% Encoding: UTF-8
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With a subtle choice of the initial ensemble, we obtain an
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abstract = {In the last decades the Moore-Penrose pseudoinverse
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became a useful tool for physicists dealing, for instance, with optimization
problems, with data analysis, with the solution of linear integral equations,
etc. The existence of such applications alone should attract the interest
of students and researchers in the Moore-Penrose pseudoinverse and
in related subjects, like the singular values decomposition theorem for
matrices. In this note we present a tutorial review of the theory of the
Moore-Penrose pseudoinverse. We present the first definitions and
some motivations and, after obtaining some basic results, we center
our discussion on the Spectral Theorem and present an algorithmically
simple expression for the computation of the Moore-Penrose
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We present a complete and self-contained
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those more theoretically oriented and for those who already have
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year={2013},
abstract={Using path-integral methods, a formula is deduced for the
noise-induced escape rate from an attracting fixed point across an unstable
fixed point in one-dimensional maps. The calculation starts from the trace
formula for the eigenvalues of the Frobenius�Perron operator ruling the
time evolution of the probability density in noisy maps. The escape rate is
determined from the loop formed by two heteroclinic orbits connecting back
and forth the two fixed points of the one-dimensional map extended to a
two-dimensional symplectic map. The escape rate is obtained with the
expression of the prefactor to the Arrhenius�van�t�Hoff exponential
factor.}
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stretched exponential to log-normal to algebraic growth. The set of all
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journal = {Nature},
volume = {404},
pages = {733},
year = {2000},
abstract = {Numerical study on Boussinesq equation(pseudospectral method
with time splitting scheme). Demonstrate the extensivity of Lyapunov
dimension of the extended system and showed that the generating mechanism
is breaking or connecting rolls mediated by defects at the domain boundary
and exsisting for only a short time.}
}
@ARTICLE{egolf94,
author = {D. A. Egolf and H. S. Greenside},
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pages = {129},
year = {1994},
abstract = {Numerical study on CGLe. Show that even though the fractal
dimension increases with system size the simple scaling relation is not
correct, i.e. the correlation length can not be used to directly calculate
the fractal dimension.}
}
@ARTICLE{egolf98,
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expansions for the transport coefficients that can be evaluated
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abstract = {The problem of understanding the nature of fluid
flow through a circular straight pipe remains one of the
oldest problems in fluid mechanics. So far no explanation
has been substantiated to rationalize the transition
process by which the steady unidirectional laminar flow
state gives way to a temporally and spatially disordered
three-dimensional (turbulent) solution as the flow rate
increases. Recently, new travelling wave solutions have
been discovered which are saddle points in phase space.
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of past work and discuss its implications for future
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}
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year = {1988},
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pages = {405--420},
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year = {1992},
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pages = {899--919},
abstract = {Some properties of the KSe are discussed. A unified argument
about the connections and periodic orbits is given. In particular, the
K-bifurcation of the system is conjectured to arises in the 1:n resonances of
a fixed point.}
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pages = {25--42},
abstract = {Using asymptotic analysis (beyond all orders)in the weak
shock limit, the oscillatory
shocks and the solitary saves are constructed. It shows that the oscillatory
shocks can be only be anti-symmetric and numurical results support the
anlytical calculation.}
}
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volume = {33},
number = {1},
pages = {674--685}
}
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AUTHOR = {I. Gumowski and C. Mira},
TITLE = {Recurrances and Discrete Dynamical Systems},
YEAR = 1980,
PUBLISHER = {Springer},
ADDRESS = {Berlin}
}
@Article{mis01ks,
author = {P. Zgliczynski and K. Mischaikow},
title = {Rigorous Numerics for Partial Differential Equations: the {Kuramoto-Sivashinsky} equation},
journal = {Foundations of Comp. Math.},
year = {2001},
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pages = {255--288}
}
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year={1980},
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@Article{mislor1,
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pages = {1023--1046}
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title = {Steady solutions of the {Kuramoto-Sivashinsky} equation},
