@article{MR1701385,
author = {Warwick Tucker},
title = {The {L}orenz attractor exists},
journal = {C. R. Acad. Sci. Paris S{\'e}r. I Math.},
fjournal = {Comptes Rendus de l'Acad{\'e}mie des Sciences. S{\'e}rie I. Math{\'e}matique},
volume = {328},
year = {1999},
number = {12},
pages = {1197--1202},
issn = {0764-4442},
coden = {CASMEI},
mrclass = {37D45 (34C28 37G05 37M20)},
mrnumber = {MR1701385 (2001b:37051)},
mrreviewer = {Maria Jos{\'e} Pacifico},
abstract = {We prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. The proof is based on a combination of normal form theory and rigorous numerical computations.},
doi = {10.1016/S0764-4442(99)80439-X}
}
@article{MR1793416,
author = {Stefano Luzzatto and Warwick Tucker},
title = {Non-uniformly expanding dynamics in maps with singularities and criticalities},
journal = {Inst. Hautes {\'E}tudes Sci. Publ. Math.},
fjournal = {Institut des Hautes {\'E}tudes Scientifiques. Publications Math{\'e}matiques},
number = {89},
year = {1999},
pages = {179--226 (2000)},
issn = {0073-8301},
coden = {PMIHA6},
mrclass = {37E05 (37D25 37D50)},
mrnumber = {MR1793416 (2002b:37047)},
mrreviewer = {Hans Thunberg},
abstract = {We investigate a one-parameter family of interval maps arising in the study of the geometric Lorenz flow for non-classical parameter values. Our conclusion is that for all parameters in a set of positive Lebesgue measure the map has a positive Lyapunov exponent. Furthermore, this set of parameters has a density point which plays an important dynamic role. The presence of both singular and critical points introduces interesting dynamics, which have not yet been fully understood.},
doi = {10.1007/BF02698857}
}
@incollection{MR1870096,
author = {Warwick Tucker},
title = {Computational algorithms for ordinary differential equations},
booktitle = {International {C}onference on {D}ifferential {E}quations, {V}ol. 1, 2 ({B}erlin, 1999)},
pages = {71--76},
publisher = {World Sci. Publ., River Edge, NJ},
year = {2000},
mrclass = {34A45 (65L05)},
mrnumber = {MR1870096},
abstract = {We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic. We illustrate the presented method by computing solution sets for two explicit systems.},
ps = {http://www.math.uu.se/%7Ewarwick/main/papers/iv1.ps}
}
@article{1008.37051,
title = {{Computing accurate Poincar{\'e} maps.}},
author = {Warwick Tucker},
journal = {Physica D },
pages = {127--137},
volume = {171},
number = {3},
year = {2002},
abstract = {We present a numerical method particularly suited for computing Poincar{\'e} maps for systems of ordinary differential equations. The method is a generalization of a stopping procedure described by H{\'e}non [Physica D 5 (1982) 412], and it applies to a wide family of systems.},
doi = {10.1016/S0167-2789(02)00603-6},
url = {http://www.math.uu.se/%7Ewarwick/main/papers/poincare.cc}
}
@article{WT09,
title = {{A rigorous lower bound for the stability regions of the quadratic map.}},
author = {Warwick Tucker and Daniel Wilczak},
journal = {Physica D },
pages = {1923--1936},
volume = {238},
number = {18},
year = {2009},
abstract = {We establish a lower bound on the measure of the set of stable parameters a for the quadratic map $Q_a(x)=ax(1−x)$. For these parameters, we prove that $Q_a$ either has a single stable periodic orbit or a period-doubling bifurcation. From this result, we also obtain a non-trivial upper bound on the set of stochastic parameters for $Q_a$.},
doi = {http://dx.doi.org/10.1016/j.physd.2009.06.020},
url = {supplements/logMap.tgz}
}
@article{MR1870856,
author = {Warwick Tucker},
title = {A rigorous {ODE} solver and {S}male's 14th problem},
journal = {Found. Comput. Math.},
fjournal = {Foundations of Computational Mathematics. The Journal of the Society for the Foundations of Computational Mathematics},
volume = {2},
year = {2002},
number = {1},
pages = {53--117},
issn = {1615-3375},
mrclass = {37D45 (37-04 37C70 37M99 76R99)},
mrnumber = {MR1870856 (2003b:37055)},
mrreviewer = {Maria Jos{\'e} Pacifico},
