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Predrag Cvitanovic

Publications and source records attributed to Predrag Cvitanovic.

At least 19 recordsLinked to original sources
arXiv2016

Perturbation theory for the Fokker-Planck operator in chaos

The stationary distribution of a fully chaotic system typically exhibits a fractal structure, which dramatically changes if the dynamical equations are even slightly modified. Perturbative techniques are not expected to work in this situation. In contrast, the presence of additive noise smooths out the stationary distribution, and perturbation theory becomes applicable. We show that a perturbation expansion for the Fokker-Planck evolution operator yields surprisingly accurate estimates of long-time averages in an otherwise unlikely scenario.

nlin.CD↗
arXiv2015

Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system

The finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise. In the weak-noise approximation the neighborhoods of deterministic periodic orbits can be computed as distributions stationary under the action of a local Fokker-Planck operator and its adjoint. We derive explicit formulae for widths of these distributions in the case of chaotic dynamics, when the periodic orbits are hyperbolic. The resulting neighborhoods form a basis for functions on the attractor. The global stationary distribution, needed for calculation of long-time expectation values of observables, can be expressed in this basis.

nlin.CD↗
arXiv2012

Cartography of high-dimensional flows: A visual guide to sections and slices

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 symmetry-reduced `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.

nlin.CD↗
arXiv2012

Knowing when to stop: how noise frees us from determinism

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 global averages over a 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.

nlin.CD↗
arXiv2012

Revealing the state space of turbulent pipe flow by symmetry reduction

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. Projections of these solutions and their unstable manifolds from their $\infty$-dimensional symmetry-reduced state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organising turbulence in wall-bounded shear flows. Visualisations of the flow within the slice and its linearisation at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions, and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic attractor, which capture turbulent dynamics at transitional Reynolds numbers.

physics.flu-dyn↗
arXiv2011

Reduction of continuous symmetries of chaotic flows by the method of slices

We study continuous symmetry reduction of dynamical systems by the method of slices (method of moving frames) and show that a `slice' defined by minimizing the distance to a single generic `template' intersects the group orbit of every point in the full state space. Global symmetry reduction by a single slice is, however, not natural for a chaotic / turbulent flow; it is better to cover the reduced state space by a set of slices, one for each dynamically prominent unstable pattern. Judiciously chosen, such tessellation eliminates the singular traversals of the inflection hyperplane that comes along with each slice, an artifact of using the template's local group linearization globally. We compute the jump in the reduced state space induced by crossing the inflection hyperplane. As an illustration of the method, we reduce the SO(2) symmetry of the complex Lorenz equations.

nlin.CD↗
arXiv2009

How well can one resolve the state space of a chaotic map?

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. For chaotic, locally hyperbolic flows, this resolution depends on the interplay of the local stretching/contraction and the smearing due to noise. We propose to determine the `finest attainable' partition for a given hyperbolic dynamical system and a given weak additive white noise, by computing the local eigenfunctions of the adjoint Fokker-Planck operator along each periodic point, and using overlaps of their widths as the criterion for an optimal partition. The Fokker-Planck evolution is then represented by a finite transition graph, whose spectral determinant yields time averages of dynamical observables. Numerical tests of such `optimal partition' of a one-dimensional repeller support our hypothesis.

nlin.CD↗
arXiv2008

Unstable periodic orbits and heteroclinic connections in plane Couette flow

Equilibrium, traveling wave, and periodic orbit solutions of pipe, channel, and plane Couette flows can now be computed precisely at Reynolds numbers above the onset of turbulence. These invariant solutions capture the complex dynamics of wall-bounded rolls and streaks and provide a framework for understanding low-Reynolds turbulent shear flows as dynamical systems. We present fluid dynamics videos of plane Couette flow illustrating periodic orbits, a close pass of turbulent flow to a periodic orbit, and heteroclinic connections between unstable equilibria.