journal = {Physica D},
year = {1986},
volume = {19},
pages = {89--111},
doi = {10.1016/0167-2789(86)90055-2},
abtract = {The variety of steady solution of the KSe is discussed. For large c, there is only one odd front-like bounded solution. In decreasing c, odd solutions with more zeros are born until finally a periodic solution is born. Associated with the periodic solution, infinite many tori will appear in the elliptic case and Cantor-type set of chaotic solutions, with infinite many homoclinic odd solutions.}
}
@MISC{MoMiAo14,
author = {{Morinaga}, K. and {Miyata}, R. and {Aonishi}, T.},
title = {Optimal colored noise for estimating phase response curves},
note = {\arXiv{1409.1637}},
year = {2014}
}
@article{Moore81,
author = {B. Moore},
title = {Principal component analysis in linear systems:
{Controllability}, observability, and model reduction},
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year = {1981},
pages = {17--32},
}
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author = {Morkovin, M.V},
title = {On the many faces of transition},
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year = {1969},
editor = {Wells, C.S.},
pages = {1--31},
publisher = {Plenum, New York}
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volume = {88},
pages = {214501},
year = {2002},
abstract = {}
}
@Article{Morrison04,
author = {J.F. Morrison and B.J. McKeon and W. Jiang and A.J. Smits},
title = {Scaling of the streamwise velocity component in turbulent pipe flow},
journal = {J. Fluid Mech.},
year = {2004},
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pages = {99--131}
}
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author = {M. Morse},
title = {Variational Analysis: Critical Extremals and Sturmian Extensions},
publisher = {Wiley},
address = {New York},
year = {1973}
}
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author = {J. Moser},
title = {A rapidly converging iteration method and nonlinear partial differential equations -{I}},
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pages = {265--315},
url = {http://archive.numdam.org/ARCHIVE/ASNSP/ASNSP_1966_3_20_2/ASNSP_1966_3_20_2_265_0/ASNSP_1966_3_20_2_265_0.pdf}
}
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author = {J. Moser},
title = {A rapidly convergent iteration method and non-linear differential equations - {II}},
journal = {Ann. Scuola Norm. Super. Pisa},
year = {1966},
volume = {20},
pages = {499--535},
url = {http://archive.numdam.org/ARCHIVE/ASNSP/ASNSP_1966_3_20_3/ASNSP_1966_3_20_3_499_0/ASNSP_1966_3_20_3_499_0.pdf}
}
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author = {J. Moser},
title = {Stable and Random Motions in Dynamical Systems},
publisher = {Princeton Univ. Press},
address = {Princeton},
year = {1973}
}
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author = {D. W. Moore and E. A. Spiegel},
title = {A thermally excited nonlinear oscillator},
journal = {Astrophys. J.},
volume = {143},
pages = {871},
year = {1966}
}
@article{MTYYA14,
author = {Ma, Y. and Tan, Q. and Yuan, R. and Yuan, B. and Ao, P.},
title = {Potential function in a continuous dissipative chaotic system:
{Decomposition} scheme and role of strange attractor},
journal = {Int. J. Bifur. Chaos},
volume = {24},
pages = {1450015},
year = {2014},
doi = {10.1142/S0218127414500151},
}
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author = {M. van Hecke and M. Howard},
title = {Ordered and Self-disordered dynamics of holes and defects in the
one-dimensional complex {G}inzburg-{L}andau equation},
journal = {Phys. Rev. Lett.},
volume = {86},
pages = {2018},
year = {2001},
abtract = {Ordered hole-defect dynamics happens when an unstable hole invade a plane
wave state and periodically nucleates defects from which new holes are born. The
interaction between the holes and a self-disordered background are essential for the
occurrence of spatiotemporal chaos in hole-defect states.}
}
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abstract = {The article demonstrates the existence of a nonuniform stationary measure
associated with the space-filling properties of the unstable invariant foliation in 2D
differentiable area-preserving systems and derives a sequence of analytical approximations
for it. In fluid convection systems, it is just the convected material line.}
}
@article{NaSaWa13,
author = "Nakanishi, H. and Sakaue, T. and Wakou, J.",
title = "{Hamilton-Jacobi} method for molecular distribution function in
a chemical oscillator",
journal = "J. Chem. Phys.",
year = "2013",
volume = "139",
pages = "214105",
doi = "10.1063/1.4834636"
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@MISC{natmsr,
author = {P. Cvitanovi\'{c} and R. Artuso
and L. Rondoni and E. A. Spiegel},
note = {Chapter {\em Transporting densities}, in
\refref{DasBuch}},
year = {2012},
}
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title = {Thermodynamic formula for the cumulant generating function of
time-averaged current},
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}
@Article{NeuSch01,
author = "A. Neumaier and T. Schneider",
title = "Estimation of parameters and eigenmodes of
multivariate autoregressive models",
journal = {A.C.M. Trans. Math. Softw.},
volume = "27",
pages = "27--57",
year = "2001",
abstract = "Dynamical characteristics of a complex system can
often be inferred from analyses of a stochastic time
series model fitted to observations of the system.