abstract = {We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic with directed rounding. As an application, we prove that the Lorenz equations support a strange attractor, as conjectured by Edward Lorenz in 1963. This conjecture was recently listed by Steven Smale as one of several challenging problems for the 21st century. We also prove that the attractor is robust, i.e., it persists under small perturbations of the coefficients in the underlying differential equations. Furthermore, the flow of the equations admits a unique SRB measure, whose support coincides with the attractor. The proof is based on a combination of normal form theory and rigorous computations.},
pdf = {http://www.springerlink.com/content/myglw1pwkhu9r5g2/fulltext.pdf},
url = {http://www.math.uu.se/~warwick/main/rodes.html}
}
@article{1073.45500,
author = {Irina Mitrea and Warwick Tucker},
title = {{Some counterexamples for the spectral-radius conjecture.}},
journal = {Differ. Integral Equ. },
volume = {16},
number = {12},
pages = {1409--1439},
year = {2003},
abstract = {The goal of this paper is to produce a series of
counterexamples for the Lp spectral radius conjecture, 1 < p < ∞,
for double-layer potential operators associated to a distinguished
class of elliptic systems in polygonal domains in $R^2$. More specifically the class under discussion is that of second-order elliptic systems in two dimensions whose coefficient tensor (with constant real entries) is symmetric and strictly positive definite. The general techniques employed are those of the Mellin transform and Calder´on-Zygmund theory. For the case p ∈ (1, 4), we construct a computer-aided proof utilizing validated numerics based on interval analysis.},
pdf = {http://www.math.uu.se/~warwick/main/papers/counterExPaper.pdf},
url = {http://www.math.uu.se/%7Ewarwick/main/papers/radius.cc}
}
@article{MR2086158,
author = {Warwick Tucker},
title = {Robust normal forms for saddles of analytic vector fields},
journal = {Nonlinearity},
fjournal = {Nonlinearity},
volume = {17},
year = {2004},
number = {5},
pages = {1965--1983},
issn = {0951-7715},
coden = {NONLE5},
mrclass = {37G05 (34C20 37M99)},
mrnumber = {MR2086158 (2005g:37094)},
mrreviewer = {Vincent Naudot},
abstract = {The aim of this paper is to introduce a technique for describing trajectories of
systems of ordinary differential equations (ODEs) passing near saddle-fixed
points. In contrast to classical linearization techniques, the methods of this
paper allow for perturbations of the underlying vector fields. This robustness
is vital when modelling systems containing small uncertainties, and in the
development of numerical ODE solvers producing rigorous error bounds.}
}
@incollection{MR2226836,
author = {Warwick Tucker and Vincent Moulton},
title = {Reconstructing metabolic networks using interval analysis},
booktitle = {Algorithms in bioinformatics},
series = {Lecture Notes in Comput. Sci.},
volume = {3692},
pages = {192--203},
publisher = {Springer},
address = {Berlin},
year = {2005},
mrclass = {92B20 (34A60 92C40)},
mrnumber = {MR2226836 (2007f:92005)},
mrreviewer = {Brian D. Sleeman},
abstract = {Recently, there has been growing interest in the modelling
and simulation of biological systems. Such systems are often modelled in
terms of coupled ordinary differential equations that involve parameters
whose (often unknown) values correspond to certain fundamental properties
of the system. For example, in metabolic modelling, concentrations
of metabolites can be described by such equations, where parameters correspond
to the kinetic rates of the underlying chemical reactions. Within
this framework, the increasing availability of time series data opens up
the attractive possibility of reconstructing approximate parameter values,
thus enabling the in silico exploration of the behaviour of complex
dynamical systems. The parameter reconstruction problem, however, is
very challenging – a fact that has resulted in a plethora of heuristics
methods designed to fit parameters to the given data.
