Random Dynamical Systems and Network Dynamics

University of São Paulo (USP)

One-dimensional reversible random walks on ℤ driven by a bounded potential v : ℤ → ℝ are studied through the growth of the second moment M₂(n,v) = E(|Xₙ|²), for which upper and lower polynomial growth exponents are defined. For a generic potential in the product topology (in the sense of Baire), the upper exponent equals 1/2 while the lower exponent equals 0. Topologically typical dynamics therefore exhibit an intermittent regime in which arbitrarily long diffusive time scales coexist with arbitrarily long trapping time scales. This raises the problem of constructing explicit deterministic potentials realising prescribed intermittent time scales, as well as the possibility of potentials for which the two exponents coincide at a value in (0, 1/2).

University of São Paulo (USP)

Reconstructing the network interaction structure from multivariate time series is an important problem across many fields. When the network dynamics is written as a linear combination of multivariate polynomials, the reconstruction becomes an optimisation problem — but for large networks this problem does not always have a unique solution, leading to wrong reconstructions. The Ergodic Basis Pursuit (EBP) method leverages the statistical properties of the network dynamics to accurately reconstruct sparse networks: the restricted isometry property of the associated library matrix, a crucial condition for unique reconstruction, can be derived from the ergodic properties of the dynamics. Reconstruction is unique when the data length scales quadratically with node degree and logarithmically with network size. Compared to traditional methods, EBP needs significantly less data and is robust to noise, as validated on experimental time series from optoelectronic networks.

UNSW Sydney

A gentle introduction to network dynamics, the role randomness can play in this setting, some of the open challenges, an overview of the state of the art and pointers to current research directions.

Imperial College London

Products of i.i.d. random matrices sampled according to a probability measure on GL(d,ℝ) are considered. The transition to a zero top Lyapunov exponent regime is studied, showing that the expected number of back-and-forth escapes from a neighbourhood scales as 1/|λ| as λ approaches 0. Regularity of Lyapunov moments and invariant measures with respect to the underlying probability measure is also discussed.

Imperial College London

Differential equations with bounded noise of the form ẋ = f(x) + ξ(t), with |ξ(t)| < ε, are studied. The resulting set-valued dynamics exhibit finite-speed propagation, but determining reachable sets requires optimisation over an infinite-dimensional family of admissible noise realisations. Interpreting the noise as a control input, techniques from optimal control — specifically a Pontryagin-type maximum principle — can be used to select the extremal realisations that govern the boundary evolution of reachable sets. This leads to a higher-dimensional but deterministic dynamical system closely related to the original random differential equation, establishing a correspondence between invariant objects in the stochastic system and invariant manifolds of the deterministic one.

University of São Paulo (USP)

Results on the existence and stability of stationary measures for random dynamical systems within the framework of Markov operators. In this setting, stationary measures are fixed points of the associated Markov operator, and the stability properties of these fixed points provide information about the ergodic behaviour of the system.

Imperial College London

Methodologies to learn random iterated function systems from time series of partial observations, using delay embeddings.

Imperial College London

A geometric theory for random dynamical systems with bounded noise, based on the boundary map (or flow). The focus is on structural properties of deterministic attractors and how these reflect statistical features of the orbits through stationary probability measures, including regularity and singularities, stability under perturbations, local and global bifurcation scenarios, and directions for coupled and networked systems.

Imperial College London

State University of Campinas (UNICAMP)

Conditions under which thermodynamic formalism can be built for families of random hyperbolic diffeomorphisms are presented, together with the uniqueness of equilibrium states and their statistical properties. Joint work with Lucas Amorim, Benoît Saussol and Sandro Vaienti.

University of São Paulo (USP)

A synchronisation phenomenon distinctive of hypernetworks, called hyperlocking, is presented. A system of three coupled oscillators with nearly identical frequencies interacts through a triadic hypernetwork motif in which one node modulates the interaction between the other two. Using normal form transformations and phase reduction, it is derived analytically how a specific choice of coupling functions induces hyperlocking; the predictions are confirmed with numerical simulations and chemical experiments.

University of São Paulo (USP)

Random dynamical systems generated by the independent and identically distributed iteration of contractions — Iterated Function Systems — are studied. The asymptotic behaviour of random orbits is investigated and a characterisation of their attractors is provided. The main result establishes the existence and uniqueness of stationary measures for these systems, a property commonly known as the Letac principle.

University of São Paulo (USP)

Random Iterated Function Systems in random environments are studied as a framework for modelling adaptive systems. Unlike classical i.i.d. IFS, where maps are chosen independently from a fixed distribution, this model introduces an underlying environment that evolves over time: the choice of the map depends both on the current environmental state and on a source of independent innovation. While the specific choice at each step is random, the available options are dictated by the shifting background of the environment.

Imperial College London

Numerical explorations of bifurcations of attractors in the Hénon map with additive bounded noise. The bifurcations are analysed via a finite-dimensional boundary map that preserves the normal bundles of the boundaries of the attractors, and the challenges arising in the presence of chaotic dynamics are discussed.