Oscillations in geophysical systems, for example, are
sometimes characterized by principal oscillation
patterns, eigenmodes of estimated autoregressive (AR)
models of first order. This paper describes the
estimation of eigenmodes of AR models of arbitrary
order. AR processes of any order can be decomposed into
eigenmodes with characteristic oscillation periods,
damping times, and excitations. Estimated eigenmodes
and confidence intervals for the eigenmodes and their
oscillation periods and damping times can be computed
from estimated model parameters. As a computationally
efficient method of estimating the parameters of AR
models from high-dimensional data, a stepwise least
squares algorithm is proposed. This algorithm computes
model coefficients and evaluates criteria for the
selection of the model order stepwise for AR models of
successively decreasing order. Numerical simulations
indicate that, with the least squares algorithm, the AR
model coefficients and the eigenmodes derived from the
coefficients are estimated reliably and that the
approximate 95\% confidence intervals for the
coefficients and eigenmodes are rough approximations of
the confidence intervals inferred from the
simulations.",
}
@MISC{NeuSch97,
author = {A. Neumaier and T. Schneider},
title = {Multivariate autoregressive and {Ornstein-Uhlenbeck} processes:
{Estimates} for order, parameters, spectral information,
and confidence regions},
note = {Submitted to A.C.M. Trans. Math. Softw., unpublished},
}
@Article{newcgl1,
author = {P. K. Newton and L. Sirovich},
title = {Instabilities of {G}inzburg-{L}andau equation: periodic solutions},
journal = {Quart. Appl. Math.},
year = {1986},
volume = {XLIV},
pages = {49--58}
}
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title = {Instabilities of {Ginzburg-{L}andau} equation: secondary bifurcation},
journal = {Quart. Appl. Math.},
year = {1986},
volume = {XLIV},
pages = {367--374}
}
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year = {1974},
volume = {15},
pages = {157--163}
}
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author = {S. E. Newhouse and D. Ruelle and F. Takens},
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@MISC{Nico90,
author = {G. Nicolis},
title = {Chaotic dynamics and markovian coarse-graining in
nonlinear dynamical systems},
note = {in \textit{Noise and Chaos in Nonlinear Dynamical Systems},
edited by F. Moss, L. A. Lugiato, and W. Schleich
(Cambridge Univ. Pres)},
year = {1990}
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year = {1986},
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pages = {1109--1196}
}
@Article{nlin0001013,
author = {{Weber}, J. and {Haake}, F. and {{\v S}eba}, P.},
title = {{Frobenius-Perron} resonances for maps with a mixed phase space},
journal = {Phys. Rev. Lett.},
year = {2000},
volume = {85},
pages = {3620--3623},
note = {\arXiv{nlin/0001013}}
}
@Article{nlin0102039,
title = "Periodic orbit quantization of a {Hamiltonian} map on
the sphere",
author = "A. J. Scott and G. J. Milburn",
year = "2001",
volume = "34",
abstract = "In a previous paper we introduced examples of
Hamiltonian mappings with phase space structures
resembling circle packings. It was shown that a vast
number of periodic orbits can be found using special
properties. We now use this information to explore the
semiclassical quantization of one of these maps.",
journal = "J. Phys. A",
pages = "7541",
note = {\arXiv{nlin/0102039}},
}
@Article{nlin0105047,
author = {{Weber}, J. and {Haake}, F. and {Braun}, P.~A. and {Manderfeld}, C. and {Seba}, P.},
title = {Resonances of the {Frobenius-Perron} operator for a {Hamiltonian} map with a mixed phase space},
journal = {J. Phys. A},
year = {2001},
volume = {34},
pages = {7195--7211},
note = {\arXiv{nlin/0105047}}
}
@Article{nlin0107020,
author = {{Manderfeld}, C. and {Weber}, J. and {Haake}, F.},
title = {Classical versus quantum time evolution of (quasi-) probability densities at limited phase-space resolution},
journal = {J. Phys. A},
year = {2001},
volume = {34},
pages = {9893--9905},
note = {\arXiv{nlin/0107020}}
}
@Article{nlin0312062,
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journal = {Phys. Rev. E},
year = {2004},
volume = {69},
pages = {056211},
note = {\arXiv{nlin/0312062}}
}
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year = {1991},
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pages = {1149--1153},
