slides**: 1-intro.pdf, 0-info-processing.pdf

**videos**:**1-1-intro** (motivation, rough contents, and operation of the course)**1-2-intro** (spaceship Earth and its operation)**1-3-intro** (what is different with environmental systems and aspects of their representation)**0-info-processing** (operational: how to deal with the information flood that likely comes with this course; “likely” because it is not just the material presented but all the lines that you may be enticed to follow on your own)

**unit 2 – nonlinear dynamical systems I**

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

**slides**: 2-nds.pdf

**videos**:**2-1-nds** (definitions and discussion of different examples)**2-2-nds** (one-dimensional, continuous, and autonomous systems)

**unit 3 – nonlinear dynamical systems II**

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

**slides**: 3-nds.pdf

**videos**:**3-1-nds** (characterization and linear stability of fixpoints)**3-2-nds** (invariant sets and manifolds)

**unit 4 – nonlinear dynamical systems III**

Topological limitations for differentiable dynamical systems, system transitions with bifurcations.

**slides**: 4-nds.pdf

**videos**:**4-1-nds** (topological limitations)**4-2-nds** (saddle node, transcritical, and pitchfork bifurcations in 1 and 2d)**4-3-nds** (Hopf and homoclinic bifurcations in 2d)

**unit 5 – discrete chaotic systems I**

Deterministic chaos in dynamical systems that are discrete in time. Workhorse: logistic map.

**slides**: 5-dch.pdf

**videos**:**5-1-dch** (discrete is different, iterated functions, linear stability)**5-2-dch** (logistic map I: bifurcation diagram, stationary regime, period-doubling cascade with Feigenbaum numbers and universality)

**unit 6 – discrete chaotic systems II**

Chaotic regime of logistic map with universality also in this regime.

**slides**: 6-dch.pdf

**videos**:**6-1-dch** (transition into chaos, strange attractors, structures in chaotic regime: supertracks, supercycles)**6-2-dch** (supertracks and supercycles more quantitatively, windows and Sharkovskii’s theorem, crises)

**unit 7 – discrete chaotic systems III**

The horseshoe map as a simplification of the logistic map, and a bare bones strange attractor. The phenomenology of fluid flow reflected on the logistic map. How common is chaos in discrete and in continuous systems? Two-dimensional discrete chaotic systems.

**slides**: 7-dch.pdf

**videos**:**7-1-dch** (horseshoe map, logistic map and fluid dynamics)**7-2-dch** (how common is chaos? discrete chaos in two dimensions)

**unit 8 – continuous chaotic systems I**

Deterministic chaos in systems that are continuous in time: the physical pendulum, periodically driven and viscously damped.

**slides**: 8-cch.pdf

**videos**:**8-1-cch** (physics of the driven and viscously damped pendulum; sensitive dependence of trajectories on initial conditions and noise)**8-2-cch** (compact descriptions: trajectories → phase diagrams → Poincaré maps)

**unit 9 – continuous chaotic systems II**

Conservative vs dissipative systems; the nature of strange attractors, temporal analysis (instead of our common spatial analysis), and transitions in parameter space, all illustrated with the periodically driven and viscously damped pendulum.

**slides**: 9-cch.pdf

**videos**:**9-1-cch** (attractors and the nature of strange attractors)**9-2-cch** (temporal analysis)**9-3-cch** (transitions in parameter space I: hysteresis, fragile regimes, symmetry bifurcatiion, transition to chaos)**9-4-cch** (transitions in parameter space II: larger parameter ranges, multiplicative and additive perturbations)

**unit 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).

**slides**: 10-cch.pdf

**videos**:**10-1-cch** (Rayleigh-Bénard convection and its lowest-dimensional approximation)**10-2-cch** (L63: fixpoints, stability, non-chaotic regime)

**unit 11 – continuous chaotic systems IV**

The chaotic regime of the L63 system with a focus on the transition from stable fixpoints into chaos as parameter r increases from 0, 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. Still deeper understanding of the nature of strange attractors.

**slides**: 11-cch.pdf

**videos**:**11-1-cch** (general operation of L63 system with u3-axis playing the key role; transient chaos, introduction of Lorenz map)**11-2-cch** (transition from regime with stable fixpoints to fully developed chaos; grand view of attracting set, cut along r-axis)

**unit 12 – complexity fundamentals I**

Complex vs chaotic systems; phenomenology of landscapes: critical slopes; more general geometries: fractal sets

**slides**: 12-cof.pdf

**videos**:**12-1-cof** (phenomenology of landscapes: critical slopes)**12-2-cof** (fractal sets)

**unit 13 – discrete complex systems I**

Cellular automata, the workhorse for discrete complex systems and far beyond; the BTW sand pile model; basics of contact processes with percolation and identification of clusters

**slides**: 13-dco.pdf

**videos**:**13-1-dco** (cellular automata)**13-2-dco** (BTW sand pile model)**13-3-dco** (basics of contact processes)

**unit 14 – discrete complex systems II**

Forest fire models and contagious disease model.

