"Computer-aided proofs in analysis"


Sunset over Uppsala


D. Gaideshev, M. Yampolsky and R. Radu. Renormalization and Siegel disks for complex Henon maps. [ preprint ]

D. Gaideshev and M. Yampolsky. Renormalization for almost commuting pairs. [ preprint ]

D. Gaideshev and M. Yampolsky. Golden mean Siegel disk universality and renormalization. [ preprint ]

J.-Ll. Figueras and D. Strängberg. Non-Smooth Bifurcations of Uniformly Hyperbolic Invariant Manifolds in Skew Product Systems: Rigorous Results. [ preprint ]

O. Fogelklou. Estimation of the diffusion coefficient by interval methods, level curves and bicubic splines. [ preprint ]

A. Danis, S. Ueckert, A. Hooker and W. Tucker. Rigorous parameter estimation for noisy mixed-effects models. [ preprint ]

To appear

T. Johnson and W. Tucker. On a fast and accurate method to enclose all zeros of an analytic function on a triangulated domain. To appear in Lecture Notes in Computer Science. [ bib | preprint code ]


Z. Galias and W. Tucker. Rigorous integration of smooth vector fields around spiral saddles with an application to the cubic Chua's attractor, Journal of Differential Equations, 2018. [ doi ]

Z. Galias and W. Tucker. On the existence of the double scroll attractor for the Chua's circuit with a smooth nonlinearity. In Proc. IEEE Int. Symp. Circuits Syst. (ISCAS), pages 1-5, Florence, May 2018. [ pdf ]

J.-Ll. Figueras, A. Haro and A. Luque. On the sharpness of the Rüssmann estimates. Communications in Nonlinear Science and Numerical Simulation, 55:42-55, 2018. [ doi ]


I. Mitrea, K. Ott and W. Tucker. Invertability properties of singular integral operators associated with the Lamé and Stokes systems on infinite sectors in two dimensions. Integral Equations and Operator Theory, 1-57, 2017. [ doi  | preprint ]

D. Gaideshev. Renormalization for Lorenz maps of long monotone combinatorial types. Ergodic Theory and Dynamical Systems, 1-27, 2017. [ doi ]

J.-Ll. Figueras, M. Gameiro, J.-P. Lessard and R. de la Llave. A Framework for the Numerical Computation and A Posteriori Verification of Invariant Objects of Evolution Equations. SIAM Journal on Applied Dynamical Systems, 16:2, 1070-1088, 2017. [ doi ]

J.-Ll. Figueras and R. de la Llave. Numerical Computations and Computer Assisted Proofs of Periodic Orbits of the Kuramoto--Sivashinsky Equation. SIAM Journal on Applied Dynamical Systems, 16:2, 834-852, 2017. [ doi ]

T. Kapela and C. Simó. Rigorous KAM results around arbitrary periodic orbits for Hamiltonian Systems. Nonlinearity, 3:30, 965-986, 2017. [ doi ]


M. Joldes, J-M. Muller, V. Popescu and W. Tucker. CAMPARY: Cuda Multiple Precision Arithmetic Library and Applications. Mathematical Software -- ICMS 2016: 5th International Conference, 232-240, 2016. [ doi  ]

D. Gaideshev, T. Johnson, and M. Martens. Rigidity for infinitely renormalizable area-preserving maps. Duke Math. J., 165:1, 129-159, 2016. [ doi ]

J.-Ll. Figueras, A. Haro and A. Luque. Rigorous Computer-Assisted Application of KAM Theory: A Modern Approach. Foundations of Computational Mathematics, 1-71, 2016. [ doi ]

D. Gaideshev and T. Johnson. Spectral properties of the renormalization for area-preserving maps. DCDS-A, 36:7, 3651-3675, 2016. [ doi ]

M. Nehmeier, J. W. von Gudenberg and W. Tucker [editors]. Scientific Computing, Computer Arithmetic, and Validated Numerics: 16th International Symposium, SCAN 2014. Springer-Verlag, 2016. [ doi  ]

M. Canadell, J.-Ll. Figueras, A. Haro, A. Luque and J.M. Mondelo. The Parameterization method for Invariant Manifolds: From Rigorous Results to Effective Computations. Applied Mathematical Sciences, Springer, (2016)., ISBN 9783319296609. [ doi ]

W. Tucker. Interval Methods. In Uncertainty in Biology, pages 199-211. Volume 17 of the series Studies in Mechanobiology, Tissue Engineering and Biomaterials, 2016. [ chapter ]

