English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

In this paper, we propose a general sparse decomposition of dynamical systemsprovided that the vector field and constraint set possess certain sparsestructures, which we call subsystems. This notion is based on causal dependencein the dynamics between the different states. This results in sparsedescriptions for fundamental problems from nonlinear dynamical systems: regionof attraction, maximum positively invariant set, and global attractor. Thedecompositions can be paired with any method for computing (outer)approximations of these sets to reduce the computation to lower dimensionalsystems. This is illustrated by methods from previous work based oninfinite-dimensional linear programming. This exhibits one example where thecurse of dimensionality is present and hence dimension reduction is crucial. Inthis context, for polynomial dynamics, we show that these problems admit asparse sum-of-squares (SOS) approximation with guaranteed convergence such thatthe number of variables in the largest SOS multiplier is given by the dimensionof the largest subsystem appearing in the decomposition. The dimension of suchsubsystems depends on the sparse structure of the vector field and theconstraint set; if the dimension of the largest subsystem is small compared tothe ambient dimension, this allows for a significant reduction in thecomputation time of the SOS approximations. Numerical examples accompany theapproach.

Sparse decompositions of nonlinear dynamical systems and applications to moment-sum-of-squares relaxations