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Predrag Cvitanović: periodic orbit theory publications

	Periodic orbit research overview
	Abstract of a general colloquium on the periodic orbit theory

Program whose goal is a theory of turbulent dynamics of classical, stochastic and quantum fields. Chaotic field theory is developed in two directions: 1) explorations of the feasibility of describing weak turbulence in terms of spatiotemporally recurrent patterns, and 2) new perturbation theory methods for computing corrections about nontrivial saddles of path integrals.

Read the sketch paper first, perhaps click through the trace formulas seminar next.

Predrag Cvitanović Chaotic Field Theory: a sketch, Physica A 288 , 61 (2000) [ arXiv:nlin.CD/0001034 | DOI | NSF review panel critique ]

We have been doing it all wrong. Junk your forward-in time integrators: a deterministic, from the equations up, no statistical assumptions theory of turbulence requires a spatiotemporal formulation.

	A deep dive into turbulence in spacetime: seminars, papers and notes
	H. Liang and P Cvitanović A chaotic lattice field theory in one dimension J. Phys. A 55, 304002 (2022) [ arXiv:2201.11325 | DOI ]
	Our spacetime discretized model field theory, modern formulation.
	B Gutkin, P Cvitanović, R Jafari, A K Saremi and L Han Linear encoding of the spatiotemporal cat map Nonlinearity 34, 2800--2836 (2021) [ arXiv:1912.02940 | DOI ]
	Our spacetime discretized model field theory, pre-1969 Soviet style approach.

Getting hang of turbulence for plumbers: pipes and planes. Program manifesto, gentle multimedia introduction, a few talks.

	Predrag Cvitanović You know how to cycle? That's rich J. Fluid Mech. Focus on Fluids 722, (2013)
	Predrag Cvitanović and John F. Gibson Geometry of state space in plane Couette flow in Bruno Eckhardt, ed., Adv. in Turbulence XII, Proc. 12th EUROMECH Eur. Turb. Conf., Marburg, 78 (Springer, New York 2009)
	John F. Gibson and Predrag Cvitanović Geometry of turbulence in wall-bounded shear flows: a stroll through 61,506 dimensions recommended
	Predrag Cvitanović and John F. Gibson Geometry of turbulence in wall-bounded shear flows: Periodic orbits Phys. Scr. T142, 014007 (2010)
	3 streaming videos of Newton Inst presentations

Planes and pipes: technical papers

	N. Burak Budanur, Kimberly Y. Short, Mohammad Farazmand, Ashley P. Willis and Predrag Cvitanović Relative periodic orbits form the backbone of turbulent pipe flow [ arXiv:1705.03720 ]
	Ashley P. Willis, Kimberly Y. Short, and Predrag Cvitanović Symmetry reduction in high dimensions, illustrated in a turbulent pipe Phys. Rev. E 93, 022204 (2016) [ arXiv:1504.05825 ]
	Ashley P. Willis, Predrag Cvitanović and Marc Avila Revealing the state space of turbulent pipe flow by symmetry reduction J. Fluid Mech. 721, 514-540 (2013) [ arXiv:1203.3701 ]
	John F. Gibson, Jonathan Halcrow and Predrag Cvitanović Visualizing the geometry of state space in plane Couette flow J. Fluid Mech. 611, 107 (2008) [ arXiv:0705.3957 ]
	John F. Gibson, Jonathan Halcrow,and Predrag Cvitanović Equilibrium and traveling-wave solutions of plane Couette flow J. Fluid Mech. 638, 243 (2009) [ arXiv:0808.3375 ]
	Jonathan Halcrow, John F. Gibson, Predrag Cvitanović and Divakar Viswanath Heteroclinic connections in plane Couette flow J. Fluid Mech. 621, 365 (2009) [ arXiv:0808.1865 ]
	Divakar Viswanath and Predrag Cvitanović Stable manifolds and the transition to turbulence in pipe flow J. Fluid Mech. 627, 215 (2009) [ arXiv:0801.1918 ]

State space geometry of Kuramoto-Sivashinsky flow

Explorations of weak turbulence described in terms of spatiotemporally recurrent patterns.