**slides**: 14-dco.pdf

**videos**:**14-1-dco** (forest fires)**14-2-dco** (contagious diseases)

**unit 15 – complexity fundamentals II**

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

**slides**: 15-cof.pdf

**videos**:**15-1-cof** (evidence for power law distributions in our environment)**15-2-cof** (possible causes for power law distributions)**15-3-cof** (characterizing complexity)

**unit 16 – patterns I**

Patterned landscapes and some physics behind them & the representation and classification of patterns and their dynamics

**slides**: 16-pat.pdf

**videos**:**16-1-pat** (sand ripples, crack networks, and vegetation patterns)**16-2-pat** (representation and classification)

**unit 17 – patterns II**

The Swift-Hohenberg model, its construction and phenomenology

**slides**: 17-pat.pdf

**videos**:**17-1-pat** (Swift-Hohenberg model – construction)**17-2-pat** (Swift-Hohenberg model – phenomenology)

**unit 18 – patterns III**

The reaction-diffusion model, a large class of process-based representations.

**slides**: 18-pat.pdf

**videos**:**18-1-pat** (reaction-diffusion model – construction & linear stability analysis)**18-2-pat** (Turing instability, Gray-Scott model)

**unit 19 – population dynamics I**

Fundamentals of population dynamics and the case of well-mixed stochastic systems.

**slides**: 19-pop.pdf

**videos**:**19-1-pop** (definitions and fundamentals of population dynamics)**19-2-pop** (dynamics of one- and two-species population in well-mixed (non-spatial) stochastic regime)

**unit 20 – population dynamics II**

Deterministic two-species interactions in non-spatial domains. Hence, two large populations in a common domain, with each population easily capable to traverse the whole domain.

**slides**: 20-pop.pdf

**videos**:**20-1-pop** (competition-cooperation or p-process)**20-2-pop** (predator-prey or s-process)

**unit 21 – population dynamics III**

Spatial domain with two and species coupled through s-process, hence a food chain.

**slides**: 21-pop.pdf

**videos**:**21-1-pop** (model formulation and setting of initial state)**21-2-pop** (2 species predator-prey system)**21-3-pop** (2 species predator-prey-grass system)

**unit 22 – evolution I**

Introduction to evolution with some definition, physical and chemical self-organization, necessities for and consequences of life, principles of Darwinian evolution.

**slides**: 22-evo.pdf

**videos**:**22-1-evo** (definitions and overview of Grand Unfolding)**22-2-evo** (physical aggregation & self-organization; library of Babel)**22-3-evo** (necessities for the emergence of life; Darwinian evolution; chemical self-organization and evolution; autocatalytic sets)

**unit 23 – evolution II**

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

**slides**: 23-evo.pdf

**videos**:**23-1-evo** (building blocks and operation of modern biomolecular machinery; from autocatalytic sets to the RNA world)**23-2-evo** (a scenario for the origin of life: warm submarine hydrothermal vents; diversity of extant life)

**unit 24 – evolution III**

Oxygenation of system Earth with its major reorganization of global biogeochemical regimes and cycles, the subsequent emergence and radiation of animals, and major evolutionary transitions in individuality.

**slides**: 24-evo.pdf

**videos**:**24-1-evo** (dynamic Earth; the innovation of photosynthesis with the subsequent Great Oxygenation Event (GOE) and the final Neoproterozoic Oxygenation Event (NOE))**24-2-evo** (the Ediacaran emergence of multicellular life, its rapid unfolding in the Cambrian radiation, and a major setback in the Permian-Triassic (PT) mass extinction as manifestations for: completion of a toolkit (invention) → delayed ecological impact (innovation) → saturation of accessible ecospace; overview of major evolutionary transitions in individuality)

**Utopia**

Did you want to play, maybe even work and do research with the models used in this course, and with quite a few more along the line? Check out the Utopia framework, which now has a beautiful homepage.

**unit 25 – evolution IV**

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

**slides**: 25-evo.pdf

**videos**:**25-1-evo** (some thoughts on evolutions)**25-2-evo** (imperfect self-replication leads to finite complexity; hierarchical modularity as way out)**25-**3**-evo** (adaptive evolution of microorganisms; evolutionary spiral)

**unit 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.

**slides**: 26-evo.pdf

**videos**:**26-1-evo** (the hierarchy of living beings; parasites, an evolutionary drive to the bottom while simultaneously driving the advancing front, with a glimpse on COVID-19)**26-2-evo** (the individual as an ecosystems, a holobiont composed of the multicellular host and its symbiotic microbiota)**26-**3**-evo** (culture – no, not humankind’s –, the line-up for the next major evolutionary transition with eusocial groups and superorganismal individuals)