J.-Ll. Figueras and A. Haro. A note on the fractalization of saddle invariant curves in quasiperiodic systems. DCDS-S, 9:4, 1095-1107, 2016. [ doi ]


F.A. Bartha and W. Tucker. Fixed points of a destabilized Kuramoto-Sivashinsky equation. Applied Mathematics and Computation, 266, 339-349, 2015. [ doi  | preprint code ]

J.-Ll. Figueras and A. Haro. Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps. Chaos, 25:123119, 2015. [ doi ]

W. Tucker. Interval Analysis. In The Princeton Companion to Applied Mathematics, pages 105-106. Princeton University Press, 2015. [ book ]

W. Tucker. Computer-Aided Proofs via Interval Analysis. In The Princeton Companion to Applied Mathematics, pages 790-795. Princeton University PressV, 2015. [ book ]

D. Gaideshev. On the Scaling Ratios for Siegel Disks. Comm. Math. Phys., 333, 931-957, 2015 [ doi  | paper ]

R. Barrio, A. Dena and W. Tucker. A database of rigorous and high-precision periodic orbits of the Lorenz model. Computer Physics Communications, 194, 76-83, 2015. [ doi  | paper | code ]

Z. Galias and W. Tucker. Is the Hénon map chaotic? Chaos 25, 033102, 2015. [ doi  | paper ]


M. Joldes, V. Popescu and W. Tucker. Searching for sinks of Hénon map using a multiple-precision GPU arithmetic library. SIGARCH Comput. Archit. News, 42(4), 63-68, 2014. [ doi  | paper ]

F.A. Bartha and A. Garab. Necessary and sufficient condition for the global stability of a delayed discrete-time single neuron model. Journal of Computational Dynamics, 1:2, 213-232, 2014. [ paper ]

Z. Galias and W. Tucker. A systematic approach to find periodic sinks of the Hénon map close to the classical case. In Proc. Int. Symposium on Nonlinear Theory and its Applications, NOLTA'14, pages 771-774, Luzern, 2014. [preprint ]

Z. Galias and W. Tucker. On the structure of existence regions for sinks of the Hénon map. Chaos, 24:013120, 2014. [ doi  | preprint ]

H. Härdin, A. Zagaris, A. Willms, and H. Westerhoff. Clusters of reaction rates and concentrations in protein networks such as the phosphotransferase system. the FEBS Journal, 281:2, 531-548, 2014. [ doi preprint ]

F.A. Bartha and H. Munthe-Kaas. Computing of B-series by Automatic Differentiation. Discrete and Continuous Dynamical Systems - Series A. 34(3):903-914, 2014. [ doi  | code ]

P. Gennemark P and D. Wedelin. ODEion - a software tool for structural identification of ordinary differential equations. J. Bioinform. Comput. Biol.12:01, 2014. [ doi html ]


Z. Galias and W. Tucker. Combination of exhaustive search and continuation method for the study of sinks in the Hénon map. Proceedings of the IEEE International Symposium on Circuits and Systems. 2751-2754, 2013. [ doi preprint ]

R. Sainudiin, J. Harlow and W. Tucker. There and Back Again: Split and Prune to Tighten. Proceedings of the 2013 IEEE International Conference on Fuzzy Systems. 1-7, 2013. [ doi preprint ]

J.-L. Figueras and À. Haro. Triple collisions of invariant bundles. Journal of Differential Equations 18:8, 2069-2082, 2013. [ doi ]

F.A. Bartha, A. Garab, and T. Krisztin. Local stability implies global stability for the 2-dimensional Ricker map. Journal of Difference Equations and Applications. 36 pp, 2013. [ doi ]

Z. Galias and W. Tucker. Numerical study of coexisting attractors for the Hénon map. International Journal of Bifurcation and Chaos. 23(7), 2013 [ doi ]

C. Kühn and P. Gennemark. Modeling yeast osmoadaptation at different levels of resolution. J. Bioinform. Comput. Biol. 11(2),330001 (17 pages), 2013. [ doi ]

J.-L. Figueras, W. Tucker and J. Villadelprat. Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals. Journal of Differential Equations 254(8):3647-3663, 2013. [ doi code ]


J. Harlow, R. Sainudiin and W. Tucker. Mapped Regular Pavings. Reliable Computing 16:252-282, 2012. [ paper ]

R. Calleja and J.-L. Figueras. Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map. Chaos 22:3, 033114, 201s. [ doi ]

D. Gaidashev and B. Winckler. Existence of a Lorenz renormalization fixed point of an arbitrary critical order. Nonlinearity 25:6, 1819-1841, 2012. [ doi ]

