Koopman Operator and its Approximations for Systems with Symmetries

Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, including complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces block diagonal structure in operator approximations, revealing hidden organization. Practically, if the data is symmetric, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise

Conservation Laws and Integrability of a One-dimensional Model of Diffusing Dimers

We study a model of assisted diffusion of hard-core particles on a line. The model shows strongly ergodicity breaking : configuration space breaks up into an exponentially large number of disconnected sectors. We determine this sector-decomposion exactly. Within each sector the model is reducible to the simple exclusion process, and is thus equivalent to the Heisenberg model and is fully integrable. We discuss additional symmetries of the equivalent quantum Hamiltonian which relate observables in different sectors. In some sectors, the long-time decay of correlation functions is qualitatively different from that of the simple exclusion process. These decays in different sectors are deduced from an exact mapping to a model of the diffusion of hard-core random walkers with conserved spins, and are also verified numerically. We also discuss some implications of the existence of an infinity of conservation laws for a hydrodynamic description.Comment: 39 pages, with 5 eps figures, to appear in J. Stat. Phys. (March 1997

Application of Shemesh theorem to quantum channels

Completely positive maps are useful in modeling the discrete evolution of quantum systems. Spectral properties of operators associated with such maps are relevant for determining the asymptotic dynamics of quantum systems subjected to multiple interactions described by the same quantum channel. We discuss a connection between the properties of the peripheral spectrum of completely positive and trace preserving map and the algebra generated by its Kraus operators $\mathcal{A}(A_1,\ldots A_K)$. By applying the Shemesh and Amitsur - Levitzki theorems to analyse the structure of the algebra $\mathcal{A}(A_1,\ldots A_K)$ one can predict the asymptotic dynamics for a class of operations

The robustness of democratic consensus

In linear models of consensus dynamics, the state of the various agents converges to a value which is a convex combination of the agents' initial states. We call it democratic if in the large scale limit (number of agents going to infinity) the vector of convex weights converges to 0 uniformly. Democracy is a relevant property which naturally shows up when we deal with opinion dynamic models and cooperative algorithms such as consensus over a network: it says that each agent's measure/opinion is going to play a negligeable role in the asymptotic behavior of the global system. It can be seen as a relaxation of average consensus, where all agents have exactly the same weight in the final value, which becomes negligible for a large number of agents.Comment: 13 pages, 2 fig