abtract = {A Hamiltonian equation is proposed for the generalized nonlinear Schr\"{o}dinger equation. Soliton solutions are found and the stability of the plane waves is investigated.}
}
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abstract = {A nonlinear Schr\"{o}dinger equation is found with growth and damping
terms in the weakly nonlinear regime. An exact solitary solution is found for the
equation.}
}
@Article{NLSstn03,
author = {L. Stenflo and M. Y. Yu},
title = {Oscillons at a plasma surface},
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year = {2003},
volume = {10},
pages = {912--913},
abstract = {Under the external driving force, an oscillon solutions is found.}
}
@Article{NockelStone,
author = {Nockel, J. U. and Stone, A. D.},
title = {Ray and wave chaos in asymmetric resonant optical cavities},
journal = {Nature},
year = {1997},
volume = {385},
number = {6611},
pages = {45--47},
doi = {Doi 10.1038/385045a0},
abstract = {OPTICAL resonators are essential components of lasers and
other optical devices, A resonator is characterized by a set of modes,
each with a resonant frequency omega and resonance width delta omega =
1/tau, where tau is the lifetime of a photon in the mode, Cylindrical
or spherical dielectric resonators have extremely long-lived
resonances(1) due to 'whispering gallery' modes in which light
circulates around the perimeter trapped by total internal reflection,
These resonators emit light isotropically, Recently a new category of
asymmetric resonant cavities has been proposed in which substantial
deformation of the cavity from cylindrical or spherical symmetry leads
to partially chaotic ray dynamics, This has been predicted(2-4) to
give rise to a universal, frequency-independent broadening of
the,whispering-gallery resonances, and to highly anisotropic emission,
Here we present solutions of the wave equation for asymmetric resonant
cavities which confirm these predictions but also reveal interesting
frequency-dependent effects characteristic of quantum chaos, For small
deformations the lifetime is controlled by evanescent leakage, the
optical analogue of quantum tunnelling(5); here the lifetime is
significantly shortened by a process known as 'chaos-assisted
tunnelling'(6-7). In contrast, for large deformations (similar to 10%)
some resonances are found to have longer lifetimes than predicted by
the ray chaos model due to the phenomenon of 'dynamical
localization'(8).}
}
@Inproceedings{NoeckChang,
author = {N\"ockel, J. U. and Chang, R. K.},
title = {{2D} microcavities: {Theory} and experiemtns},
booktitle = {Cavity-Enhanced Spectroscopies},
year = {2002},
editor = {Van Zee, R. D. and Looney, J. P.},
isbn = {9780124759879},
pages = {185--226},
publisher = {Academic}
}
@ARTICLE{noisy_Fred,
author = {P. Cvitanovi\'{c} and C. P. Dettmann and R. Mainieri
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pages = {981},
year = {1998},
note = {\arXiv{chao-dyn/9807034}},
}
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abstract = {nonlinear {G}\"{a}lerkin methods}
}
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author = {W. H. Press and S. A. Teukolsky and W. T. Vetterling and
B. P. Flannery},
title = {Numerical Recipes in {C}},
publisher = {Cambridge Univ. Press},
address = {Cambridge, England},
year = {1992}
}
@BOOK{ns,
author = {J. W. Thomas},
title = {Numerical Partial Differential Equations},
publisher = {Spring-Verlag},
address = {New York},
year = {1995}
}
@Article{NSTks85,
author = {B. Nicolaenko and B. Scheurer and R. Temam},
title = {Some global dynamical properties of the {Kuramoto-Sivashinsky} equations: nonlinear stability and attractors},
journal = {Physica D},
year = {1985},
volume = {16},
pages = {155--183},
abstract = {The KSe shows coherent spatial patterns but temporal chaos. The
anti-diffusion term destablizes the solutin while stability is achived by the collective
action of all modes through the nonlinear convection term. An estimate of the L^2 norm of
the solution in the antisymmetric periodic function space is given. The number of
determining Fourier modes is calculated together with fractal dimension of the strange
attractor.}
}
@Article{OcToHe92,
author = {P. W. Oconnor and S. Tomsovic and Heller, E. J. },
title = {Accuracy of semiclassical dynamics in the presence of chaos},
journal = {J. Stat. Phys.},
year = {1992},
volume = {68},
pages = {131--152}
}
@ARTICLE{OHSZKCKY04,
title = {A local ensemble {Kalman} filter for atmospheric data assimilation},
author = {Ott, E. and Hunt, B. and Szunyogh, I. and Zimin, A. and Kostelich, E.