physics.flu-dyn↗
arXiv2008

Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics

We undertake a systematic exploration of recurrent patterns in a 1-dimensional Kuramoto-Sivashinsky system. For a small, but already rather turbulent system, the long-time dynamics takes place on a low-dimensional invariant manifold. A set of equilibria offers a coarse geometrical partition of this manifold. A variational method enables us to determine numerically a large number of unstable spatiotemporally periodic solutions. The attracting set appears surprisingly thin - its backbone are several Smale horseshoe repellers, well approximated by intrinsic local 1-dimensional return maps, each with an approximate symbolic dynamics. The dynamics appears decomposable into chaotic dynamics within such local repellers, interspersed by rapid jumps between them.

nlin.PS↗
arXiv2006

Periodic orbits in scattering from elastic voids

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.

nlin.CD↗
arXiv2006

Variational method for locating invariant tori

We formulate a variational fictitious-time flow which drives an initial guess torus to a torus invariant under given dynamics. 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.

nlin.CD↗
arXiv2005

Wave Chaos in Elastodynamic Cavity Scattering

The exact elastodynamic scattering theory is constructed to describe the spectral properties of two- and more-cylindrical cavity systems, and compared to an elastodynamic generalization of the semi-classical Gutzwiller unstable periodic orbits formulas. In contrast to quantum mechanics, complex periodic orbits associated with the surface Rayleigh waves dominate the low-frequency spectrum, and already the two-cavity system displays chaotic features.

nlin.CD↗
arXiv2004

Modulated Amplitude Waves in Bose-Einstein Condensates

We analyze spatio-temporal structures in the Gross-Pitaevskii equation to study the dynamics of quasi-one-dimensional Bose-Einstein condensates (BECs) with mean-field interactions. A coherent structure ansatz yields a parametrically forced nonlinear oscillator, to which we apply Lindstedt's method and multiple-scale perturbation theory to determine the dependence of the intensity of periodic orbits (``modulated amplitude waves'') on their wave number. We explore BEC band structure in detail using Hamiltonian perturbation theory and supporting numerical simulations.

nlin.CD↗
arXiv2003

Cycle expansions for intermittent maps

In a generic dynamical system chaos and regular motion coexist side by side, in different parts of the phase space. The border between these, where trajectories are neither unstable nor stable but of marginal stability, manifests itself through intermittency, dynamics where long periods of nearly regular motions are interrupted by irregular chaotic bursts. We discuss the Perron-Frobenius operator formalism for such systems, and show by means of a 1-dimensional intermittent map that intermittency induces branch cuts in dynamical zeta functions. Marginality leads to long-time dynamical correlations, in contrast to the exponentially fast decorrelations of purely chaotic dynamics. We apply the periodic orbit theory to quantitative characterization of the associated power-law decays.

nlin.CD↗
arXiv2003

Variational method for finding periodic orbits in a general flow

A variational principle for determining unstable periodic orbits of flows as well as unstable spatio-temporally periodic solutions of extended systems is proposed and implemented. 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 ``\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 Hénon-Heiles system, the circular restricted three-body problem, and the Kuramoto-Sivashinsky system in a weakly turbulent regime.

nlin.CD↗
arXiv2003

Turbulent fields and their recurrences

We introduce a new variational method for finding periodic orbits of flows and spatio-temporally periodic solutions of classical field theories, a generalization of the Newton method to a flow in the space of loops. The feasibility of the method is demonstrated by its application to several dynamical systems, including the Kuramoto-Sivashinsky system.

nlin.CD↗
arXiv2002

Stationary modulated-amplitude waves in the 1-D complex Ginzburg-Landau equation

We reformulate the one-dimensional complex Ginzburg-Landau equation as a fourth order ordinary differential equation in order to find stationary spatially-periodic solutions. Using this formalism, we prove the existence and stability of stationary modulated-amplitude wave solutions. Approximate analytic expressions and a comparison with numerics are given.

nlin.PS↗