D. Gaidashev and T. Johnson. A numerical study of infinitely renormalizable area-preserving maps. Dynamical Systems: An International Journal, 27:3, 283-301, 2012. [ doi ]

O. Fogelklou, T. Konstantopoulos and W. Tucker. On the global stability of a peer-to-peer network model. OR Letters, 40(3):190-194, 2012. [ doi  ]


R. Jörnsten, T. Abenius, T. Kling, L. Schmidt, E. Johansson, T. Nordling, B. Nordlander, C. Sander, P. Gennemark, K. Funa, B. Nilsson, L. Lindahl and S. Nelander. Network modeling of the transcriptional effects of copy number aberrations in glioblastoma. Molecular Systems Biology 7:486, 2011. [ doi ]

O. Fogelklou, G. Kreiss and W. Tucker. A Computer--assisted Proof of the Existence of Traveling Wave Solutions to the Scalar Euler Equations with Artificial Viscosity. Nonlinear Differential Equations and Applications, 19(1):97-13, 2011. [ doi ]

D. Gaidashev . Period Doubling Renormalization for Area-Preserving Maps and Mild Computer Assistence in Contraction Mapping Principle. Int. J. Bifurcation Chaos 21, 3217, 2011. [ doi ]

D. Gaidashev . Period doubling in area-preserving maps: an associated one-dimensional problem. Erg. Theory and Dyn. Sys. 31(4), 1193-1228, 2011. [ doi ]

T. Johnson and W. Tucker. On a computer-aided approach to the computation of Abelian integrals. BIT, 51(3):653-667, 2011. [ doi ]

P. Gennemark, A. Danis, J. Nyberg, A. Hooker and W. Tucker. Optimal design in population kinetic experiments by set-valued methods. AAPS Journal, 13(4):495-507, 2011. [ code | doi ]

W. Tucker. Validated Numerics: A Short Introduction to Rigorous Computations. Princeton University Press, ISBN-13: 978-0691147819, 2011. [ www ]

Z. Galias and W. Tucker. Validated Study of Short Cycles for Chaotic Systems using Symbolic Dynamics and Interval Tools. International Journal of Bifurcation and Chaos, 21(2):551-563, 2011. [ doi ]

T. Johnson and W. Tucker. A note on the convergence of parametrised non-resonant invariant manifolds. Qualitative Theory of Dynamical Systems, 10(1):107-121, 2011. [ bib | doi ]

O. Fogelklou, G. Kreiss, M. Siklosi and W. Tucker. A computer-assisted proof of the existence of solutions to a boundary value problem with an integral boundary condition. Communications in Nonlinear Science and Numerical Simulation, 16(3):1227-1243, 2011. [ bib | doi ]


D. Gaidashev . On analytic perturbations of a family of Feigenbaum-like equations. J. Math. Anal. and Appl. 374(2), 355-373, 2010. [ doi ]

D. Wilczak. Uniformly hyperbolic attractor of the Smale-Williams type for a Poincaré map in the Kuznetsov system. SIAM Journal on Applied Dynamical Systems 9(4):1263-1283, 2010. [ doi ]

A. Danis, A. Hooker and W. Tucker. Rigorous parameter estimation for noisy mixed-effects models. Proceedings of International Symposium on Nonlinear Theory and its Applications 67-70, 2010. [ bib | article ]

T. Johnson and W. Tucker. An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 20(5):1-8, 2010. [ bib | doi code ]

T. Johnson and W. Tucker. An improved lower bound on the number of limit cycles bifurcating from a quintic hamiltonian planar vector field under quintic perturbation. International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 20(1):63-70, 2010. [ bib  | doi | code ]


D. Wilczak, P. Zgliczyński. Computer assisted proof of the existence of homoclinic tangency for the Hénon map and for the forced-damped pendulum. SIAM Journal on Applied Dynamical Systems 8(4):1632-1663, 2009. [ doi ]

Z. Galias and W. Tucker. Symbolic Dynamics Based Method for Rigorous Study of the Existence of Short Cycles for Chaotic Systems. Proceedings of IEEE Int. Symposium on Circuits and Systems, ISCAS'09, 1907-1910, 2009. [ paper ]

D. Gaidashev and T. Johnson. Dynamics of the universal area-preserving map associated with period doubling: Stable sets. Journal of Modern Dynamics 3(4):555-587, 2009. [ bib | doi ]

D. Wilczak. Abundance of heteroclinic and homoclinic orbits for the hyperchaotic Rössler system. Discrete and Continuous Dynamical System - Series B 11(4):1039-1055, 2009. [ doi ]