In this paper we propose a completely deterministic method for parameter
reconstruction that is based on interval analysis. We illustrate its
utility by applying it to reconstruct metabolic networks using S-systems.
Our method not only estimates the parameters very precisely, it also
determines the appropriate network topologies. A major strength of the
proposed method is that it proves that large portions of parameter space
can be disregarded, thereby avoiding spurious solutions.},
doi = {10.1007/11557067_16}
}
@incollection{MR2185786,
author = {Warwick Tucker},
title = {Validated numerics for pedestrians},
booktitle = {European {C}ongress of {M}athematics},
pages = {851--860},
publisher = {Eur. Math. Soc., Z{\"u}rich},
year = {2005},
mrclass = {65G20 (65H05)},
mrnumber = {MR2185786 (2006g:65071)},
abstract = {The aim of this paper is to give a very brief introduction to the emerging
area of validated numerics. This is a rapidly growing field of research faced
with the challenge of interfacing computer science and pure mathematics. Most
validated numerics is based on interval analysis, which allows its users to account
for both rounding and discretization errors in computer-aided proofs. We will illustrate
the strengths of these techniques by converting the well-known bisection
method into a efficient, validated root finder.},
url = {http://www.ems-ph.org/books/show_abstract.php?proj_nr=23&vol=1&rank=54},
pdf = {http://www.math.uu.se/%7Ewarwick/main/papers/ECM04Tucker.pdf}
}
@article{1098.65082,
author = {Warwick Tucker and Vincent Moulton},
title = {{Parameter reconstruction for biochemical networks using interval analysis.}},
journal = {Reliab. Comput. },
volume = {12},
number = {5},
pages = {389--402},
year = {2006},
doi = {10.1007/s11155-006-9009-2},
abstract = {In recent years, themodeling and simulation of biochemical networks has attracted increasing
attention. Such networks are commonly modeled by systems of ordinary differential equations,
a special class of which are known as S-systems. These systems are specifically designed to mimic
kinetic reactions, and are sufficiently general to model genetic networks, metabolic networks, and
signal transduction cascades. The parameters of an S-system correspond to various kinetic rates of
the underlying reactions, and one of the main challenges is to determine approximate values of these
parameters, given measured (or simulated) time traces of the involved reactants.
Due to the high dimensionality of the problem, a straight-forward optimization strategy will rarely
produce correct parameter values. Instead, almost all methods available utilize genetic/evolutionary
algorithms to perform the non-linear parameter fitting. We propose a completely deterministic
approach, which is based on interval analysis. This allows us to examine entire sets of parameters,
and thus to exhaust the global search within a finite number of steps. The proposed method
can in principle be applied to any system of finitely parameterized differential equations, and, as we demonstrate, yields encouraging results for low dimensional S-systems.}
}
@article{1117.65163,
author = {Irina Mitrea and Warwick Tucker},
title = {{Interval analysis techniques for boundary value problems of elasticity in two dimensions.}},
journal = {J. Differ. Equations },
volume = {233},
number = {1},
pages = {181--198},
year = {2007},
doi = {10.1016/j.jde.2006.10.010},
abstract = {In this paper we prove that the $L^2$ spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in $R^2$ is within $10^{−2}$ from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón–Zygmund theory, as well as interval analysis—resulting in a computer-aided proof.}
}
@article{PubMed_17591176_3,
issn = {1751-8849},
volume = {1},
number = {3},
year = {2007},
month = may,
journal = {IET Syst Biol},
title = {S-system parameter estimation for noisy metabolic profiles using newton-flow analysis.},
pages = {174--80},
