arXiv subjects

Predrag Cvitanović

Publications and source records attributed to Predrag Cvitanović.

At least 19 recordsLinked to original sources

arXiv2026

A chaotic lattice field theory in two dimensions

We describe spatiotemporally chaotic (or turbulent) field theories discretized over d-dimensional lattices in terms of sums over their multi-periodic orbits. `Chaos theory' is here recast in the language of statistical mechanics, field theory, and solid state physics, with the traditional periodic orbits theory of low-dimensional, temporally chaotic dynamics a special, one-dimensional case. In the field-theoretical formulation, there is no time evolution. Instead, treating the temporal and spatial directions on equal footing, one determines the spatiotemporally periodic orbits that contribute to the partition sum of the theory, each a solution of the system's defining deterministic equations, with sums over time-periodic orbits of dynamical systems theory replaced here by sums of d-periodic orbits over d-dimensional spacetime geometries, the weight of each orbit given by the Jacobian of its spatiotemporal orbit Jacobian operator. The weights, evaluated by application of the Bloch theorem to the spectrum of periodic orbit's Jacobian operator, are multiplicative for spacetime orbit repeats, leading to a spatiotemporal zeta function formulation of the theory in terms of prime orbits.

nlin.CD↗

arXiv2025

Learning dissipation and instability fields from chaotic dynamics

To make predictions or design control, information on local sensitivity of initial conditions and state-space contraction is both central, and often instrumental. However, it is not always simple to reliably determine instability fields or local dissipation rates, due to computational challenges or ignorance of the governing equations. Here, we construct an alternative route towards that goal, by estimating the Jacobian of a discrete-time dynamical system locally from the entries of the transition matrix that approximates the Perron-Frobenius operator for a given state-space partition. Numerical tests on one- and two-dimensional chaotic maps show promising results.

nlin.CD↗

arXiv2024

Chaotic mixing in plane Couette turbulence

Lagrangian tracer particle trajectories for invariant solutions of the Navier-Stokes equations confined to the three-dimensional geometry of plane Couette flow are studied. Treating the Eulerian velocity field of an invariant solution as a dynamical system, the transport of these passive scalars along Lagrangian flow trajectories reveals a rich repertoire of different types of motion that can occur, including stagnation points, for which there is no fluid movement, and invariant tori, which obstruct chaotic mixing across the full volume of the plane Couette flow minimal cell. We determine the stability of these stagnation points, along with their stable and unstable manifolds, and find heteroclinic connections between them. These topological features produce a skeleton that shapes the passive tracer flow for a turbulent fluid, providing a first step to elucidating Lagrangian particle transport and mixing in three-dimensional Navier-Stokes turbulent flows.

physics.flu-dyn↗

arXiv2022

A chaotic lattice field theory in one dimension

Motivated by Gutzwiller's semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB `skeleton' for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no `Lyapunov horizon'; there is only an enumeration of lattice states that contribute to the theory's partition sum, each a global spatiotemporal solution of system's deterministic Euler-Lagrange equations. The reformulation aligns `chaos theory' with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the time-periodic orbits of dynamical systems theory are replaced by periodic $d$-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells. The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already 1-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, `time'-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

nlin.CD↗

arXiv2020

Linear encoding of the spatiotemporal cat

The dynamics of an extended, spatiotemporally chaotic system might appear extremely complex. Nevertheless, the local dynamics, observed through a finite spatiotemporal window, can often be thought of as a visitation sequence of a finite repertoire of finite patterns. To make statistical predictions about the system, one needs to know how often a given pattern occurs. Here we address this fundamental question within a spatiotemporal cat, a 1-dimensional spatial lattice of coupled cat maps evolving in time. In spatiotemporal cat, any spatiotemporal state is labeled by a unique 2-dimensional lattice of symbols from a finite alphabet, with the lattice states and their symbolic representation related linearly (hence "linear encoding"). We show that the state of the system over a finite spatiotemporal domain can be described with exponentially increasing precision by a finite pattern of symbols, and we provide a systematic, lattice Green's function methodology to calculate the frequency (i.e., the measure) of such states.

nlin.CD↗

arXiv2017

Relative periodic orbits form the backbone of turbulent pipe flow

Chaotic dynamics of low-dimensional systems, such as Lorenz or Rössler flows, is guided by the infinity of periodic orbits embedded in their strange attractors. Whether this also be the case for the infinite-dimensional dynamics of Navier--Stokes equations has long been speculated, and is a topic of ongoing study. Periodic and relative periodic solutions have been shown to be involved in transitions to turbulence. Their relevance to turbulent dynamics---specifically, whether periodic orbits play the same role in high-dimensional nonlinear systems like the Navier--Stokes equations as they do in lower-dimensional systems---is the focus of the present investigation. We perform here a detailed study of pipe flow relative periodic orbits with energies and mean dissipations close to turbulent values. We outline several approaches to reduction of the translational symmetry of the system. We study pipe flow in a minimal computational cell, and report a library of invariant solutions found with the aid of the method of slices. Detailed study of the unstable manifolds of a sample of these solutions is consistent with the picture that relative periodic orbits are embedded in the chaotic saddle and that they guide the turbulent dynamics.