	Xiong Ding, H. Chaté, Predrag Cvitanović, Evangelos Siminos, and K. A. Takeuchi Estimating dimension of inertial manifold from unstable periodic orbits Phys. Rev. Lett. 117, 024101 (2017) [ arXiv:1604.01859 | DOI ]
	Related links: Turbulence: How fat is it? , a video of a KITP talk
	Predrag Cvitanović, Ruslan L. Davidchack and Evangelos Siminos On state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain SIAM J. Appl. Dyn. Syst. 9, 1 (2010) [ arXiv:0709.2944 ]
	Yueheng Lan and Predrag Cvitanović Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics Phys. Rev. E 78, 026208 (2008) [ arXiv:0804.2474 ]
	Predrag Cvitanović, Freddy Christiansen and Vahtang Putkaradze Hopf's last hope: spatiotemporal chaos in terms of unstable recurrent patterns Nonlinearity 10, 50 (1997) [ arXiv:chao-dyn/9606016 | NSF critique ]

	Predrag Cvitanović and Y. Lan Turbulence and what to do about it (Bristol, Gottingen 2004 - PDF overheads)
	Predrag Cvitanović Spatiotemporal chaos in terms of unstable recurrent patterns (overheads + streaming video (!), seminar abstract)
	Related links: Vachtang Putkaradze PhD thesis The initial attempt: Vachtang's term paper
	Why ``Hopf's last hope'' in the title?

How to quotient a symmetry of an equvariant dynamical system, rewrite the dynamics in the reduced, invariant coordinates.

This was meant to be a brief explanatory section in the ChaosBook.org chapter on trace formulas for systems with a continuous symmetry. Instead, I have gone the way of Marsdenites and am writing one paper after the other, each one looking very much like the preceeding one to a vulgar eye. Hopefully this all will eventually get digested into a single chapter of ChaosBook.org, and be done with.

	N. Burak Budanur and Predrag Cvitanović Unstable manifolds of relative periodic orbits in the symmetry-reduced state space of the Kuramoto-Sivashinsky system J. Stat. Phys. , (2016) [ arXiv:1509.08133 | ReadCube | DOI ]
	N. Burak Budanur, D. Borrero-Echeverry and Predrag Cvitanović Periodic orbit analysis of a system with continuous symmetry - A tutorial Chaos 25, 073112 (2015) [ arXiv:1411.3303 | DOI ]
	N. Burak Budanur, Predrag Cvitanović, Ruslan L. Davidchack and Evangelos Siminos Reduction of the SO(2) symmetry for spatially extended dynamical systems [ arXiv:1405.1096 ]
	Predrag Cvitanović, Daniel Borrero-Echeverry, Keith Carroll, Bryce Robbins and Evangelos Siminos Cartography of high-dimensional flows: A visual guide to sections and slices Chaos 22, 047506 (2012) [ arXiv:1209.4915 ]
	Predrag Cvitanović Got symmetry? here is how you slice it Classics Illustrated version of the above paper.
	Stefan Froehlich and Predrag Cvitanović Reduction of continuous symmetries of chaotic flows by the method of slices Comm. Nonlinear Sci. and Numer. Simul. 17, 2074–2084 (2012) [ DOI | arXiv:1101.3037 ]
	Predrag Cvitanović, Ashley P. Willis and Marc Avila Revealing the geometry of turbulent pipe flow attractor by symmetry reduction in Jianxiang Wang, ed., Proceed. ICTAM 2012 Intern. Congr. Theor. and Appl. Mech. (2012) [ the long paper is here ]
	Evangelos Siminos and Predrag Cvitanović Continuous symmetry reduction and return maps for high-dimensional flows Physica D 240, 187-198 (2011) [ arXiv:1006.2362 ]
	Predrag Cvitanović Relativity for cyclists ChaosBook.org A (hopefully) pedagogical overview of the symmetry reduction methods.
	Predrag Cvitanović Continuous symmetry reduction for high-dimensional flows Attempt to motivate the need for symmetry reduction, and the available methods in a few slides.
	Predrag Cvitanović Continuous symmetry reduced trace formulas (in preparation, July 2006) Why is this paper not published yet? I would like at least one person out there in the universe to understand it before submitting it. Seems to be a high treshold.
	Predrag Cvitanović and Bruno Eckhardt Symmetry decomposition of chaotic dynamics Nonlinearity 6, 277 (1993) [ arXiv:chao-dyn/9303016 ] in part superceeded by ChaosBook.org: World in a mirror

Getting hang of spatiotemporal dynamics in nearly integrable regimes, as a warmup to turbulent dynamics...

	Mason A. Porter and Predrag Cvitanović A perturbative analysis of Modulated Amplitude Waves in Bose-Einstein Condensates CHAOS 14, 739 (2004) [ arXiv:nlin.CD/0308024 ]
	Mason A. Porter and Predrag Cvitanović Modulated Amplitude Waves in Bose-Einstein Condensates Phys. Rev. E 69, 047201 (2004) [ arXiv:nlin.CD/0307032 ]
	Yueheng Lan, Nicola Garnier and Predrag Cvitanović Modulated solutions of the complex Ginzburg-Landau equation Physica D 188 193, (2004) [ arXiv:nlin.ps/0208001 ]

The noise that physical systems are affected by limits the resolution that can be attained in partitioning their state space. We determine the `finest attainable' partition and replace the Fokker-Planck evolution is by a finite matrix.