and Corrazza, M. and Kalnay, E. and Yorke, J.},
journal = {Tellus A},
volume = {56},
year = {2004},
pages = {415--428},
}
@article{OkEckThiDoe08,
title={Mixing effectiveness depends on the source--sink structure: simulation results},
author={Okabe, T. and Eckhardt, B. and Thiffeault, J.-L. and Doering, C. R.},
journal={J. Stat. Phys},
volume={2008},
pages={P07018},
year={2008},
doi = {10.1088/1742-5468/2008/07/P07018}
}
@BOOK{Oksendal03,
author = {{\O}ksendal, Bernt},
title ={Stochastic Differential Equations},
year = {2003},
publisher = {Springer},
address = {New York},
}
@article{OlGo92,
author = {Olmsted, P. D. and Goldbart, P. M.},
title = "Isotropic-nematic transition in shear flow:
{State} selection, coexistence, phase transitions, and critical behavior",
year = {1992},
pages = {4966--4993},
journal = {Phys. Rev. A},
volume = {46},
doi = {10.1103/PhysRevA.46.4966},
}
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author = {Olmsted, P. D.},
title = {Rotational invariance and {Goldstone modes} in nematic elastomers and gels},
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year = {1994},
volume = {4},
pages = {2215--2230},
doi = {10.1051/jp2:1994257}
}
@article{OnKa79,
author = "A. Onuki and K. Kawasaki",
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journal = "Ann. Phys.",
volume = "121",
pages = "456--528",
year = "1979",
doi = "10.1016/0003-4916(79)90105-2",
}
@ARTICLE{Onsager53,
author = {L. Onsager and S. Machlup},
title = {Fluctuations and irreversible processes},
journal = {Phys. Rev.},
volume = {91},
pages = {1505, 1512},
year = {1953}
}
@article{Onuki87,
author = {A. Onuki},
title = {Homogeneous block copolymer systems under shear flow},
year = {1987},
journal = {J. Chem. Phys.},
volume = {87},
pages = {3692--3697},
doi = {10.1063/1.452967}
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@ARTICLE{optp1,
author = {B. R. Hunt and E. Ott},
title = {Optimal periodic orbits of chaotic systems},
journal = {Phys. Rev. Lett.},
volume = {76},
pages = {2254},
year = {1996}
}
@ARTICLE{optp2,
author = {T.-H. Yang and B. R. Hunt and E. Ott},
title = {Optimal periodic orbits of continuous time chaotic systems},
journal = {Phys. Rev. E},
volume = {62},
pages = {1950},
year = {2000}
}
@Article{Ordonez96,
author = "G. E. Ordonez and D. J. Driebe",
title = "Spectral decomposition of tent maps using symmetry
considerations",
journal = "J. Stat. Phys.",
volume = "84",
pages = "269--276",
year = "1996",
keywords = "Frobenius-Perron, coupled systems, tent map,
symmetry",
}
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pages = {174114},
doi = {10.1063/1.3586036}
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@Article{OrlMaj15,
author = {S. N. Majumdar and H. Orland},
title = {Effective Langevin equations for constrained stochastic processes},
journal = {J. Stat. Mech.},
year = {2015},
pages = {P06039},
doi = {10.1088/1742-5468/2015/06/P06039}
}
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journal = {Phys. Rev.},
year = {1930},
volume = {36},
pages = {823--841},
doi = {10.1103/PhysRev.36.823}
}
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title = {Boundary conditions for incompressible flows},
journal = {J. Sci. Comput.},
volume = {1},
pages = {75--111},
year = {1986},
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author = {Steven A.\ Orszag},
title = {Accurate solution of the {O}rr-{S}ommerfeld stability equation},
journal = {J. Fluid Mech.},
year = {1971},
volume = {50},
pages = {689--703}
}
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volume = {4},
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@ARTICLE{ostruszka01,
author = {A. Ostruszka and K. Zyczkowski},
title = {Spectrum of the {Frobenius-Perron} operator for systems
with stochastic perturbation},
journal = {Phys. Lett. A},
volume = {289},
pages = {306},
note = {\arXiv{nlin/0106003}},
year = {2001}
}
@BOOK{ottbook,
author = {E. Ott},
title ={Chaos and Dynamical Systems},
publisher = {Cambridge Univ. Pres},
address = {},
year = {2002}
}
@ARTICLE{Ozorio,
author = {A.M. Ozorio de Almeida and J.H. Hannay},
title = {Resonant periodic orbits in the energy spectrum},
journal = {J. Phys. A},
volume = {20},
pages = {5873},
year = {1987},
abstract = {}
}
@Article{PaCrFaSh80,
title = "Geometry from a Time Series",
author = {Packard, N. H. and Crutchfield, J. P. and Farmer, J. D. and Shaw, R. S.},