D. Gaidashev and T. Johnson. Dynamics of the universal area-preserving map associated with period doubling: Hyperbolic sets. Nonlinearity, 22:2487-2520, 2009. [ bib | doi | code ]

W. Tucker and D. Wilczak. A rigorous lower bound for the stability regions of the quadratic map. Physica D, 238(18):1923-1936, 2009. [ bib | doi | code ]

T. Johnson and W. Tucker. Automated computation of robust normal forms of planar analytic vector fields. Discrete and Continuous Dynamical Systems: Series B, 12(4):769-782, 2009. [ bib | doi | code ]

T. Johnson and W. Tucker. Enclosing all zeros of an analytic function - a rigorous approach. J. Comput. Appl. Math., 228(1):418-423, 2009. [ bib | doi | code ]

D. Wilczak, P. Zgliczyński. Period doubling in the Rössler system - a computer assisted proof. Foundations of Computational Mathematics 9(5):611-649, 2009. [ doi ]

W. Tucker. Fundamental of chaos. Kocarev, Ljupco (ed.) et al., Intelligent computing based on chaos. Berlin: Springer. Studies in Computational Intelligence 184, 1-23, 2009. [ bib | doi | book ]

T. Johnson and W. Tucker. A rigorous study of possible configurations of limit cycles bifurcating from a hyper-elliptic Hamiltonian of degree five. Dyn. Syst., 24(2):237-247, 2009. [ bib | doi | code ]


Z. Galias and W. Tucker. Rigorous study of short periodic orbits for the Lorenz system. In Proc. IEEE Int. Symposium on Circuits and Systems, ISCAS'08, pages 764-767, Seattle, May 2008. [ bib | doi ]

T. Johnson and W. Tucker. Rigorous parameter reconstruction for differential equations with noisy data. Automatica, 44(9):2422-2426, 2008. [ bib | doi | code ]

Z. Galias and W. Tucker. Short periodic orbits for the Lorenz system. In Proc. Int. Conference on Signals and Electronic Systems, ICSES'08, pages 285-288, Krakow, 2008. [ bib | doi ]

T. Johnson. Lp spectral radius estimates for the Lamé system on an infinite sector. Experiment. Math., 17(3):333-339, 2008. [ bib | paper ]


W. Tucker, Z. Kutalik and V. Moulton. Estimating parameters for generalized mass action models using constraint propagation. Math Biosci, 208(2):607-20, Aug. 2007. [ bib | doi | code ]

Z. Kutalik, W. Tucker and V. Moulton. S-system parameter estimation for noisy metabolic profiles using newton-flow analysis. IET Syst Biol, 1(3):174-80, May 2007. [ bib | doi | supplements ]

I. Mitrea and W. Tucker. Interval analysis techniques for boundary value problems of elasticity in two dimensions. J. Differ. Equations, 233(1):181-198, 2007. [ bib | doi ]

2006 and older

W. Tucker and V. Moulton. Parameter reconstruction for biochemical networks using interval analysis. Reliab. Comput., 12(5):389-402, 2006. [ bib | doi ]

W. Tucker and V. Moulton. Reconstructing metabolic networks using interval analysis. In Algorithms in bioinformatics, volume 3692 of Lecture Notes in Comput. Sci., pages 192-203. Springer, Berlin, 2005. [ bib | doi ]

W. Tucker. Validated numerics for pedestrians. In European Congress of Mathematics, pages 851-860. Eur. Math. Soc., Zürich, 2005. [ bib | book | paper ]

W. Tucker. Robust normal forms for saddles of analytic vector fields. Nonlinearity, 17(5):1965-1983, 2004. [ bib | doi ]

I. Mitrea and W. Tucker. Some counterexamples for the spectral-radius conjecture. Differ. Integral Equ., 16(12):1409-1439, 2003. [ bib | paper | code ]

W. Tucker. Computing accurate Poincaré maps. Physica D, 171(3):127-137, 2002. [ bib | doi | code ]

W. Tucker. A rigorous ODE solver and Smale's 14th problem. Found. Comput. Math., 2(1):53-117, 2002. [ bib | doi | html | paper ]

W. Tucker. Computational algorithms for ordinary differential equations. In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 71-76. World Sci. Publ., River Edge, NJ, 2000. [ bib | paper ]

W. Tucker. The Lorenz attractor exists. C. R. Acad. Sci. Paris Sér. I Math., 328(12):1197-1202, 1999. [ bib | doi ]

S. Luzzatto and W. Tucker.  Non-uniformly expanding dynamics in maps with singularities and criticalities. Inst. Hautes Études Sci. Publ. Math., (89):179-226 (2000), 1999. [ bib | doi ]