# Dynamics, stability, and statistics on lattices and networks

## Abstract

These lectures aim at surveying some dynamical models that have been widely explored in the recent scientific literature as case studies of complex dynamical evolution, emerging from the spatio-temporal organization of several coupled dynamical variables. The first message is that a suitable mathematical description of such models needs tools and concepts borrowed from the general theory of dynamical systems and from out-of-equilibrium statistical mechanics. The second message is that the overall scenario is definitely reacher than the standard problems in these fields. For instance, systems exhibiting complex unpredictable evolution do not necessarily exhibit deterministic chaotic behavior (i.e., Lyapunov chaos) as it happens for dynamical models made of a few degrees of freedom. In fact, a very large number of spatially organized dynamical variables may yield unpredictable evolution even in the absence of Lyapunov instability. Such a mechanism may emerge from the combination of spatial extension and nonlinearity. Moreover, spatial extension allows one to introduce naturally disorder, or heterogeneity of the interactions as important ingredients for complex evolution. It is worth to point out that the models discussed in these lectures share such features, despite they have been inspired by quite different physical and biological problems. Along these lectures we describemore »

- Authors:

- Institut d' Etudes Avancées (IEA), Université de Cergy-Pontoise (France)

- Publication Date:

- OSTI Identifier:
- 22306193

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 7; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; DEGREES OF FREEDOM; EQUILIBRIUM; LECTURES; LYAPUNOV METHOD; NONLINEAR PROBLEMS; STATISTICAL MECHANICS; STATISTICS; STOCHASTIC PROCESSES

### Citation Formats

```
Livi, Roberto.
```*Dynamics, stability, and statistics on lattices and networks*. United States: N. p., 2014.
Web. doi:10.1063/1.4881526.

```
Livi, Roberto.
```*Dynamics, stability, and statistics on lattices and networks*. United States. doi:10.1063/1.4881526.

```
Livi, Roberto. Tue .
"Dynamics, stability, and statistics on lattices and networks". United States.
doi:10.1063/1.4881526.
```

```
@article{osti_22306193,
```

title = {Dynamics, stability, and statistics on lattices and networks},

author = {Livi, Roberto},

abstractNote = {These lectures aim at surveying some dynamical models that have been widely explored in the recent scientific literature as case studies of complex dynamical evolution, emerging from the spatio-temporal organization of several coupled dynamical variables. The first message is that a suitable mathematical description of such models needs tools and concepts borrowed from the general theory of dynamical systems and from out-of-equilibrium statistical mechanics. The second message is that the overall scenario is definitely reacher than the standard problems in these fields. For instance, systems exhibiting complex unpredictable evolution do not necessarily exhibit deterministic chaotic behavior (i.e., Lyapunov chaos) as it happens for dynamical models made of a few degrees of freedom. In fact, a very large number of spatially organized dynamical variables may yield unpredictable evolution even in the absence of Lyapunov instability. Such a mechanism may emerge from the combination of spatial extension and nonlinearity. Moreover, spatial extension allows one to introduce naturally disorder, or heterogeneity of the interactions as important ingredients for complex evolution. It is worth to point out that the models discussed in these lectures share such features, despite they have been inspired by quite different physical and biological problems. Along these lectures we describe also some of the technical tools employed for the study of such models, e.g., Lyapunov stability analysis, unpredictability indicators for “stable chaos,” hydrodynamic description of transport in low spatial dimension, spectral decomposition of stochastic dynamics on directed networks, etc.},

doi = {10.1063/1.4881526},

journal = {Journal of Mathematical Physics},

number = 7,

volume = 55,

place = {United States},

year = {Tue Jul 15 00:00:00 EDT 2014},

month = {Tue Jul 15 00:00:00 EDT 2014}

}