Global potential, topology, and pattern selection in a noisy stabilized Kuramoto–Sivashinsky equation

Abstract

We formulate a general method to extend the decomposition of stochastic dynamics developed by Ao et al. [J. Phys. Math. Gen. 37, L25–L30 (2004)] to nonlinear partial differential equations which are nonvariational in nature and construct the global potential or Lyapunov functional for a noisy stabilized Kuramoto–Sivashinsky equation. For values of the control parameter where singly periodic stationary solutions exist, we find a topological network of a web of saddle points of stationary states interconnected by unstable eigenmodes flowing between them. With this topology, a global landscape of the steady states is found. We show how to predict the noise-selected pattern which agrees with those from stochastic simulations. Our formalism and the topology might offer an approach to explore similar systems, such as the Navier Stokes equation.

Topology, Vorticity and Limit Cycle in a Stabilized Kuramoto-Sivashinsky Equation

“…Also in the SI [42] we correct an error in our earlier work where there is an erroneous factor h in the expression (h a k−k ′ ) in Eqs. (34) and (35) of [23]. The global landscape Φ(κ) can be verified by comparing with Φ s (κ) from direct stochastic simulations for the probability distribution P (κ) in the presence of strong external noise with the algorithm of [34].…”

“…This approach can be refined by noticing that the entire set of fixed points forms an interconnected web [23]. There is always a pair of dominant eigenmodes of R 0 leaving from one state and flowing towards another state.…”

“…Complex systems far from equilibrium can rarely be described by well-established potentials or thermodynamic functions [1][2][3][4][5][6][7][8][9]. Real world problems such as the Navier-Stokes (NS) equation [8][9][10][11][12] and artificial deep neural networks (DNN) [13][14][15][16] are examples of such systems.…”

“…The methodology can be extended to nonlinear partial differential equations (PDEs) where the dynamical variables are labelled by continuous spatial coordinate(s). In an earlier work [23], a noisy one-dimensional stabilized Kuramoto-Sivashinsky (SKS) equation [24][25][26] was used to demonstrate the application of this. The SKS equation is derived formally [10] from an NS equation and it can describe a variety of physical phenomena with bifurcation instabilities [27][28][29][30].…”

“…Eq. ( 19) is known as a continuous Lyapunov equation [39][40][41] for which there exist efficient numerical algorithms [18,23]. From Eq.…”