Topology, vorticity, and limit cycle in a stabilized Kuramoto–Sivashinsky equation

Abstract

A noisy stabilized Kuramoto–Sivashinsky equation is analyzed by stochastic decomposition. For values of the control parameter for which periodic stationary patterns exist, the dynamics can be decomposed into diffusive and transverse parts which act on a stochastic potential. The relative positions of stationary states in the stochastic global potential landscape can be obtained from the topology spanned by the low-lying eigenmodes which interconnect them. Numerical simulations confirm the predicted landscape. The transverse component also predicts a universal class of vortex-like circulations around fixed points. These drive nonlinear drifting and limit cycle motion of the underlying periodic structure in certain regions of parameter space. Our findings might be relevant in studies of other nonlinear systems such as deep learning neural networks.

Computation of Large-Dimension Jordan Normal Transform via Popular Platforms

“…More recently, Kwon, Ao and Thouless proved in stochastic system near fixed point that the strict Lyapunov function can be obtained with the help of Jordan transform [9]. The same can be extended to the work of Chen et al since 2020 [10,11].…”

“…On a very general consideration, the new algorithm proposed realizes Jordan transform without the limitation by the order of the matrix. It quickly gives the Jordan normal form J and the corresponding transform matrix P. When dealing with practical problems, such as the high-dimensional complex, nonlinear systems presented in [10,11,16,17], the method may play a crucial role and improve the computational efficiency. are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material.…”

Stochastic Gradient Descent and Anomaly of Variance-Flatness Relation in Artificial Neural Networks

“…The latter is found to be closely related to the vortexlike circulations in similar systems. [22] Such features may be further explored in the current field. It may be used to construct a more natural strategy targeting the catastrophic forgetting problem in DNNs.…”

“…Together they offer near a fixed point a Boltzmann-like probability distribution as well as flux-carrying stationary states without detailed balance. This methodology has been applied to a number of practical systems, [20][21][22] in particular, complex networks for biological studies. [23] In our approach, the covariance matrix for the dynamics is in fact inversely proportional to the energy matrix 𝑈 for the stochastic potential.…”

“…The stochastic decomposition itself can reveal rich structural features. In a related study, [22] the eigenvalues and eigenvectors of the 𝑆 and 𝐴 matrices [cf. Eq.…”