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/

Global: Zendy Tools annual

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

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We consider a nonlinear autonomous random dynamical system of $N$ degrees offreedom coupled by Gaussian random interactions and characterized by acontinuous spectrum $n_{\mu}(\lambda)$ of real positive relaxation rates. UsingKac-Rice formalism, the computation of annealed complexities (both of stableequilibria and of all types of equilibria) is reduced to evaluating theaverages involving the modulus of the determinant of the random Jacobianmatrix. In the limit of large system $N\gg 1$ we derive exact analyticalresults for the complexities for short-range correlated coupling fields,extending results previously obtained for the "homogeneous" relaxation spectrumcharacterised by a single relaxation rate. We show the emergence of a "topologytrivialisation" transition from a complex phase with exponentially manyequilibria to a simple phase with a single equilibrium as the magnitude of therandom field is decreased. Within the complex phase the complexity of stableequilibria undergoes an additional transition from a phase with exponentiallysmall probability to find a single equilibrium to a phase with exponentiallymany stable equilibria as the fraction of gradient component of the field isincreased. The behaviour of the complexity at the transition is found only todepend on the small $\lambda$ behaviour of the spectrum of relaxation rates$n_{\mu}(\lambda)$ and thus conjectured to be universal. We also provide someinsights into a counting problem motivated by a paper by B. Spivak and A.Zyuzin of 2004 about wave scattering in a disordered nonlinear medium.

Counting equilibria in a random non-gradient dynamics with heterogeneous relaxation rates