Papers by Predrag Cvitanovic
Periodic orbits in scattering from elastic voids
Abstract: The scattering determinant for the scattering of waves from several obstacles is consid... more Abstract: The scattering determinant for the scattering of waves from several obstacles is considered in the case of elastic solids with voids. The multi-scattering determinant displays contributions from periodic ray-splitting orbits. A discussion of the weights of such orbits is presented.
Sometimes a solution to a mathematical problem is so beautiful that it can impede further progres... more Sometimes a solution to a mathematical problem is so beautiful that it can impede further progress for a whole century. So is the case with the Killing-Cartan classification of semi-simple Lie algebras [Killing 1888; Cartan 1952]. It is elegant, it is beautiful, and it says that the 3 classical families and 5 exceptional algebras are all there is, but what does that mean?
(yy s+ yy~ t+ yyu) I and all s, t, and u dependence is contained in the three-dimensional integra... more (yy s+ yy~ t+ yyu) I and all s, t, and u dependence is contained in the three-dimensional integral d.(An exact evaluation of Jis given in the Appendix.) In the s channel the denominator has the form u'-yy s+ y, y. I t I+ yiy. Iu I and it vanishes along a two-sheet hyperboloid in y space.(There the integration contour for Z is defined by the Feynman prescription pp—te.) We are always able to go to a new set of y axes for which (10) diagonalizes.
Abstract. The scattering determinant for the scattering of waves from several obstacles is consid... more Abstract. The scattering determinant for the scattering of waves from several obstacles is considered in the case of elastic solids with voids. The scattering determinant displays contributions from closed ray splitting orbits. A discussion of the weights of such orbits is presented. Keywords: semiclassics, zeta function, scattering determinant, elastodynamics PACS: 03.65. Sq, 05.45. Mt, 46.40. Cd, 62.30.+ d
Volume 6513, number 3 PHYSICS LETTERS INFRA-RED STRUCTURE OF YANG-MILLS THEORIES Predrag CVITANOV... more Volume 6513, number 3 PHYSICS LETTERS INFRA-RED STRUCTURE OF YANG-MILLS THEORIES Predrag CVITANOVIC CERN, Geneva, Switzerland and Institute for Advanced Study, Princeton, USA Received 17 September 1976 22 November 1976 An analysis of QCD magnetic moment shows that all infra-red divergences are contained in the coupling constant re normalization. They are controlled by the tenor malization function for the pure Yang-Mills field.
Symbolic dynamics and Markov partitions for the stadium billiard
Abstract: We investigate the Bunimovich stadium dynamics and find that in the limit of infinitely... more Abstract: We investigate the Bunimovich stadium dynamics and find that in the limit of infinitely long stadium the symbolic dynamics is a subshift of finite type. For a stadium of finite length the Markov partitions are infinite, but the inadmissible symbol sequences can be determined exactly by means of the appropriate pruning front. We outline a construction of a sequence of finite Markov graph approximations by means of approximate pruning fronts with finite numbers of steps.
Revealing the state space of turbulent pipe flow by symmetry reduction
Abstract: Symmetry reduction by the method of slices is applied to pipe flow in order to quotient... more Abstract: Symmetry reduction by the method of slices is applied to pipe flow in order to quotient the stream-wise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits.
Abstract The dynamical theory of moderate Reynolds number turbulence triangulates the infinite-di... more Abstract The dynamical theory of moderate Reynolds number turbulence triangulates the infinite-dimensional Navier–Stokes state space by sets of exact solutions (equilibria, relative equilibria, periodic orbits, etc), which form a rigid backbone that enables us to describe and predict the sinuous motions of a turbulent fluid. We report on the determination of a set of unstable periodic orbits from close recurrences of the turbulent flow.
More than 60 million people worldwide are afflicted with epilepsy, and over 30% of these people a... more More than 60 million people worldwide are afflicted with epilepsy, and over 30% of these people are not sufficiently helped by medications [1]. Some of these patients have the option of surgically removing the seizure-generating part of the brain (the focus). While surgery is often successful in preventing seizures, it can have serious side effects such as memory loss–in up to 35% of patients [2]–or speech deficits, not to mention adverse events such as infections and hemorrhage [3].
1. INTRODUCTION AND SUMMARY The 19th century study of invariance groups reached its peak with Car... more 1. INTRODUCTION AND SUMMARY The 19th century study of invariance groups reached its peak with Cartan's classification of ccroplex Lie algebras. He gave an explicit construction of generators of all possible complex Lie algebras, but did not give the invariants associated with each algebra.