physics.flu-dyn↗

arXiv2017

Symmetry reduction in high dimensions, illustrated in a turbulent pipe

Equilibrium solutions are believed to structure the pathways for ergodic trajectories in a dynamical system. However, equilibria are atypical for systems with continuous symmetries, i.e. for systems with homogeneous spatial dimensions, whereas relative equilibria (traveling waves) are generic. In order to visualize the unstable manifolds of such solutions, a practical symmetry reduction method is required that converts relative equilibria into equilibria, and relative periodic orbits into periodic orbits. In this article we extend the fixed Fourier mode slice approach, previously applied 1-dimensional PDEs, to a spatially 3-dimensional fluid flow, and show that is substantially more effective than our previous approach to slicing. Application of this method to a minimal flow unit pipe leads to the discovery of many relative periodic orbits that appear to fill out the turbulent regions of state space. We further demonstrate the value of this approach to symmetry reduction through projections (projections only possible in the symmetry-reduced space) that reveal the interrelations between these relative periodic orbits and the ways in which they shape the geometry of the turbulent attractor.

physics.flu-dyn↗

arXiv2016

Unstable manifolds of relative periodic orbits in the symmetry-reduced state space of the Kuramoto-Sivashinsky system

Systems such as fluid flows in channels and pipes or the complex Ginzburg-Landau system, defined over periodic domains, exhibit both continuous symmetries, translational and rotational, as well as discrete symmetries under spatial reflections or complex conjugation. The simplest, and very common symmetry of this type is the equivariance of the defining equations under the orthogonal group O(2). We formulate a novel symmetry reduction scheme for such systems by combining the method of slices with invariant polynomial methods, and show how it works by applying it to the Kuramoto-Sivashinsky system in one spatial dimension. As an example, we track a relative periodic orbit through a sequence of bifurcations to the onset of chaos. Within the symmetry-reduced state space we are able to compute and visualize the unstable manifolds of relative periodic orbits, their torus bifurcations, a transition to chaos via torus breakdown, and heteroclinic connections between various relative periodic orbits. It would be very hard to carry through such analysis in the full state space, without a symmetry reduction such as the one we present here.

nlin.CD↗

arXiv2015

Periodic eigendecomposition and its application to Kuramoto-Sivashinsky system

Periodic eigendecomposition, to be formulated in this paper, is a numerical method to compute Floquet spectrum and Floquet vectors along periodic orbits in a dynamical system. It is rooted in numerical algorithms advances in computation of 'covariant vectors' of the linearized flow along an ergodic trajectory in a chaotic system. Recent research on covariant vectors strongly strongly suggests that the physical dimension of inertial manifold of a dissipative PDE can be characterized by a finite number of 'entangled modes', dynamically isolated from the residual set of transient degrees of freedom. We anticipate that Floquet vectors display similar properties as covariant vectors. In this paper we incorporate periodic Schur decomposition to the computation of dynamical Floquet vectors, compare it with other methods, and show that the method can yield the full Floquet spectrum of a periodic orbit at every point along the orbit to high accuracy. Its power, and in particular its ability to resolve eigenvalues whose magnitude differs by hundreds of orders magnitude, is demonstrated by applying the algorithm to computation of the full linear stability spectrum of several periodic solutions in one dimensional Kuramoto-Sivashinsky flow.

nlin.CD↗

arXiv2015

Periodic orbit analysis of a system with continuous symmetry - a tutorial

Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid in terms of a Fourier series truncated to a finite number of modes. Here, we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar to those that appear in fluid dynamics problems. A crucial step in the analysis of such a system is symmetry reduction. We use the model to illustrate different symmetry-reduction techniques. Its relative equilibria are conveniently determined by rewriting the dynamics in terms of a symmetry-invariant polynomial basis. However, for the analysis of its chaotic dynamics, the `method of slices', which is applicable to very high-dimensional problems, is preferable. We show that a Poincaré section taken on the `slice' can be used to further reduce this flow to what is for all practical purposes a unimodal map. This enables us to systematically determine all relative periodic orbits and their symbolic dynamics up to any desired period. We then present cycle averaging formulas adequate for systems with continuous symmetry and use them to compute dynamical averages using relative periodic orbits. The convergence of such computations is discussed.

