# DYNAMIC SYSTEMS, CHAOS AND COMPLEXITY

**Academic Year 2023/2024**- Teacher:

**ALESSANDRO PLUCHINO**

## Expected Learning Outcomes

Acquisition of the fundamental concepts of the theory of dynamical systems and classical statistical mechanics. The course aims to provide students with a gradual introduction to the science of complex systems through a path that, starting from dynamical systems - both dissipative and conservative, both continuous and discrete - with a few degrees of freedom, already capable of manifesting chaotic behaviors, then moves to the study of systems with many degrees of freedom, to be addressed by means of a statistical approach, with particular attention to non-equilibrium phenomena, systems with long-range interactions and those at the edge of chaos. In this regard, alongside the standard classical statistical mechanics (Boltzmann-Gibbs) one of its most important generalizations will also be introduced, the so-called "non-extensive" statistical mechanics of Constantino Tsallis, particularly suited to the description of complex systems in the physical, biological or socio-economic context. In addition to the theoretical notions, the course provides for the detailed presentation of numerous examples of practical applications, so that the student can acquire the skills relating to mathematical methods and statistical or computational techniques necessary for the analysis of simple dynamical systems at few or many degrees of freedom. Finally, to the student will also be given notions of programming in the NetLogo environment, a free cross-platform software oriented towards agent simulations and here adapted to the exploration of dynamic systems with few and many degrees of freedom.

**With reference to the Dublin Descriptors, this course contributes to acquiring the following transversal skills:**

__Knowledge and understanding:__

- Inductive and deductive reasoning skills.
- Ability to schematize a natural phenomenon in terms of scalar and vector physical quantities.
- Ability to set up a problem using appropriate relationships between physical quantities (algebraic, integral or differential) and to solve it with analytical or numerical methods.
- Ability to perform statistical analysis of data.

__Ability to apply knowledge:__

- Ability to apply the acquired knowledge for the description of physical phenomena using rigorously the scientific method.
- Ability to design simple experiments and perform the analysis of experimental data obtained in all areas of interest of physics, including those with technological implications.

__Autonomy of judgment:__

- Critical reasoning skills.
- Ability to identify the most appropriate methods to critically analyze, interpret and process experimental data.
- Ability to identify the predictions of a theory or model.

__Communication skills:__

- Good skills in tools for the management of scientific information and for data processing and bibliographic research.
- Ability to present orally, with properties of language and terminological rigor, a scientific topic, illustrating its motivations and results.

__Learning ability:__

- Ability to know how to update their knowledge through the reading of scientific publications, in Italian or English, in the various fields of physical disciplines, even if not specifically studied during their training.

## Course Structure

The course includes 6 CFU (42 hours) of formal lectures, with applications, exercises and notes on the NetLogo environment for the development of numerical simulations. In addition to the lessons slides, students will be provided with appropriate software and audiovisual material.

## Required Prerequisites

You need to know:

(Indispensable) the contents of the mathematical analysis (1 and 2) and geometry courses.

(Important) the contents of physics 1 course .

(Useful) Hamiltonian formalism and variational principles.

## Attendance of Lessons

## Detailed Course Content

**Part I - Introduction to the new Science of Complexity**

From chaos theory to the new science of complexity - Catastrophe theory - Scale invariance, fractals and power laws

Self-organized criticality (SOC) - Complex networks: Small World and Scale Free - Wolfram and Conway cellular automata

Synchronization and Kuramoto model - Emerging phenomena on the edge of chaos - Synergetics - Sociophysics and Econophysics

Computational Social Science - Introduction to agent simulations: the NetLogo development environment. User Interface and Programming Language

**Part II - Dynamical systems with few degrees of freedom. Chaos and Fractals**

Continuous dissipative dynamical systems (flows) - Space of states - Non-intersection theorem

Flows in one dimension - Fixed point attractors - Stable fixed points (nodes) and unstable points (repellors) - Saddle points - The Logistic Equation

Flows in two dimensions - Fixed point and limit cycle attractors - Poincarè-Bendixson theorem - Lotka-Volterra and Brussellator equations - Poincarè section - Bifurcations Theory - Tangent bifurcation or "saddle-node" bifurcation - Hopf bifurcation

Flows in three dimensions - Fixed points and limit cycles in three dimensions - Poincarè plane - Floquet matrix - Stability of limit cycles - Quasi-periodic attractors - Routes towards chaos - Homoclinical and heteroclinical chaos - The Lorenz model - Lyapunov exponents

Discrete dissipative dynamical systems (maps) - One-dimensional maps - The Logistic Map - Attractors and bifurcation diagram - The Feigenbaum constants - Chaos and Lyapunov exponents - Stretching and folding - The edge of chaos

Two-dimensional dissipative maps - Hénon's map - Self-similarity and fractals - Koch's curve - Fractal dimensions: box-counting and correlation - Haussdorf dimension