abstract = {Biochemical systems are commonly modelled by systems of ordinary differential equations (ODEs). A particular class of such models called S-systems have recently gained popularity in biochemical system modelling. The parameters of an S-system are usually estimated from time-course profiles. However, finding these estimates is a difficult computational problem. Moreover, although several methods have been recently proposed to solve this problem for ideal profiles, relatively little progress has been reported for noisy profiles. We describe a special feature of a Newton-flow optimisation problem associated with S-system parameter estimation. This enables us to significantly reduce the search space, and also lends itself to parameter estimation for noisy data. We illustrate the applicability of our method by applying it to noisy time-course data synthetically produced from previously published 4- and 30-dimensional S-systems. In addition, we propose an extension of our method that allows the detection of network topologies for small S-systems. We introduce a new method for estimating S-system parameters from time-course profiles. We show that the performance of this method compares favorably with competing methods for ideal profiles, and that it also allows the determination of parameters for noisy profiles.},
affiliation = {Department of Medical Genetics, University of Lausanne, Rue de Bugnon 27, Lausanne 1005, Switzerland. zoltan.kutalik@unil.ch},
author = {Z Kutalik and W Tucker and V Moulton},
url = {http://www.math.uu.se/%7Ewarwick/main/papers/ktmSup.pdf},
doi = {10.1049/iet-syb:20060064}
}
@article{PubMed_17306307,
issn = {0025-5564},
volume = {208},
number = {2},
year = {2007},
month = aug,
journal = {Math Biosci},
title = {Estimating parameters for generalized mass action models using constraint propagation.},
pages = {607--20},
abstract = {As modern molecular biology moves towards the analysis of biological systems as opposed to their individual components, the need for appropriate mathematical and computational techniques for understanding the dynamics and structure of such systems is becoming more pressing. For example, the modeling of biochemical systems using ordinary differential equations (ODEs) based on high-throughput, time-dense profiles is becoming more common-place, which is necessitating the development of improved techniques to estimate model parameters from such data. Due to the high dimensionality of this estimation problem, straight-forward optimization strategies rarely produce correct parameter values, and hence current methods tend to utilize genetic/evolutionary algorithms to perform non-linear parameter fitting. Here, we describe a completely deterministic approach, which is based on interval analysis. This allows us to examine entire sets of parameters, and thus to exhaust the global search within a finite number of steps. In particular, we show how our method may be applied to a generic class of ODEs used for modeling biochemical systems called Generalized Mass Action Models (GMAs). In addition, we show that for GMAs our method is amenable to the technique in interval arithmetic called constraint propagation, which allows great improvement of its efficiency. To illustrate the applicability of our method we apply it to some networks of biochemical reactions appearing in the literature, showing in particular that, in addition to estimating system parameters in the absence of noise, our method may also be used to recover the topology of these networks.},
affiliation = {Department of Mathematics, Uppsala University, Box 480, Uppsala, Sweden. warwick@math.uu.se},
author = {Warwick Tucker and Zolt{\'a}n Kutalik and Vincent Moulton},
doi = {10.1016/j.mbs.2006.11.009},
url = {http://www.math.uu.se/%7Ewarwick/main/papers/publicGMAcode.tar}
}
@article{1153.93035,
author = {Tomas Johnson and Warwick Tucker},
title = {{Rigorous parameter reconstruction for differential equations with noisy data.}},
journal = {Automatica },
volume = {44},
number = {9},
pages = {2422--2426},
year = {2008},
abstract = {We present a method that–given a data set, a finitely parametrized system of ordinary differential equations (ODEs), and a search space of parameters–discards portions of the search space that are inconsistent with the model ODE and data. The method is completely rigorous as it is based on validated integration of the vector field. As a consequence, no consistent parameters can be lost during the pruning phase. For data sets with moderate levels of noise, this yields a good reconstruction of the underlying parameters. Several examples are included to illustrate the merits of the method.},
