Variational method for finding periodic orbits in a general flow.
TL;DR: A variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems by an initial loop approximating a periodic solution by a minimization of local errors along the loop.
Abstract: A variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems. An initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop. The "Newton descent" partial differential equation that governs this evolution is an infinitesimal step version of the damped Newton-Raphson iteration. The feasibility of the method is demonstrated by its application to the Henon-Heiles system, the circular restricted three-body problem, and the Kuramoto-Sivashinsky system in a weakly turbulent regime.
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TL;DR: In this paper, the authors describe some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows.
Abstract: Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. A significant advance in this respect has been the numerical discovery of simple invariant sets, such as nonlinear equilibria and periodic solutions, in well-resolved Navier-Stokes flows. This review describes some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows. It is shown that the near-wall regeneration cycle of coherent structures can be reproduced by such solutions. The typical similarity laws of turbulence, i.e., the Prandtl wall law and the Kolmogorov law for the viscous range, as well as the pattern and intensity of turbulence-driven secondary flow in a square duct can also be represented by these simple invariant solutions.
282 citations
TL;DR: In this article, the authors describe some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows.
Abstract: Recent remarkable progress in computing power and numerical analysis is enabling us to fill a gap in the dynamical systems approach to turbulence. One of the significant advances in this respect has been the numerical discovery of simple invariant sets, such as nonlinear equilibria and periodic solutions, in well-resolved Navier--Stokes flows. This review describes some fundamental and practical aspects of dynamical systems theory for the investigation of turbulence, focusing on recently found invariant solutions and their significance for the dynamical and statistical characterization of low-Reynolds-number turbulent flows. It is shown that the near-wall regeneration cycle of coherent structures can be reproduced by such solutions. The typical similarity laws of turbulence, i.e. the Prandtl wall law and the Kolmogorov law for the viscous range, as well as the pattern and intensity of turbulence-driven secondary flow in a square duct can also be represented by these simple invariant solutions.
208 citations
Cites background from "Variational method for finding peri..."
...Lan & Cvitanović (2004) formulated a global Newton step for periodic solutions....
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TL;DR: In this paper, the authors consider long-time simulations of two-dimensional turbulence body forced by on the torus with the purpose of extracting simple invariant sets or "exact recurrent flows" embedded in this turbulence.
Abstract: We consider long-time simulations of two-dimensional turbulence body forced by on the torus with the purpose of extracting simple invariant sets or ‘exact recurrent flows’ embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics using periodic orbit theory. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not fully in its asymptotic (strongly) turbulent regime. Here, a total of 50 recurrent flows are found with the majority buried in the part of phase space most populated by the turbulence giving rise to a good reproduction of the energy and dissipation p.d.f. However, at higher forcing amplitudes now in the asymptotic turbulent regime, the generated turbulence data set proves insufficiently long to yield enough recurrent flows to make viable predictions. Despite this, the general approach seems promising providing enough simulation data is available since it is open to extensive automation and naturally generates dynamically important exact solutions for the flow.
185 citations
TL;DR: In this paper, the trajectories of Kolmogorov flows are derived whose trajectories converge asymptotically to the equilibrium and travelling-wave solutions of the Navier-Stokes equations.
Abstract: We consider the incompressible Navier–Stokes equations with periodic boundary conditions and time-independent forcing. For this type of flow, we derive adjoint equations whose trajectories converge asymptotically to the equilibrium and travelling-wave solutions of the Navier–Stokes equations. Using the adjoint equations, arbitrary initial conditions evolve to the vicinity of a (relative) equilibrium at which point a few Newton-type iterations yield the desired (relative) equilibrium solution. We apply this adjoint-based method to a chaotic two-dimensional Kolmogorov flow. A convergence rate of is observed, leading to the discovery of new steady-state and travelling-wave solutions at Reynolds number . Some of the new invariant solutions have spatially localized structures that were previously believed to exist only on domains with large aspect ratios. We show that one of the newly found steady-state solutions underpins the temporal intermittencies, i.e. high energy dissipation episodes of the flow. More precisely, it is shown that each intermittent episode of a generic turbulent trajectory corresponds to its close passage to this equilibrium solution.
64 citations
TL;DR: In this paper, the authors present an analysis of the properties as well as the diverse applications and extensions of the method of stabilisation transformation, which was originally invented to detect unstable periodic orbits in chaotic dynamical systems.
62 citations
References
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01 Jan 1994
TL;DR: The Diskette v 2.06, 3.5''[1.44M] for IBM PC, PS/2 and compatibles [DOS] Reference Record created on 2004-09-07, modified on 2016-08-08.
Abstract: Note: Includes bibliographical references, 3 appendixes and 2 indexes.- Diskette v 2.06, 3.5''[1.44M] for IBM PC, PS/2 and compatibles [DOS] Reference Record created on 2004-09-07, modified on 2016-08-08
19,881 citations
Book•
01 Jan 2004
TL;DR: The fourth volume in a series of volumes devoted to self-contained and up-to-date surveys in the theory of ODEs was published by as discussed by the authors, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience.
Abstract: This handbook is the fourth volume in a series of volumes devoted to self contained and up-to-date surveys in the theory of ordinary differential equations, with an additional effort to achieve readability for mathematicians and scientists from other related fields so that the chapters have been made accessible to a wider audience. It covers a variety of problems in ordinary differential equations. It provides pure mathematical and real world applications. It is written for mathematicians and scientists of many related fields.
7,749 citations
Book•
16 Feb 2013
TL;DR: This well written book is enlarged by the following topics: B-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for theLR and QR algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations and preconditioning techniques.
Abstract: This well written book is enlarged by the following topics:
$B$-splines and their computation, elimination methods for
large sparse systems of linear equations, Lanczos algorithm for
eigenvalue problems, implicit shift techniques for the $LR$ and
$QR$ algorithm, implicit differential equations, differential
algebraic systems, new methods for stiff differential
equations, preconditioning techniques and convergence rate of
the conjugate gradient algorithm and multigrid methods for
boundary value problems. Cf. also the reviews of the German
original editions.
6,270 citations
TL;DR: A prototype equation to the Lorenz model of turbulence contains just one (second-order) nonlinearity in one variable as mentioned in this paper, which allows for a "folded" Poincare map (horseshoe map).
3,334 citations
Book•
23 Oct 1990
TL;DR: In this article, the three-body problem: Moon-Earth-Sun, Three Methods of Section, Periodic Orbits, and Surface of Solution is considered, as well as the Diamagnetic Kepler Problem.
Abstract: Contents: Introduction- The Mechanics of Lagrange- The Mechanics of Hamilton and Jacobi- Integrable Systems- The Three-Body Problem: Moon-Earth-Sun- Three Methods of Section- Periodic Orbits- The Surface of Solution- Models of the Galaxy and of Small Molecules- Soft Chaos and the KAM Theorem- Entropy and Other Measures of Chaos- The Anisotropic Kepler Problem- The Transition From Classical to Quantum Mechanics- The New World of Quantum Mechanics- The Quantization of Integrable Systems- Wave Functions in Classically Chaotic Systems- The Energy Spectrum of a Classically Chaotic System- The Trace Formula- The Diamagnetic Kepler Problem- Motion on a Surface of Constant Negative Curvature- Scattering Problems, Coding and Multifractal Invariant Measures- References- Index
3,239 citations
"Variational method for finding peri..." refers background in this paper
...The periodic orbit theory of classical and quantum chaos [1, 2] is one of the major advances in the study of long-time behavior of chaotic dynamical systems....
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