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volume = {45},
pages = {712--716},
year = {1980},
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@Article{Paul05,
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pages = {157--175},
year = {2005},
}
@Article{Pauls04,
title = "Summing up noise in gene expression",
author = {Paulsson, J.},
journal = {Nature},
volume = {427},
pages = {415--418},
year = {2004},
}
@ARTICLE{pchaot,
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author = {Sapsis, T. P.},
title = {Attractor local dimensionality, nonlinear energy transfers
and finite-time instabilities in unstable dynamical systems with
applications to two-dimensional fluid flows},
volume = {469},
year = {2013},
doi = {10.1098/rspa.2012.0550},
abstract = {We examine the geometry of the finite-dimensional
attractor associated with fluid flows described by
Navier-Stokes equations and relate its nonlinear
dimensionality to energy exchanges between dynamical components (modes)
of the flow. Specifically, we use a stochastic framework based on the
dynamically orthogonal equations to perform efficient order-reduction and
describe the stochastic attractor in the reduced-order phase space in
terms of the associated probability measure. We introduce the notion of
local fractal dimensionality to describe the geometry of the attractor
and we establish a connection with the number of positive finite-time
Lyapunov exponents. Subsequently, we illustrate in specific fluid flows
that the low dimensionality of the stochastic attractor is caused by the
synergistic activity of linearly unstable and stable modes as well as the
action of the quadratic terms. In particular, we illustrate the
connection of the low-dimensionality of the attractor with the
circulation of energy: (i) from the mean flow to the unstable modes (due
to their linearly unstable character), (ii) from the unstable modes to
the stable ones (due to a nonlinear energy transfer mechanism) and (iii)
from the stable modes back to the mean (due to the linearly stable
character of these modes).},
journal = {Proc. R. Soc. A}
}
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author = {Cvitanovi{\'c}, P. and Davidchack, R. L. and Siminos, E.},
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volume = {9},
pages = {1--33},
note = {\arXiv{0709.2944}}
}
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journal = {Science},
year = {2004},
volume = {305},
pages = {1594--1598},
abstract = {Transition to turbulence in pipe flow is one of
the most fundamental and longest-standing problems in fluid
dynamics. Stability theory suggests that the flow remains
laminar for all flow rates, but in practice pipe flow becomes
turbulent even at moderate speeds. This transition
drastically affects the transport efficiency of mass,
momentum, and heat. On the basis of the recent discovery of
unstable traveling waves in computational studies of the
Navier-Stokes equations and ideas from dynamical systems
theory, a model for the transition process has been
suggested. We report experimental observation of these
traveling waves in pipe flow, confirming the proposed
transition scenario and suggesting that the dynamics
associated with these unstable states may indeed capture the
nature of fluid turbulence.
},
eprint = {http://www.sciencemag.org/cgi/reprint/305/5690/1594.pdf},
url = {http://www.sciencemag.org/cgi/content/abstract/305/5690/1594}
}
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}
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doi = {10.1143/PTP.51.1731},
abstract = {The notion of irreversible circulation of fluctuation around
the steady state, or cyclic balance, is proposed. In addition to the
shape of the probability distribution function this concept is necessary
in describing the behaviour of the thermodynamically coupled degrees of
freedom at off-equilibrium situation. When the circulation entails
instability, a macroscopic orbital revolution, i.e. limit cycle, may
appear, which is characteristic of a far from equilibrium situation.
Examples are mentioned, including chemical oscillations and undamped
spiking in lasers.}
}
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abstract = {{N}ewton method is used to construct periodic orbits of longer
and longer period to approach a invariant torus with specific rotation vector.