	Jeffrey M. Heninger, Domenico Lippolis and Predrag Cvitanović Perturbation theory for the Fokker-Planck operator in chaos Comm. Nonlinear Sci. and Numer. Simul. 55, 16-28 (2017) [ DOI | arXiv:1507.00462 ]
	Jeffrey M. Heninger, Domenico Lippolis and Predrag Cvitanović Neighborhoods of periodic orbits and the stationary distribution of a stochastic chaotic systems Phys. Rev. E. 92 062922, (2015) [ arXiv:1507.00462 ]
	Predrag Cvitanović and Domenico Lippolis Knowing when to stop: How noise frees us from determinism in M. Robnik and V.G. Romanovski, eds., Let's Face Chaos through Nonlinear Dynamics, pp. 82--126 (Am. Inst. of Phys., Melville, New York, 2012) [ arXiv:1206.5506 ]
	Predrag Cvitanović Physicist's life is intractable Intractability workshop: "Counting, Inference, and Optimization on Graphs", 2 nov 2011 (talk aimed at computer scientists)
	video of the intractable lecture (pretty clear on the interplay of noise and determinism - recommended)
	Domenico Lippolis and Predrag Cvitanović How well can one resolve the state space of a chaotic flow? Phys. Rev. Lett. 104, 014101 (2010) [ arXiv:0902.4269 ]
	Predrag Cvitanović and Domenico Lippolis How well can one resolve the state space of a chaotic flow? SIAM Snowbird Dynamical Systems 2011 talk.

A triptych of technical papers whose goal is to develop improved methods of computing higher order corrections to nontrivial saddles of path integrals:

	Predrag Cvitanović, Carl P. Dettmann, Ronnie Mainieri and Gábor Vattay Trace formulas for stochastic evolution operators: Weak noise perturbation theory J. Stat. Phys. 93, 981 (1998) [ chao-dyn/9807034 ]
	Predrag Cvitanović, Carl P. Dettmann, Ronnie Mainieri and Gábor Vattay Trace formulas for stochastic evolution operators: Smooth conjugation method Nonlinearity 12, 939 (1999) [ ps.gz published version , ps.gz , chao-dyn/9811003 ]
	Predrag Cvitanović, Carl P. Dettmann, G. Palla, Niels Søndergaard and Gábor Vattay Spectrum of stochastic evolution operators: Local matrix representation approach Phys. Rev. E 60, 3936 (1999) [ ps.gz , chao-dyn/9904027 ]
	Predrag Cvitanović Trace formulas for stochastic evolution operators (seminar abstract, 1999).
	Predrag Cvitanović Noisy Chaos a talk at Hans C. Fogedby 60 symposium.

A step toward generalizing Gutzwiller semiclassical theory to wave chaos in elastodynamical systems (professionals do not allow us to call this "acoustics")

Predrag Cvitanović, Niels Søndergaard and Andreas Wirzba Wave Chaos in Elastodynamic Cavity Scattering Europhysics Letters 72, 534 (2005) Brief history: submitted to Phys Rev Letters Aug 28 2001; revised version submitted May 18 2004; referee offended by the "In elastodynamics, period two implies chaos" title - rejected; submitted to Europhysics Letters, May 2005, revised version with more resonances included Sep 23 2005, accepted Sep 26 2005. [ arXiv:nlin/0108053 ]

Related links: Niels Søndergaard, Predrag Cvitanović, and Andreas Wirzba, Closed complex rays in scattering from elastic voids in B. Nilsson, ed., Mathematical Modelling of Wave Phenomena 2005, AIP Conference Proceedings (2006) Niels Søndergaard, PhD thesis (Dec 2000) Andreas Wirzba, Georgia Tech seminar (Aug 2001)

How to compute Floquet exponents that differ by 1000's of orders of magnitude (a step toward determining physical dimension of inertial manifolds):

Xiong Ding and Predrag Cvitanović Periodic eigendecomposition and its application in Kuramoto-Sivashinsky system [ arXiv:1406.4885 ]

A variational principle for robust periodic orbit and invariant tori searches:

	Yueheng Lan, Cristel Chandre, and Predrag Cvitanović Variational method for locating invariant tori Phys. Rev. E 74, 046206 (2006) [ arXiv:nlin.CD/0508026 ]
	Predrag Cvitanović and Yueheng Lan Turbulent fields and their recurrences in N. Antoniou, ed., Proceed. of 10. Intern. Workshop on Multiparticle Production: Correlations and Fluctuations in QCD (World Scientific, Singapore 2003); [ arXiv:nlin.CD/0308006 ]
	Yueheng Lan and Predrag Cvitanović Variational method for finding periodic orbits in a general flow Phys. Rev. E 69, 016217 (2004) [ arXiv:nlin.CD/0308008 | ps.gz ]

An early proposal on how to fish for periodic orbits by looking for near recurrences:

D. Auerbach, Predrag Cvitanović, Jean-Pierre Eckmann, Gemunu Gunaratne and Itamar Procaccia Exploring chaotic motion through periodic orbits Phys. Rev. Lett. 58, 2387-2389 (1987)

An attempt to demonstrate existence of unstable periodic orbits and chaos in a slices of rat brains, very noisy neurophysiology experimental data

	Marc W. Slutzky, Predrag Cvitanović and David J. Mogul Deterministic chaos and noise in three in vitro hippocampal models of epilepsy Annals of Biomedical Engineering 29, 607 (2001).
	Marc W. Slutzky, Predrag Cvitanović and David J. Mogul Manipulating epileptiform bursting in the rat hippocampus using chaos control and adaptive techniques IEEE Transactions on Biomedical Engineering 50, 559 (2001).
	Marc W. Slutzky, Predrag Cvitanović and David J. Mogul Identification of determinism in noisy neuronal systems J. Neuroscience Methods 118, 153 (2002)
	Marc W. Slutzky, David J. Mogul and Predrag Cvitanović Chaos control of epileptiform bursting in the brain in Control of chaos in nonlinear circuits and systems, by B. W.-K. Ling, H. H.-C. Iu, and H.-K. Lam, eds., (World Scientific, Singapore 2009)

Two papers demonstrating that periodic orbits must satisfy infinitely many sum rules

	Predrag Cvitanović, Kim Hansen, Juri Rolf and Gabor Vattay Beyond periodic orbit theory Nonlinearity 11, 1209 (1998) [ ps.gz ]
	Sune F. Nielsen, Per Dahlqvist and Predrag Cvitanović Periodic orbit sum rules for billiards: Accelerating cycle expansions J. Phys. A 32, 6757 (1999) [ ps.gz , chao-dyn/9901001 , working notes ]

Non-hyperbolicity, intermittency, power-law correlations: convergence of cycle expansions, analyticity of dynamical zeta functions, approach to the border of order by renormalization methods.

	Roberto Artuso, Predrag Cvitanović and Gregor Tanner Cycle expansions for intermittent maps Proc. Theo. Phys. Supp. 150, 1 (2003) [ nlin.CD/0305008 ] [most of this paper is incorporated in ChaosBook.org]
	Carl P. Dettmann and Predrag Cvitanović Cycle expansions for intermittent diffusion Phys. Rev. E 56, 6687 (1997)
	Roberto Artuso, Erik Aurell and Predrag Cvitanović Recycling of strange sets: II. applications Nonlinearity 3, 361 (1990)

Periodic orbit theory of deterministic diffusion

	Predrag Cvitanović, Jean-Pierre Eckmann and Pierre Gaspard Transport properties of the Lorentz gas in terms of periodic orbits Chaos, Solitons and Fractals 6, 113 (1995) - (61 kB) [ ps.gz ]
	Predrag Cvitanović, Pierre Gaspard and Thomas Schreiber Investigation of the Lorentz Gas in terms of periodic orbits CHAOS 2, 85 (1992)

Periodic orbit theory of diffusion extended to power spectra, with Pikovsky and Feigenbaum. Both Mitchell's draft and Arkady's draft have interesting material not in the published, abreviated version:

Predrag Cvitanović and Arkady S. Pikovsky Cycle expansion for power spectrum Proc. SPIE - Int. Soc. Opt. Eng. (USA) 2038, 290 (1997)