doi = {10.1016/j.automatica.2008.01.032}
}
@article{0810.5282v1,
author = {Tomas Johnson and Warwick Tucker},
title = {Automated computation of robust normal forms of planar analytic vector fields},
journal = {Discrete and Continuous Dynamical Systems: Series B},
volume = {12},
number = {4},
pages = {769--782},
year = {2009},
keywords = {Dynamical Systems (math.DS)},
abstract = {We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighborhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method.},
doi = {10.3934/dcdsb.2009.12.769}
}
@article{1165.65344,
author = {Tomas Johnson and Warwick Tucker},
title = {{Enclosing all zeros of an analytic function - a rigorous approach.}},
journal = {J. Comput. Appl. Math. },
volume = {228},
number = {1},
pages = {418--423},
year = {2009},
abstract = {We present a method to find all zeros of an analytic function in a rectangular domain. The approach is based on finding guaranteed enclosures rather than approximations of the zeros. Well-isolated simple zeros are determined fast and with high accuracy. Clusters of zeros can in many cases be distinguished from multiple zeros by applying the argument principle to sufficiently high-order derivatives of the function. We illustrate the proposed method through five examples of varying levels of complexity.},
doi = {10.1016/j.cam.2008.10.014}
}
@misc{pre05587366,
author = {Warwick Tucker},
title = {{Fundamentals of chaos.}},
howpublished = {{Kocarev, Ljupco (ed.) et al., Intelligent computing based on chaos. Berlin: Springer. Studies in Computational Intelligence 184, 1-23 (2009).}},
year = {2009},
doi = {10.1007/978-3-540-95972-4_1}
}
@misc{JTpre2010b,
abstract = {Truncated Taylor series representations of invariant manifolds are abundant in numerical computations. We present an aposteriori method to compute the convergence radii and error estimates of analytic parametrisations of non-resonant local invariant manifolds of a saddle of an analytic vector field, from such a truncated series. This enables us to obtain local enclosures, as well as existence results, for the invariant manifolds.},
author = {Tomas Johnson and Warwick Tucker},
title = {A note on the convergence of parametrised non-resonant invariant manifolds},
year = {2009},
month = jun,
keywords = {Dynamical Systems (math.DS)},
v1url = {http://www.arxiv.org/abs/0811.4500v1},
v1descr = {Thu, 27 Nov 2008 10:00:00 GMT (71kb,D)},
v2descr = {Tue, 16 Jun 2009 14:18:13 GMT (17kb)},
url = {http://arxiv.org/abs/0811.4500}
}
@misc{JTpre2010a,
abstract = {An accurate method to compute enclosures of Abelian integrals is developed. It is applied to the study of bifurcations of limit cycles from cubic perturbations of elliptic Hamiltonians of degree four. We give examples of perturbations such that 2, 3, 3, and 4 limit cycles bifurcate from the truncated pendulum, the saddle loop, the interior of the cuspidal loop, and the interior of the figure eight loop, respectively. Some methods to find perturbations with a given number of limit cycles are illustrated in the examples.},
author = {Tomas Johnson and Warwick Tucker},
title = {On a computer-aided approach to the computation of Abelian integrals},
year = {2009},
month = apr,
keywords = {Dynamical Systems (math.DS)},
v1url = {http://www.arxiv.org/abs/0809.2867v1},
v1descr = {Wed, 17 Sep 2008 07:53:12 GMT (124kb)},
v2descr = {Wed, 22 Apr 2009 08:41:29 GMT (78kb)},
url = {http://arxiv.org/abs/0809.2867}
}
@misc{JT10a,
abstract = {The limit cycle bifurcations of a $Z_2$ equivariant planar Hamiltonian vector field of degree 7 under $Z_2$ equivariant degree 7 perturbation is studied. We prove that the given system can have at least 53 limit cycles. This is an improved lower bound for the weak formulation of Hilbert’s 16th problem for degree 7, i.e., on the possible number of limit cycles that can bifurcate from a degree 7 planar Hamiltonian system under degree 7 perturbation.},
author = {Tomas Johnson and Warwick Tucker},
title = {An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7.},
year = {2010},
url = {papers/eq_8.pdf},
html = {supplements/AI_8.tar.gz}
}
@misc{JT10b,
abstract = {The limit cycle bifurcations of a $Z_2$ equivariant quintic planar Hamil-
tonian vector field under $Z_2$ equivariant quintic perturbation is studied.