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finite value in the PT regime and decreases to zero when the transition to defect
turbulence is approached. A modified KSe is able to reproduce the main feature of the
stable waves and to explain their origin.}
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abstract = {A periodic orbit analysis in the Lorenz system
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linear equations are performed. A set of vectors are found
that satisfy the Oseledec (1968) theorem and reduce to
Floquet eigenvectors in the particular case of a periodic
orbit. These vectors, called Lyapunov vectors, can be
considered the generalization to aperiodic orbits of the
normal modes of the instability problem and are not
necessarily mutually orthogonal. The relation between
singular vectors and Lyapunov vectors is clarified. The
mechanism responsible for super-Lyapunov growth is shown to
be related to the nonorthogonality of Lyapunov vectors. The
leading Lyapunov vectors, as defined here, as well as the
asymptotic final singular vectors, are tangent to the
attractor, while the leading initial singular vectors, in
general, point away from it. Perturbations that are on the
attractor can be found in the subspace of the leading
Lyapunov vectors.},
}
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title={A First Course in Rational Continuum Mechanics:
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year={1991},
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address = {New York},
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@Article{TTRD93,
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abstract = {Perturbed plane Couette flow containing a thin
spanwise-oriented ribbon undergoes a subcritical bifurcation at
Re = 230 to a steady 3D state containing streamwise vortices.
This bifurcation is followed by several others giving rise to a
fascinating series of stable and unstable steady states of
different symmetries and wavelengths. First, the
backwards-bifurcating branch reverses direction and becomes
stable near Re = 200. Then, the spanwise reflection symmetry is
broken, leading to two asymmetric branches which are themselves
destabilized at Re = 420. Above this Reynolds number, time
evolution leads first to a metastable state whose spanwise
wavelength is halved and then to complicated time-dependent
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(r.s.) matrices has long been of interest. This subject is
generalized here to the following problem raised by Taussky-Tod: What
is the first simultaneous block diagonal structure of a nonsingular
pair of r.s. matrices S and T? If the number of blocks is maximal,
then one can speak of the finest simultaneous block diagonalization
of S and T."
}
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author = {Ulam, S. M.},
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@inproceedings{Varga05,
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volume={16},
pages={474--474},
year={2005}
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@inproceedings{Varga11,
title={An overview of recent developments in computational methods
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author={Varga, A.},
booktitle={Proc. of IFAC Workshop on Periodic Control Systems},
year={2011}
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title={Computational methods for periodic systems-an overview},
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editor = {S. Bittanti and P. Colaneri},
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organization = {IFAC Workshop},
publisher = {Elsevier},
address = {New York},
pages={171--176},
year={2001},
isbn = {978-0-08-043682-1},
url = {http://elib.dlr.de/11743/1/varga_como01p1.pdf}
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efficient. Under these assumptions, it is shown by means of an
arbitrage argument that the expected rate of return on any bond
in excess of the spot rate is proportional to its standard
deviation. This property is then used to derive a partial
differential equation for bond prices. The solution to that
equation is given in the form of a stochastic integral
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walk: locating the stochastic limit cycle in a reversible
{Schnakenberg} model},
volume = {466},
pages = {771--788},
year = {2010},
doi = {10.1098/rspa.2009.0346},
abstract = {Recent studies on stochastic oscillations mostly focus on
the power spectral analysis. However, the power spectrum yields
information only on the frequency of oscillation and cannot differentiate
between a stable limit cycle and a stable focus. The cycle flux,
introduced by Hill (Hill 1989 Free energy transduction and biochemical
cycle kinetics), is a quantitative measure of the net movement over a
closed path, but it is impractical to compute for all possible cycles in
systems with a large state space. Through simple examples, we introduce
concepts used to quantify stochastic oscillation, such as the cycle flux,
the Hill{\textendash}Qian stochastic circulation and rotation number. We
introduce a novel device, the Poincar{\'e}{\textendash}Hill cycle map
(PHCM), which combines the concept of Hill{\textquoteright}s cycle flux
with the Poincar{\'e} map from nonlinear dynamics. Applying the PHCM to a
reversible extension of an oscillatory chemical system, the Schnakenberg
model, reveals stable oscillations outside the Hopf bifurcation region in
which the deterministic system contains a limit cycle. Bistable behaviour
is found on the small volume scale with high probabilities around both
the fixed point and the limit cycle. Convergence to the deterministic
system is found in the thermodynamic~limit. },
journal = {Proc. Roy. Soc. Lond. Ser A}
}
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volume = {33},
pages = {4717--4724},
note = {\arXiv{chao-dyn/9904039}}