How to implement the periodic orbit theory as a computational method

	Predrag Cvitanović Trace formulas in classical dynamical systems in I.V. Lerner, J.P. Keating, D.E. Khmelnitskii, eds., Supersymmetry and Trace Formulae: Chaos and Disorder, (Plenum, New York 1998)
	Gábor Simon, Predrag Cvitanović, Mogens T. Levinsen, I. Csabai and Á. Horváth Periodic orbit theory applied to a chaotically oscillating gas bubble in water Nonlinearity 15, 25 (2002)
	Predrag Cvitanović, Gábor Vattay and Andreas Wirzba Quantum fluids and classical determinants in H. Friedrich and B. Gerhardt., eds., Classical, Semiclassical and Quantum Dynamics in Atoms - in Memory of Dieter Wintgen, Lecture Notes in Physics 485 (Springer, New York 1997) [ ps.gz , chao-dyn/9608012 ]
	Predrag Cvitanović Dynamical averaging in terms of periodic orbits Physica D 83, 109 (1995) [most of this paper is incorporated in ChaosBook.org]
	Neal J. Balmforth, Predrag Cvitanović, Glenn R. Ierley, Edward A. Spiegel and Gabor Vattay Advection of vector fields by chaotic flows in Stochastic Processes in Astrophysics, Annals of New York Academy of Sciences 706, 148 (1993) [ ps.gz | arXiv:chao-dyn/9307011 - sorry, no figures ]
	Freddy Christiansen and Predrag Cvitanović Periodic orbit quantization of the anisotropic Kepler problem CHAOS 2, 61 (1992)
	Periodic orbit theory in classical and quantum mechanics, CHAOS 2, 1 (1992)
	Kvantekaos, KVANT 5, 1 (1994) [e-print not available]
	Kvantes Lykkelige Dag (with Kenneth Krabat), Naturligvis 20 (1991) [e-print not available]
	Predrag Cvitanović and Bruno Eckhardt Periodic orbit expansions for classical smooth flows J. Phys A 24, L237 (1991) [ preprint ]
	Freddy Christiansen, Predrag Cvitanović and Hans Henrik Rugh The spectrum of the period-doubling operator in terms of cycles J. Phys A 23, L713 (1990)
	Predrag Cvitanović and Bruno Eckhardt Periodic orbit quantization of chaotic systems Phys. Rev. Lett. 63, 823 (1989)

The two in-depth papers on cycle expansions combine Artuso and Aurell PhD theses with several years of my own work - mostly incorporated into ChaosBook.org since. Nobody reads the second paper, which is a pity - there is some good stuff there.

Roberto Artuso, Erik Aurell and Predrag Cvitanović Recycling of strange sets: I. cycle expansions Nonlinearity 3, 325 (1990)

The first paper on cycle expansions:

Predrag Cvitanović Invariant measurement of strange sets in terms of cycles Phys. Rev. Lett. 61, 2729 (1988)

For nice hyperbolic systems Fredhollm determinats are entire, but the Gutzwiller-Voros zeta functions have poles. In 1992 we conjectured a "quantum Fredholm determinant" and showed that it has a larger radius of analiticity than the Gutzwiller-Voros zeta function. Extension of the evolution from phase space to phase space together with the tangent space enables us to construct a multiplicative "quasiclassical evolution operator" and the associated (entire) "quasiclassical zeta function". Unfortunatelly this zeta function has extraneous "classical" eigenvalues, and is not useful in practice. Restriciting its function space to purely quantum spectrum remains an open problem.

	Predrag Cvitanović, Gábor Vattay and Andreas Wirzba Quantum fluids and classical determinants in H. Friedrich and B. Gerhardt., eds., Classical, Semiclassical and Quantum Dynamics in Atoms - in Memory of Dieter Wintgen, Lecture Notes in Physics 485 (Springer, New York 1997) [ ps.gz , chao-dyn/9608012 ]
	Predrag Cvitanović, Bruno Eckhardt, Per E. Rosenqvist, Gunnar Russberg and P. Scherer Pinball scattering in G. Casati and B. Chirikov, eds., Quantum Chaos, (Cambridge University Press, Cambridge 1994) [ ps.gz - sorry, no figures ]
	Predrag Cvitanović and Gábor Vattay Entire Fredholm determinants for evaluation of semi-classical and thermodynamical spectra Phys. Rev. Lett. 71, 4138 (1993) [ arXiv:chao-dyn/9307012 ]
	Predrag Cvitanović, Per E. Rosenqvist, Gábor Vattay, and Hans H. Rugh A Fredholm determinant for semi-classical quantization CHAOS 3, 619 (1993) [ ps | arXiv:chao-dyn/9307014 ]
	Predrag Cvitanović and Per E. Rosenqvist A new determinant for quantum chaos in G.F. Dell'Antonio, S. Fantoni and V.R. Manfredi, eds., From Classical to Quantum Chaos, Soc. Italiana di Fisica Conf. Proceed. 41, pp. 57-64 (Ed. Compositori, Bologna 1993)