We prove that the given system can have at least 27 limit cycles. This
is an improved lower bound on the possible number of limit cycles that
can bifurcate from a quintic planar Hamiltonian system under quintic
perturbation.},
author = {Tomas Johnson and Warwick Tucker},
title = {An improved lower bound on the number of limit cycles bifurcating from a quintic Hamiltonian planar vector field under quintic perturbation.},
year = {2010},
url = {papers/equivariant_V4.pdf},
html = {supplements/AI_6.tar.gz}
}
@misc{FKST08,
abstract = {In this paper, we present a computer–aided method (based on [Ya98]) that establishes
the existence and local uniqueness of a stationary solution to the viscous Burgers’
equation. The problem formulation involves a left boundary condition and one integral
boundary condition, which is a variation of the approach taken in [Si04].},
author = {Oswald Fogelklou and Gunilla Kreiss and Malin Siklosi and Warwick Tucker},
title = {A Computer-assisted Proof of the Existence of Solutions to a Boundary Value Problem with an Integral Boundary Condition.},
year = {2008},
url = {papers/burgers.pdf}
}
@article{pre05579298,
author = {Tomas Johnson and Warwick Tucker},
title = {{A rigorous study of possible configurations of limit cycles bifurcating from a hyper-elliptic Hamiltonian of degree five.}},
journal = {Dyn. Syst. },
volume = {24},
number = {2},
pages = {237--247},
year = {2009},
abstract = {We consider a hyper-elliptic Hamiltonian of degree five, chosen from a generic set of parameters, and study what configurations of limit cycles can bifurcate from the corresponding differential system under quartic perturbations. Perturbations of Lienard type are considered separately. Several different configurations with seven (four) limit cycles, bifurcating from the given system for general (Lienard type) quartic perturbations, are constructed. We also discuss how to construct perturbations yielding a given configuration, and how to validate the correctness of such a candidate perturbation. },
doi = {10.1080/14689360802641206}
}
@inproceedings{ZLorenzPOISCAS08,
author = {Z.~Galias and W.~Tucker},
title = {Rigorous study of short periodic orbits for the {L}orenz system},
booktitle = { Proc. IEEE Int.~Symposium on Circuits and Systems, ISCAS'08},
year = 2008,
pages = {764--767},
address = {Seattle},
month = {May},
field = {INT PER LORENZ},
abstract = {The existence of short periodic orbits for the Lorenz
system is studied rigorously. We describe a method for finding
all short cycles embedded in a chaotic singular attractor (i.e.
an attractor containing an equilibrium). The method uses an
interval operator for proving the existence of periodic orbits in
regions where it can be evaluated, and bounds for the return
time in other regions. The six shortest periodic orbits for the
Lorenz system are found.},
http = {http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4541530&isnumber=4541329}
}
@inproceedings{ZLorenzPOICSES08,
author = {Z.~Galias and W.~Tucker},
title = {Short Periodic Orbits for the {L}orenz System},
booktitle = { Proc. Int. Conference on Signals and Electronic Systems, ICSES'08},
pages = {285--288},
year = 2008,
address = {Krakow},
field = {INT PER LORENZ},
abstract = {The existence of short periodic orbits for the Lorenz
system is studied rigorously. We describe a method for finding
all short cycles embedded in the chaotic attractor. We use the
method of close returns to find initial points for the Newton
operator, combined with interval tools for proving the existence
of periodic orbits in a neighborhood of a pseudo-periodic orbit.
All periodic orbits with period $p ≤ 8$ of the Poincar{\'e} map for
the Lorenz system are found.},
http = {http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4673416&isnumber=4673327}
}
@inproceedings{DHT10,
author = {A.~Danis, A.~Hooker, and W.~Tucker},
title = {Rigorous parameter estimation for noisy mixed-effects models},
booktitle = { Proc. International Symposium on Nonlinear Theory and its Applications, NOLTA'10},
pages = {67--70},
year = 2010,
address = {Krakow},
field = {},
abstract = {We describe how constraint propagation
techniques can be used to reliably reconstruct model pa-
rameters from noisy data. The main algorithm combines a
branch and bound procedure with a data inflation step; it is
robust and insensitive to noise. The set–valued results are
transformed into point clouds, after which statistical prop-
erties can be retrieved. We apply the presented method to a
mixed-effects model.},
}
@misc{JG09,
abstract = {It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $R^2$. A renormalization approach has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer-assisted proof of existence of a “universal” area-preserving map $F*$ — a map with orbits of all binary periods $2^k$, $k ∈ N$. In this paper, we consider maps in some neighbourhood of $F*$ and study their dynamics.
