scholarly journals Stabilized Kuramoto-Sivashinsky equation: A useful model for secondary instabilities and related dynamics of experimental one-dimensional cellular flows

2007 ◽  
Vol 76 (1) ◽  
Author(s):  
P. Brunet
Keyword(s):  
One Dimensional ◽  
Cellular Flows ◽  
Secondary Instabilities ◽  
Sivashinsky Equation

Topological classification of periodic orbits in the Kuramoto–Sivashinsky equation

2018 ◽  
Vol 32 (15) ◽  
pp. 1850155 ◽  
Author(s):  
Chengwei Dong
Keyword(s):  
Nonlinear System ◽  
Variational Method ◽  
Periodic Orbits ◽  
Symbolic Dynamics ◽  
Chaotic Systems ◽  
Building Blocks ◽  
Topological Classification ◽  
One Dimensional ◽  
Sivashinsky Equation

In this paper, we systematically research periodic orbits of the Kuramoto–Sivashinsky equation (KSe). In order to overcome the difficulties in the establishment of one-dimensional symbolic dynamics in the nonlinear system, two basic periodic orbits can be used as basic building blocks to initialize cycle searching, and we use the variational method to numerically determine all the periodic orbits under parameter [Formula: see text] = 0.02991. The symbolic dynamics based on trajectory topology are very successful for classifying all short periodic orbits in the KSe. The current research can be conveniently adapted to the identification and classification of periodic orbits in other chaotic systems.

Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation

1994 ◽  
Vol 49 (1) ◽  
pp. 166-183 ◽  
Author(s):  
Chaouqi Misbah ◽  
Alexandre Valance
Keyword(s):  
Secondary Instabilities ◽  
Sivashinsky Equation

scholarly journals LYAPUNOV EXPONENTS OF THE KURAMOTO–SIVASHINSKY PDE

2019 ◽  
Vol 61 (3) ◽  
pp. 270-285 ◽  
Author(s):  
RUSSELL A. EDSON ◽  
J. E. BUNDER ◽  
TRENT W. MATTNER ◽  
A. J. ROBERTS
Keyword(s):  
Differential Equation ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Large Range ◽  
Nonlinear Partial Differential Equation ◽  
Bifurcation Parameter ◽  
One Dimensional ◽  
Lyapunov Spectra ◽  
Sivashinsky Equation

The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence.

scholarly journals Nonlinear Forecasting of the Generalized Kuramoto–Sivashinsky Equation

2015 ◽  
Vol 25 (05) ◽  
pp. 1530015 ◽  
Author(s):  
Hiroshi Gotoda ◽  
Marc Pradas ◽  
Serafim Kalliadasis
Keyword(s):  
Chaotic Dynamics ◽  
Control Parameter ◽  
Deterministic Chaos ◽  
High Dimensional ◽  
One Dimensional ◽  
Forecasting Method ◽  
Nonlinear Forecasting ◽  
The One ◽  
Low Dimensional ◽  
Sivashinsky Equation

The emergence of pattern formation and chaotic dynamics is studied in the one-dimensional (1D) generalized Kuramoto–Sivashinsky (gKS) equation by means of a time-series analysis, in particular, a nonlinear forecasting method which is based on concepts from chaos theory and appropriate statistical methods. We analyze two types of temporal signals, a local one and a global one, finding in both cases that the dynamical state of the gKS solution undergoes a transition from high-dimensional chaos to periodic pulsed oscillations through low-dimensional deterministic chaos while increasing the control parameter of the system. Our results demonstrate that the proposed nonlinear forecasting methodology allows to elucidate the dynamics of the system in terms of its predictability properties.

Circular Nodes in Neural Networks

1996 ◽  
Vol 8 (2) ◽  
pp. 390-402 ◽  
Author(s):  
Michael J. Kirby ◽  
Rick Miranda
Keyword(s):  
Neural Network ◽  
Network Architecture ◽  
Dimensional Manifold ◽  
Almost Periodic ◽  
One Dimensional ◽  
Trefoil Knot ◽  
Hidden Layer ◽  
The Individual ◽  
Backward Propagation ◽  
Sivashinsky Equation

In the usual construction of a neural network, the individual nodes store and transmit real numbers that lie in an interval on the real line; the values are often envisioned as amplitudes. In this article we present a design for a circular node, which is capable of storing and transmitting angular information. We develop the forward and backward propagation formulas for a network containing circular nodes. We show how the use of circular nodes may facilitate the characterization and parameterization of periodic phenomena in general. We describe applications to constructing circular self-maps, periodic compression, and one-dimensional manifold decomposition. We show that a circular node may be used to construct a homeomorphism between a trefoil knot in ℝ3 and a unit circle. We give an application with a network that encodes the dynamic system on the limit cycle of the Kuramoto-Sivashinsky equation. This is achieved by incorporating a circular node in the bottleneck layer of a three-hidden-layer bottleneck network architecture. Exploiting circular nodes systematically offers a neural network alternative to Fourier series decomposition in approximating periodic or almost periodic functions.

scholarly journals Lyapunov exponents of the Kuramoto--Sivashinsky PDE

2019 ◽  
Vol 61 ◽  
pp. 270-285
Author(s):  
Russell A. Edson ◽  
Judith E. Bunder ◽  
Trent W. Mattner ◽  
Anthony J. Roberts
Keyword(s):  
Differential Equation ◽  
Lyapunov Exponents ◽  
Chaotic Dynamics ◽  
Large Range ◽  
Nonlinear Partial Differential Equation ◽  
Bifurcation Parameter ◽  
One Dimensional ◽  
Lyapunov Spectra ◽  
Sivashinsky Equation

The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence. doi:10.1017/S1446181119000105

scholarly journals The Role of Scales in the Dynamics of Parameterization Uncertainties

2006 ◽  
Vol 63 (6) ◽  
pp. 1659-1671 ◽  
Author(s):  
S. Vannitsem
Keyword(s):  
Spectral Characteristics ◽  
Atmospheric Model ◽  
Model Error ◽  
Error Growth ◽  
Model Uncertainties ◽  
One Dimensional ◽  
Spatially Distributed ◽  
The One ◽  
Sivashinsky Equation

Abstract The dynamics of model error due to parameterization uncertainties are investigated in the context of two spatially distributed systems: the one-dimensional convection system known as the extended Kuramoto–Sivashinsky equation and a quasigeostrophic atmospheric model. In addition to the different phases of error growth already reported for low-order systems, unexpected behaviors associated with the spectral characteristics of the model perturbation sources have been brought out. Notably, the predictability of the system is less affected by model uncertainties acting at small scales than at larger ones. An interpretation in terms of the spectral properties of the Lyapunov vectors is advanced.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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