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note = {\arXiv{nlin/1102.3202}}
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note = "See additions \cite{vonNeum32}. Reprinted in
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@Article{vonNeum32,
author = "J. von Neumann",
title = "{Zus{\"a}tze zur Arbeit ``Zur Operatorenmethode in der
klassischen Mechanik''}. ({German}) [{Additions} to the
work ``{On} operator methods in classical
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distribution of passive scalar quantities. It seems that on these manifolds the
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author = {Waleffe, F. },
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pages = {115--128},
publisher = {National Center of Sciences, Tokyo, Japan}
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AUTHOR="Waleffe, F.",
TITLE="Proposal for a self-sustaining process in shear flows",
JOURNAL="Working paper, available at
\verb+www.math.wisc.edu/~waleffe/ECS/sspctr90.pdf+",
YEAR="1990",
VOLUME=" ",
PAGES=" "}
@Inproceedings{W90b,
author = {Waleffe, F.},
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@Article{W95a,
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@Article{W95b,
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year = {1995},
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pages = {3060--3066}
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@Article{W98,
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@UNPUBLISHED{WaKo12,
title = {Optimal resolution of state space in chaotic hyperbolic 2-dimensional maps},
author = {G. M. Wadsworth and D. T. Kovari},
note = {ChaosBook.org project},
url = {http://ChaosBook.org/projects/Lippolis/WaKo12.pdf},
year = {2012}
}
@unpublished{Waleffe90,
author = {F. Waleffe},
title = {Proposal for a self-sustaining mechanism in shear flows},
note = {Center for Turbulence Research, Stanford Univ./NASA Ames,
unpublished preprint (1990)}
}
@Article{Waleffe97,
author = {Waleffe, F.},
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}
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editor = {R.L. Panton},
pages = {385--422},
address = {Southampton},
note = {www.math.wisc.edu/~waleffe/ECS/WK97.pdf}
}
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journal = "J. Comput. Physics",
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pages = " - ",
year = "2011",
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}
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author = {Wang, Q.},
title = "Forward and adjoint sensitivity computation of chaotic
dynamical systems",
year = 2012,
note = "\arXiv{1202.5229}",
}
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author={X. Wan and X. Zhou and W. E},
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pages={475},
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@ARTICLE{weiss86,
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@Misc{WeissFeig,
author = "Weisstein, E. W.",
title = "Feigenbaum constant",
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note = {mathworld.wolfram.com/FeigenbaumConstant.html},
}
@ARTICLE{Weyl,
author = {H. Weyl},
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partial differential equations (with applications to
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@Book{wiggins2,
author = {S. Wiggins},
title = {Chaotic Transport in Dynamical Systems},
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address = {New York},
year = 1991
}
@Unpublished{wikiHermite,
author = {Wikipedia},
title = {Hermite polynomials},
note = {{en.wikipedia.org/wiki/Hermite\_polynomials}}
}
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@Article{witt99hol,
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journal = {Chaos},
year = {1999},
volume = {9},
pages = {452},
abstract = {Using a wavelet basis, the spatiotemporally chaotic regime of
the KSe is explored where a good seperation of scales is observed. In large
scales, the dynamics is Gaussian. In the intermediate scales, the dynamics is
reminiscent of travelling waves and heteroclinic cyces which is the typical
behavior for small system size. In the small scales, the dynamics is
intermittent. Through investigation of the interaction between different
scales, we see the intermediate structures give the defining shape of the
cell and the large scales trigger the spatiotemporal chaos. The small scales
dissipate energy and modify the background in a average sense.}
}
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pages = {333--371}
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@Inproceedings{WKH93,
author = {Waleffe, F. and Kim, J. and Hamilton, J.},
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@ARTICLE{WoSa07,
title = {An efficient method for recovering {Lyapunov} vectors from
singular vectors},
author = {Wolfe, C. L. and Samelson, R. M.},
journal = {Tellus A},
volume = {59},
year = {2007},
pages = {355--366},
abstract = {
Standard techniques for computing Lyapunov vectors produce
results which are norm-dependent and lack invariance under
the linearized flow. An efficient, norm-independent method
for constructing the n most rapidly growing Lyapunov
vectors from n-1 leading forward and n leading backward
asymptotic singular vectors is proposed. The Lyapunov
vectors so constructed are invariant under the linearized
flow in the sense that, once computed at one time, they are
defined, in principle, for all time through the tangent
linear propagator. An analogous method allows the
construction of the n most rapidly decaying Lyapunov
vectors from n decaying forward and n-1 decaying backward
singular vectors.}
}
@Inproceedings{WW05,
author = {Waleffe, F. and Wang, J.},
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editor = {Mullin, T. and Kerswell, R.R.},
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publisher = {Kluwer}
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