We first demonstrate that the map $F*$ admits a “bi-infinite heteroclinic
tangle”: a sequence of periodic points ${z_k}$, $k ∈ Z$ whose stable and unstable manifolds intersect transversally; and, for any N ∈ N, a compact invariant set on which F∗ is homeomorphic to a topological Markov chain on the space of all two-sided sequences composed of N symbols. A corollary of these results is the existence of unbounded and oscillating orbits. We also show that the third iterate for all maps close to F∗ admits a horseshoe. We use distortion tools to provide rigorous bounds on the Hausdorff dimension of the associated locally maximal invariant hyperbolic set:
$0.7673 \ge dimH(C_F) \ge \epsilon \ge 0.00044 e^{−1797}$.},
author = {Denis Gaidashev and Tomas Johnson},
title = {Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Hyperbolic Sets},
year = {2009},
url = {papers/Hyp_Sets.pdf}
}
@article{MR2455704,
author = {Johnson, Tomas},
title = {{$L^p$} spectral radius estimates for the {L}am\'e system
on an infinite sector},
abstract = {We prove, using interval analysis methods, that the $L^2$, $L^4$, and $L^8$ spectral radii of the traction double layer potential operator associated with the Lam\'e system on an infinite sector in ${\mathbb R}^2$ are within $2.5 \times 10^{-3}$, $10^{-2}$, and $10^{-2}$, respectively, from a certain conjectured value that depends explicitly on the aperture of the sector and the Lamé moduli of the system. We also indicate how to extend these results to $L^p$ for entire intervals of $p$, $p\ge 2$.},
journal = {Experiment. Math.},
fjournal = {Experimental Mathematics},
volume = {17},
year = {2008},
number = {3},
pages = {333--339},
issn = {1058-6458},
mrclass = {35J55 (35P15 35Q72 65J10)},
mrnumber = {MR2455704 (2009i:35081)},
url = {http://projecteuclid.org/euclid.em/1227121386}
}
@misc{JG09b,
abstract = {It is known that the famous Feigenbaum-Coullet-Tresser period doubling universality has a counterpart for area-preserving maps of $R^2$. A renormalization approach has been used in (Eckmann et al 1982) and (Eckmann et al 1984) in a computer-assisted proof of existence of a “universal” area-preserving map $F*$ — a map with orbits of all binary periods $2^k$, $k ∈ N$. In this paper, we consider infinitely renormalizable maps — maps on the renormalization stable manifold in some neighborhood of $F∗$ — and study their
dynamics.
For all such infinitely renormalizable maps in a neighborhood of the fixed
point $F∗$ we prove the existence of a “stable” invariant set $C^{∞}_F$ such that the maximal Lyapunov exponent of $F|C^{∞}_F$ is zero, and whose Hausdorff dimension satisfies $dimH(C^{∞}_F) \le 0.836$.
We also show that there exists a submanifold, $W_\omega$, of finite codimension in the renormalization local stable manifold, such that for all $F ∈ W_\omega$ the set $C^{∞}_F$ is “weakly rigid”: the dynamics of any two maps in this submanifold, restricted to the stable set $C^{∞}_F$ , is conjugated by a bi-Lipschitz transformation that preserves the Hausdorff dimension.},
author = {Denis Gaidashev and Tomas Johnson},
title = {Dynamics of the Universal Area-Preserving Map Associated with Period Doubling: Stable Sets},
year = {2009},
volume = {3},
number = {4},
pages = {555--587},
journal = {JMD},
fjournal = {Journal of Modern Dynamics},
url = {http://dx.doi.org/10.3934/jmd.2009.3.555}
}
@misc{JT10c,
author = {Tomas Johnson and Warwick Tucker},
title = {On a fast and accurate method to enclose all zeros of an analytic function on a triangulated domain.},
year = {2010},
url = {papers/trizeroFull_V2.pdf},
html = {supplements/cZero.tar.gz}
}
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