1,719 research outputs found
On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain
The continuous and discrete symmetries of the Kuramoto-Sivashinsky system
restricted to a spatially periodic domain play a prominent role in shaping the
invariant sets of its chaotic dynamics. The continuous spatial translation
symmetry leads to relative equilibrium (traveling wave) and relative periodic
orbit (modulated traveling wave) solutions. The discrete symmetries lead to
existence of equilibrium and periodic orbit solutions, induce decomposition of
state space into invariant subspaces, and enforce certain structurally stable
heteroclinic connections between equilibria. We show, on the example of a
particular small-cell Kuramoto-Sivashinsky system, how the geometry of its
dynamical state space is organized by a rigid `cage' built by heteroclinic
connections between equilibria, and demonstrate the preponderance of unstable
relative periodic orbits and their likely role as the skeleton underpinning
spatiotemporal turbulence in systems with continuous symmetries. We also offer
novel visualizations of the high-dimensional Kuramoto-Sivashinsky state space
flow through projections onto low-dimensional, PDE representation independent,
dynamically invariant intrinsic coordinate frames, as well as in terms of the
physical, symmetry invariant energy transfer rates.Comment: 31 pages, 17 figures; added references, corrected typos. Due to file
size restrictions some figures in this preprint are of low quality. A high
quality copy may be obtained from
http://www.cns.gatech.edu/~predrag/papers/preprints.html#rp
Early-Warning Signs for Pattern-Formation in Stochastic Partial Differential Equations
There have been significant recent advances in our understanding of the
potential use and limitations of early-warning signs for predicting drastic
changes, so called critical transitions or tipping points, in dynamical
systems. A focus of mathematical modeling and analysis has been on stochastic
ordinary differential equations, where generic statistical early-warning signs
can be identified near bifurcation-induced tipping points. In this paper, we
outline some basic steps to extend this theory to stochastic partial
differential equations with a focus on analytically characterizing basic
scaling laws for linear SPDEs and comparing the results to numerical
simulations of fully nonlinear problems. In particular, we study stochastic
versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a
scaling law of the covariance operator in a regime where linearization is
expected to be a good approximation for the local fluctuations around
deterministic steady states. We compare these results to direct numerical
simulation, and study the influence of noise level, noise color, distance to
bifurcation and domain size on early-warning signs.Comment: Published in Communications in Nonlinear Science and Numerical
Simulation (2014
Stability of Spatial Optical Solitons
We present a brief overview of the basic concepts of the soliton stability
theory and discuss some characteristic examples of the instability-induced
soliton dynamics, in application to spatial optical solitons described by the
NLS-type nonlinear models and their generalizations. In particular, we
demonstrate that the soliton internal modes are responsible for the appearance
of the soliton instability, and outline an analytical approach based on a
multi-scale asymptotic technique that allows to analyze the soliton dynamics
near the marginal stability point. We also discuss some results of the rigorous
linear stability analysis of fundamental solitary waves and nonlinear impurity
modes. Finally, we demonstrate that multi-hump vector solitary waves may become
stable in some nonlinear models, and discuss the examples of stable
(1+1)-dimensional composite solitons and (2+1)-dimensional dipole-mode solitons
in a model of two incoherently interacting optical beams.Comment: 34 pages, 9 figures; to be published in: "Spatial Optical Solitons",
Eds. W. Torruellas and S. Trillo (Springer, New York
Neural Lyapunov Control
We propose new methods for learning control policies and neural network
Lyapunov functions for nonlinear control problems, with provable guarantee of
stability. The framework consists of a learner that attempts to find the
control and Lyapunov functions, and a falsifier that finds counterexamples to
quickly guide the learner towards solutions. The procedure terminates when no
counterexample is found by the falsifier, in which case the controlled
nonlinear system is provably stable. The approach significantly simplifies the
process of Lyapunov control design, provides end-to-end correctness guarantee,
and can obtain much larger regions of attraction than existing methods such as
LQR and SOS/SDP. We show experiments on how the new methods obtain high-quality
solutions for challenging control problems.Comment: NeurIPS 201
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