How to partition the phase space of a chaotic dynamical system

	Mason A. Porter and Predrag Cvitanović Ground Control to Niels Bohr: Exploring Outer Space with Atomic Physics Notices Am. Math. Soc. 52, 1020 (2005) [ physics/0505085 , pdf ] Featured in: Science News 168 MSNotices200509.html (Sep 10 2005); American Mathematical Society Notices 52 (Oct 2005)
	Predrag Cvitanović Periodic orbits as the skeleton of classical and quantum chaos Physica D 51 138 (1991) [most of this paper is incorporated in ChaosBook.org]
	Predrag Cvitanović, Gemunu H. Gunaratne and Itamar Procaccia Topological and metric properties of Hénon-type attractors Phys. Rev. A 38, 1503 (1988) [the non-Procaccian, non-gibberish part of this paper is incorporated in ChaosBook.org]
	Predrag Cvitanović and Kai T. Hansen Bifurcation structures in maps of Hénon type Nonlinearity 11, 1233 (1998) [the best exposition of physicist's pruning front theory is possibly still Kai T. Hansen 1993 Ph.D. thesis, Symbolic dynamics in chaotic systems]
	Kai T. Hansen and Predrag Cvitanović Symbolic Dynamics and Markov Partitions for the Stadium Billiard J. Stat. Phys. ? (20??) [ chao-dyn/9502005 ]
	The archived version was accepted for publication by J. Stat. Phys. in 1996, but then I had a brilliant idea how to make it better, and a revised version is still waiting to be resubmitted. If you can bring it back to a publishable state - current version is Notes for Kai (sept 95) - please do it, and join us as a co-author. The paper is as good as most stuff that gets published, but neither of us has the time to finish it.

Predrag Cvitanović: transitions to chaos publications

Renormalization in chaos research overview

A very brief history of universality in period doubling

	Predrag Cvitanović Universality in chaos (or, Feigenbaum for cyclists) lectures given at 1982 Zakopane School of Theoretical Physics, Acta Phys. Polonica A65, 203 (1984) [ scan by Chi-Keung Wong ]
	Predrag Cvitanović and Mogens H. Jensen Universality in transitions to chaos in Chaos and universality (Nordita reprint selection, November 1981)
	Predrag Cvitanović and Mogens H. Jensen Universalitet i overgang til kaos Fysisk Tidsskrift 80, 82 (1982)

	Predrag Cvitanović Circle maps: irrationally winding, in C. Itzykson, P. Moussa and M. Waldschmidt, eds., Number Theory and Physics, Les Houches 1989 Spring School, (Springer, New York 1992) - sorry, no figures [most of this paper is incorporated in ChaosBook.org]
	Predrag Cvitanović, Gemunu H. Gunaratne and M.J. Vinson On the mode-locking universality for critical circle maps Nonlinearity 3, 873 (1990)
	Predrag Cvitanović, Mogens H. Jensen, Leo P. Kadanoff and Itamar Procaccia Renormalization, unstable manifolds and the fractal structure of mode locking Phys. Rev. Lett. 55, 343 (1985) [ DOI ]
	Predrag Cvitanović Farey organization of the fractional Hall effect Phys. Scr. T9, 202 (1984)
	Predrag Cvitanović Universal scaling laws for maps on the interval and circle maps in R.W. Boyd, L.M. Narducci and M.G. Raymer, eds., Instabilities and Dynamics of Lasers and Nonlinear Optical Systems, (U. of Cambridge Press, Cambridge, 1985)
	Predrag Cvitanović, B. Shraiman and Bo Söderberg Scaling laws for mode lockings in circle maps Phys. Scripta 32, 263 (1985)
	Predrag Cvitanović, Mogens H. Jensen, L.P. Kadanoff and Itamar Procaccia Circle maps in the complex plane in L. Pietronero and E. Tosatti, eds., Fractals in Physics, Trieste, July 1985 (North Holland, New York, 1985)
	Predrag Cvitanović and Tomas Bohr Chaos is good news for physics Nature 329, 391-392 (1987) [ DOI ]
	Predrag Cvitanović Chaos for cyclists in E. Moss, ed., Noise and Chaos in Nonlinear Dynamical Systems, (Cambridge Univ. Press, Cambridge 1989)
	Predrag Cvitanović Recycling chaos in A. Ferraz, F. Oliveira and R. Osorio, eds., Nonlinear Physical Phenomena, Brasilia 1989 Winter School, (World Scientific, Singapore 1990) - sorry, no figures
	Predrag Cvitanović The power of chaos in J.H. Kim and J. Stringer, eds., Applied Chaos, (John Wiley & Sons, New York 1992) - sorry, no figures

	Predrag Cvitanović and Jan Myrheim Complex universality Comm. Math. Phys. 121, 225 (1989)
	Predrag Cvitanović and Jan Myrheim Universality for period n-tuplings in complex mappings Phys. Lett. 94A, 329 (1983)
	Predrag Cvitanović Renormalization description of transitions to chaos in S. Lundquist, N.H. March and E. Tosatti, eds., Order and Chaos in Non-linear Physical Systems, pp. 73-97 (Plenum, New York 1988) - a subset of the "Complex Universality" paper [ DOI ]
	Predrag Cvitanović, Tomas Bohr and Mogens H. Jensen Fractal aggregates in the complex plane Europhys. Lett. 6, 445 (1988)

The discovery of phase transitions on "strange sets" was followed up by many other authors; such transitions were subsequently found in a variety of dynamical systems.