nlin.CD↗

arXiv2015

Reduction of SO(2) symmetry for spatially extended dynamical systems

Spatially extended systems, such as channel or pipe flows, are often equivariant under continuous symmetry transformations, with each state of the flow having an infinite number of equivalent solutions obtained from it by a translation or a rotation. This multitude of equivalent solutions tends to obscure the dynamics of turbulence. Here we describe the `first Fourier mode slice', a very simple, easy to implement reduction of SO(2) symmetry. While the method exhibits rapid variations in phase velocity whenever the magnitude of the first Fourier mode is nearly vanishing, these near singularities can be regularized by a time-scaling transformation. We show that after application of the method, hitherto unseen global structures, for example Kuramoto-Sivashinsky relative periodic orbits and unstable manifolds of travelling waves, are uncovered.

nlin.CD↗

arXiv2010

Continuous symmetry reduction and return maps for high-dimensional flows

We present two continuous symmetry reduction methods for reducing high-dimensional dissipative flows to local return maps. In the Hilbert polynomial basis approach, the equivariant dynamics is rewritten in terms of invariant coordinates. In the method of moving frames (or method of slices) the state space is sliced locally in such a way that each group orbit of symmetry-equivalent points is represented by a single point. In either approach, numerical computations can be performed in the original state-space representation, and the solutions are then projected onto the symmetry-reduced state space. The two methods are illustrated by reduction of the complex Lorenz system, a 5-dimensional dissipative flow with rotational symmetry. While the Hilbert polynomial basis approach appears unfeasible for high-dimensional flows, symmetry reduction by the method of moving frames offers hope.

nlin.CD↗

arXiv2009

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.

nlin.CD↗

arXiv2009

Equilibrium and traveling-wave solutions of plane Couette flow

We present ten new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number (Re) and two new traveling-wave solutions. The solutions are continued under changes of Re and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their 3D-physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low Re turbulence. Projections of these solutions and their unstable manifolds from their infinite-dimensional state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

physics.flu-dyn↗

arXiv2008

Heteroclinic connections in plane Couette flow

Plane Couette flow transitions to turbulence for Re~325 even though the laminar solution with a linear profile is linearly stable for all Re (Reynolds number). One starting point for understanding this subcritical transition is the existence of invariant sets in the state space of the Navier Stokes equation, such as upper and lower branch equilibria and periodic and relative periodic solutions, that are quite distinct from the laminar solution. This article reports several heteroclinic connections between such objects and briefly describes a numerical method for locating heteroclinic connections. Computing such connections is essential for understanding the global dynamics of spatially localized structures that occur in transitional plane Couette flow. We show that the nature of streaks and streamwise rolls can change significantly along a heteroclinic connection.

physics.flu-dyn↗

arXiv2008

Visualizing the geometry of state space in plane Couette flow

Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a dynamical, 10^5-dimensional state-space representation of plane Couette flow at Re = 400 in a small, periodic cell and offer a new method of visualizing invariant manifolds embedded in such high dimensions. We compute a new equilibrium solution of plane Couette flow and the leading eigenvalues and eigenfunctions of known equilibria at this Reynolds number and cell size. What emerges from global continuations of their unstable manifolds is a surprisingly elegant dynamical-systems visualization of moderate-Reynolds turbulence. The invariant manifolds tessellate the region of state space explored by transiently turbulent dynamics with a rigid web of continuous and discrete symmetry-induced heteroclinic connections.

physics.flu-dyn↗

arXiv2003

Diagrammatic Young Projection Operators for U(n)

We utilize a diagrammatic notation for invariant tensors to construct the Young projection operators for the irreducible representations of the unitary group U(n), prove their uniqueness, idempotency, and orthogonality, and rederive the formula for their dimensions. We show that all U(n) invariant scalars (3n-j coefficients) can be constructed and evaluated diagrammatically from these U(n) Young projection operators. We prove that the values of all U(n) 3n-j coefficients are proportional to the dimension of the maximal representation in the coefficient, with the proportionality factor fully determined by its S[k] symmetric group value. We also derive a family of new sum rules for the 3-j and 6-j coefficients, and discuss relations that follow from the negative dimensionality theorem.

hep-th↗

arXiv1993

A Fredholm Determinant for Semi-classical Quantization

We investigate a new type of approximation to quantum determinants, the ``\qFd", and test numerically the conjecture that for Axiom A hyperbolic flows such determinants have a larger domain of analyticity and better convergence than the \qS s derived from the \Gt. The conjecture is supported by numerical investigations of the 3-disk repeller, a normal-form model of a flow, and a model 2-$d$ map.

chao-dyn↗