Hamiltonian flows (conservative) - Hamilton equations - Phase space - Liouville theorem - Constants of motion and action-angle variables - Integrable and non-integrable systems - Hamiltonian systems in one dimension - Harmonic oscillator as a dynamic system - The rigid conservative pendulum and the forced-damped pendulum

Multi-dimensional Hamiltonian flows - The KAM theorem - Periodic, quasiperiodic and chaotic orbits - The Hénon-Heiles model

**Part III - Dynamical systems with many degrees of freedom. Thermodynamics and Statistical Mechanics**

Recalls of Thermodynamics - The equation of state of ideal gases - The first law of thermodynamics - Applications of the first law - The second law of thermodynamics - Carnot's theorem - Entropy - Thermodynamic potentials - The third law of thermodynamics

The kinetic theory according to Boltzmann: the μ space and the distribution function - Binary collisions - Classical and quantum diffusion - The Boltzmann transport equation - The Theorem H - The Maxwell-Boltzmann distribution - H Theorem and Entropy - Life and work of Ludwig Boltzmann (film)

Classical statistical mechanics - Liouville's theorem - Gibbs' ensemble theory - The ergodic theorem - Postulate of a priori equiprobability - Temporal mean and ensemble mean - Microcanonical ensemble - Additivity and extensivity of entropy - Thermodynamics and equation state of a classical ideal gas in microcanonical ensemble

The Canonical ensemble - The canonical partition function - Thermodynamics of an ideal gas in canonical ensemble - Energy fluctuations in the canonical ensemble - Equivalence between the canonical and microcanonical ensembles

Introduction to Generalized Statistical Mechanics - Complexity and long-range interactions - The Hamiltonian Mean Field model (HMF) - Kinchin and Abe axioms: generalized entropies - Equilibrium thermodynamics of the HMF model - Dynamic anomalies and quasistationary states - Dependence on the range of interaction - Generalized statistical mechanics - Synchronization and Kuramoto model - Coupled logistic maps at the Edge of Chaos

Cosmological considerations about the second law of thermodynamics, the arrow of time and the emergence of complexity in the universe - Fine tuning of fundamental constants - Weak and strong anthropic principle - Theories of Everything and Multiverse models

## Textbook Information

1) Robert C. Hilborn, “Chaos and nonlinear dynamics”, Oxford University Press, 2nd Ed. 2000

2) Steven Strogatz, “Nonlinear dynamics and chaos”, Westview Press 2001

3) K. Huang, “Meccanica Statistica”, Zanichelli 1997

4) A.Pluchino, "La firma della complessità. Una passeggiata al margine del caos", Malcor D' Edizione 2015

5) C.Tsallis, "Introduction to nonextensive statistical mechanics: approaching a complex world", Springer 2008

6) C. Gros, “*Complex and adaptive dynamical systems*”, Springer 2nd Ed. 2010

7) J.P. Sethna, “*Entropy, Order parameters and Complexity*”, Oxford University Press 2006

## Course Planning

Subjects | Text References | |
---|---|---|

1 | Introduction to the new science of complexity. (12 hours) | La Firma della Complessità, Slides |

2 | Dissipative dynamical systems, continuous (fluxes) and discrete (maps), at 1 and 2 dimensions. (11 hours) | R. Hilborn, S.Strogatz, Slides |

3 | Fluxes in three dimensions. (2 hours) | R. Hilborn, Slides |

4 | Routes towards chaos. (1 hour) | R. Hilborn, Slides |

5 | Lyapunov exponents and fractal dimension. (2 hours) | R. Hilborn, Slides |

6 | Hamiltonian Systems in 1 e 2 dimensions. (3 hours) | R. Hilborn, Slides |

7 | KAM Theorem. (1 hour) | R. Hilborn, Slides |

8 | Summary about thermodynamics. (2 hours) | K. Huang, Slides |

9 | Introduction to classic equilibrium statistical mechanics. (2 hours) | K. Huang, Slides |

10 | Boltzmann H Theorem. (2 hours) | K. Huang, Slides |

11 | Gibbs “Ensemble” Theory. (4 hours) | K. Huang, Slides |

12 | Introduction to non-extensive statistical mechanics. (2 hours) | C.Tsallis, Slides |

13 | Systems with long-range interactions. (2 hours) | C.tsallis, Articoli, Slides |

14 | Complex systems at the edge of chaos. (2 hours) | C.Tsallis, Articoli, Slides |

## Learning Assessment

### Learning Assessment Procedures

393 / 5.000

### Examples of frequently asked questions and / or exercises

The following questions are not exhaustive but represent just some examples:

- Speak about dissipative dynamical system in two dimensions;

- What are fractal dimensions and Lyapunov exponents?

- The Lorenz Attractor and the routes toward chaos;

- Entropy and second principle of thermodynamics;

- Mu space and Boltzmann distribution function;

- Obtain the partition function in microcanonical or canonical Ensemble Theory;

- Provide examples of dynamic systems at the edge of chaos;

- Provide examples of systems with long-range interactions;

- What is the difference between a complicated system and a complex one;