	Predrag Cvitanović, Roberto Artuso and Brian Kenny Phase transitions on strange irrational sets Phys. Rev. A 39, 268 (1989)
	Predrag Cvitanović Hausdorff dimension of irrational windings in R. Gilmore, ed., Proceedings of the XV International Colloquium on Group Theoretical Methods in Physics, pp. 184-198 (World Scientific, Singapore, 1987)
	Predrag Cvitanović Phase transitions on strange sets, in P. Zweifel, G. Gallavotti and M. Anile, eds., Non-linear Evolution and Chaotic Phenomena, pp. 349-361 (Plenum, New York 1988) - sorry, not scanned yet

Predrag Cvitanović: group theory publications

	overview for my group theory publications click here Group theory webbook
	Predrag Cvitanović Negative dimensions and E7 symmetry Nucl. Phys. B188, 373--396 (1981) [ SPIRES citations ]
	please click here for the background story
	Predrag Cvitanović Group theory for Feynman diagrams in non-Abelian gauge theories Phys. Rev. D 14, 1536 (1976) [ INSPIRE, Google Scholar ]

Predrag Cvitanović: quantum field theory publications

Program whose goal is a non-perturbative theory of turbulent dynamics of classical, stochastic and quantum fields. Turbulent field theory is developed in two directions:
1) explorations of the feasibility of describing weak turbulence in terms of spatiotemporally recurrent patterns, and
2) new perturbation theory methods for computing corrections about nontrivial saddles of path integrals.

Read the sketch paper first, click through the trace formulas papers next.

One big calculation, the answer

	Predrag Cvitanović and Toichiro Kinoshita Feynman-Dyson rules in parametric space Phys. Rev. D10, 3478 (1974) [ INSPIRE citations ]
	A (hopefully) pedagogical overview of the Schwinger and Feynman parametric representation of Feynman integrals, should be useful for any QFT perturbative calculation. Some new results, for example the theorem derived in the appendix.
	Predrag Cvitanović and Toichiro Kinoshita New approach to the separation of ultraviolet and infrared divergences of Feynman-parametric integrals Phys. Rev. D10, 3991 (1974) [ INSPIRE citations ]
	A new method for dealing with Feynman diagram infrared divergences is introduced. The main result are the elegant and compact formulas (3.40) and (5.14) which extract the finite part from any general mass-shell Feynman diagram, removing both ultraviolet and infrared parts of the diagram and all of its counterterms. What remains after the projection is a pointwise-convergent integrand, well suited to numerical integrations.
	Toichiro Kinoshita and Predrag Cvitanović Sixth-order radiative corrections to the electron magnetic moment Phys. Rev. Lett. 29, 1534 (1972)
	At the time perhaps the most demanding numerical computation in theoretical physics, it remained QED's most precise prediction for a number of years. A step in T. Kinoshita's heroic undertaking, but nothing you would want to read today.
	Predrag Cvitanović and Toichiro Kinoshita Sixth order magnetic moment of the electron Phys. Rev. D10, 4007 (1974) [ INSPIRE citations ]
	Most of this you can safely skip, unless you happen to be evaluating (g-2) to 3-loop level. However, the new formula (6.22) for the electron magnetic moment, Sect. VI, might be of interest: (g-2) is evaluated from a derivative of the 2-point electron self-energy, rather than a 3-point electron-photon vertex, with fewer Feynman graphs: (1) it enabled us to calculate the electron magnetic moment by two wholly independent methods.
	Predrag Cvitanović Computer generation of integrands for Feynman parametric integrals Cornell preprint CLNS-234 (June, 1973) and Proc. 3rd Coll. on Advanced Comp. Meth. in Theoretical Physics (Marseille, 1973)
	One of the ur-algebraic symbol manipulation programs

For me the conceptually most striking lesson of these long QED calculations were the amazing cancellations induced by gauge invariance. The desire to understand and exploit gauge invariance more effectively has motivated much of my subsequent research. The most interesting results of this effort were the mass-shell QCD Ward identities and the construction of the QCD gauge sets. This work also motivated the formulation of planar field theory.