1.1 Why ChaosBook? 1 1.2 Chaos ahead 2 1.3 The future as in a mirror 3 1.4 A game of pinball 7 1.... more 1.1 Why ChaosBook? 1 1.2 Chaos ahead 2 1.3 The future as in a mirror 3 1.4 A game of pinball 7 1.5 Chaos for cyclists 10 1.6 Change in time 15 1.7 From chaos to statistical mechanics 17 1.8 Chaos: what is it good for? 18 1.9 What is not in ChaosBook 20 résumé 21 further reading 23 guide to exercises 25 exercises 26 references 26
A large conceptual gap separates the theory of low-dimensional chaotic dynamics from the infinite... more A large conceptual gap separates the theory of low-dimensional chaotic dynamics from the infinite-dimensional nonlinear dynamics of turbulence. Recent advances in experimental imaging, computational methods, and dynamical systems theory suggest a way to bridge this gap and make a fundamental breakthrough in our understanding of turbulence.
Abstract. The continuous and discrete symmetries of the Kuramoto-Sivashinsky system restricted to... more Abstract. 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 spatiotemporally chaotic dynamics. The continuous spatial translation symmetry leads to relative equilibria (traveling wave) and relative periodic orbit solutions.
The main idea I would like to convey here is the inexhaustible diversity and richness of the dyna... more The main idea I would like to convey here is the inexhaustible diversity and richness of the dynamical chaos whatever description you choose: trajectories, statistics or, recently, renormalization. The importance of this relatively new phenomenon~ the dynamical chaos-is in that it presents, even in very simple models to be discussed below, the surprising complexity of the structures and evolution characteristic of a broad range of processes in nature, including the highest levels of its organization.
Page 1. On the mode-locking universality for critical circle maps This article has been downloade... more Page 1. On the mode-locking universality for critical circle maps This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1990 Nonlinearity 3 873 (http://iopscience.iop.org/0951-7715/3/3/015) Download details: IP Address: 66.249.72.245 The article was downloaded on 05/07/2011 at 06:43 Please note that terms and conditions apply.
Chaos (Woodbury, N.Y.), 2012
Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows b... more Symmetry reduction by the method of slices quotients the continuous symmetries of chaotic flows by replacing the original state space by a set of charts, each covering a neighborhood of a dynamically important class of solutions, qualitatively captured by a 'template'. Together these charts provide an atlas of the symmetryreduced 'slice' of state space, charting the regions of the manifold explored by the trajectories of interest. Within the slice, relative equilibria reduce to equilibria and relative periodic orbits reduce to periodic orbits. Visualizations of these solutions and their unstable manifolds reveal their interrelations and the role they play in organizing turbulence/chaos.
Deterministic chaotic dynamics presumes that the state space can be partitioned arbitrarily finel... more Deterministic chaotic dynamics presumes that the state space can be partitioned arbitrarily finely. In a physical system, the inevitable presence of some noise sets a finite limit to the finest possible resolution that can be attained. Much previous research deals with what this attainable resolution might be, all of it based on a global averages over stochastic flow. We show how to compute the locally optimal partition, for a given dynamical system and given noise, in terms of local eigenfunctions of the Fokker-Planck operator and its adjoint. We first analyze the interplay of the deterministic dynamics with the noise in the neighborhood of a periodic orbit of a map, by using a discretized version of Fokker-Planck formalism. Then we propose a method to determine the 'optimal resolution' of the state space, based on solving Fokker-Planck's equation locally, on sets of unstable periodic orbits of the deterministic system. We test our hypothesis on unimodal maps.
Physical Review E, 2006
We formulate a fictitious-time flow equation which drives an initial guess torus to a torus invar... more We formulate a fictitious-time flow equation which drives an initial guess torus to a torus invariant under given dynamics, provided such torus exists. The method is general and applies in principle to continuous time flows and discrete time maps in arbitrary dimension, and to both Hamiltonian and dissipative systems.
Physical Review E, 1999
A matrix representation of the evolution operator associated with a nonlinear stochastic flow wit... more A matrix representation of the evolution operator associated with a nonlinear stochastic flow with additive noise is used to compute its spectrum. In the weak noise limit a perturbative expansion for the spectrum is formulated in terms of local matrix representations of the evolution operator centered on classical periodic orbits. The evaluation of perturbative corrections is easier to implement in this framework than in the standard Feynman diagram perturbation theory. The results are perturbative corrections to a stochastic analog of the Gutzwiller semiclassical spectral determinant computed to several orders beyond what has so far been attainable in stochastic and quantum-mechanical applications.
Nuclear Physics B, 1980
Evolution equations for single species parton distributions are solved analytically. For infrared... more Evolution equations for single species parton distributions are solved analytically. For infrared-convergent theories the solutions can be expressed in a closed form as convolutions over Bessel functions. For infrared-divergent theories, such as QCD, the exact solution involves an infinite number of convolutions. The accuracy of various approximations to this solution is estimated.
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Papers by Predrag Cvitanovic