Other papers in this series are those on the QCD mass-shell infrared singularities and the diagram counting, both attempts to formulate gauge-invariant models computable to high orders, in order to investigate the nature of gauge-invariance induced cancellations.

Predrag Cvitanović Asymptotic estimates and gauge invariance Nucl. Phys. B127, 176 (1977) [ INSPIRE citations ]

On basis of very skimpy numerical evidence, I conjecture that the gauge invariance induced cancellations are so dramatic that the growth rate of high order perturbation theory corrections to mass-shell gauge-invariant quantities is slower than Dyson's asymptotic series n! estimate. In the case of QED vertex corrections, the smallest gauge invariant set contributing to (m+m'+k)th order consists of m photon "strands" attached to the incoming electron, m' photon "strands" attached to the outgoing electron, and k photon "strands" crossing the external photon vertex. Ignoring sets with electron loops and assuming that each gauge set gives a finite contribution leads to a guess that perturbation series for the electron magnetic moment sums up to approximately The story behind the conjecture.

The method that I have used to develop the planar field theory is somewhat different from what is in most field theory textbooks; consulting my Field Theory webbook might make this derivation more accessible.

	Predrag Cvitanović Planar perturbation expansion Phys. Lett. B99, 49 (1981) [ INSPIRE citations ]
	Predrag Cvitanović The planar sector of field theories (with P.G. Lauwers and P.N. Scharbach), Nucl. Phys. B203, 385 (1982) [ INSPIRE citations ]
	Though the formalism is rather different, essentially the same theory was rederived in 1994 by M.A. Douglas, R. Gopakumar and David J. Gross, D.V. Voiculescu, in 1998 by 't Hooft, and many others. R. Speicher (13 Apr 2014): "In a sense some aspects of this theory of freeness were anticipated (but mostly neglected) in the physics community in [this] paper"

	Predrag Cvitanović Yang-Mills theory of the mass-shell Phys. Rev. Lett. 37, 1528 (1976) [ INSPIRE citations ]
	Mass-shell amplitudes for both QED and QCD are defined via dimensional regularization, and shown to to be gauge invariant trough a cancellation between UL and IR singularities. Prior to this article, IR and UR were regularized by different methods which, when applied to QCD, violated gauge invariance.
	Predrag Cvitanović Infra-red structure of Yang-Mills theories Phys. Lett. 65B, 272 (1976) [ INSPIRE citations ]
	Predrag Cvitanović Quantum Chromodynamics on the mass-shell Nucl. Phys. B130, 114 (1977) [ INSPIRE citations ]
	Predrag Cvitanović, Benny Lautrup and R.B. Pearson The number and weights of Feynman diagrams Phys. Rev. D18, 1939 (1978) [ INSPIRE citations ]
	Predrag Cvitanović, Jeff Greensite and Benny Lautrup The cross-over points in lattice gauge theories with continuous gauge groups Phys. Lett. 105B, 201 (1981) [ INSPIRE citations ]
	Predrag Cvitanović, P.G. Lauwers and P.N. Scharbach Gauge invariance structure of Quantum Chromodynamics Nucl. Phys. B186, 165 (1981) [ INSPIRE citations ]

	Predrag Cvitanović Spin and parity from crossections and angular distributions, in J. Bjorken et al., Notes from the SLAC Theory Workshop on the Psi, SLAC-PUB-1515 , (Dec. 7, 1974)
	Predrag Cvitanović Wide-angle behavior of a double-scattering diagram Phys. Rev. D10, 338 (1974)
	Predrag Cvitanović, R.J. Gonsales and D.E. Neville Color charge algebras in Adler's Chromodynamics Phys. Rev. D18, 3881 (1978)
	Predrag Cvitanović, P. Hoyer and K. Konishi Partons and branching Phys. Lett. 85B 413 (1979)
	Predrag Cvitanović and R. Horsley Exact solutions of the Altarelli-Parisi equations Nucl. Phys. B173, 229 (1980)
	Predrag Cvitanović, Poul Hoyer and K. Zalewski Parton evolution as a branching process Nucl. Phys. B176, 429 (1980) [ INSPIRE citations ]

	Predrag Cvitanović et al drafts in progress (of interest only to frustrated collaborators)
	Predrag Cvitanović Midnight rider Bike World (August 1974)
	Predrag Cvitanović and Mitchell J. Feigenbaum, for David Bensimon, Thomas C. Halsey, Mogens H. Jensen, Leo P. Kadanoff, Albert Libchaber, Itamar Procaccia, Boris I. Shraiman and Joel Stavans More on microcanonical paradigm (Göteborg, 3 a.m. of 17 Nov. 1985), rejected from every proceedings submitted to.

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