# Piotr Zgliczynski

Preprints
Papers
Conference reports

## Preprints and papers pending

1. Calculus of Invariant Manifolds - topological approach
2. Estimates of eigenspaces and eigenvalues of a matrix , joint work with £ukasz Struski and Jacek Tabor
3. Symbolic dynamics for Kuramoto-Sivashinsky PDE on the line - a computer-assisted proof , a file with numerical data supplement.tgz , with Daniel Wilczak
4. Real-number Computability from the Perspective of Computer Assisted Proofs in Analysis with Ma³gorzata Moczurad
5. Computer assisted proof of the existence of the Lorenz attractor in the Shimizu-Morioka system with Maciej Capinski and Dimitry Turaev
6. On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions with Piotr Kalita

## Papers

1. Beyond the Melnikov method II: multidimensional setting with Maciej Capinski, Journal of Differential Equations
2. Stabilizing the Long-time Behavior of the Navier-Stokes Equations and Damped Euler Systems by Large Mean Flow with Jacek Cyranka , Piotr B. Mucha and Edriss S. Titi, Physica D 369 (2018) 18–-29
3. Shadowing of non-transversal heteroclinic chains with Adria Simon and Amadeu Delshams, Journal of Differential Equations 264(2018), 3619–-3663
4. Algorithm for rigorous integration of Delay Differential Equations and the computer-assisted proof of periodic orbits in the Mackey-Glass equation with Robert Szczelina, Foundations of Computational Mathematics
5. Topological shadowing and the Grobman-Hartman Theorem, Topological Methods in Nonlinear Analysis, 50 (2017), 757--785
6. Hyperbolicity and Averaging for the Srzednicki-Wójcik equation , Journal of Differential Equations , 262 (2017), 365-417
7. On the Petras algorithm for verified integration of piecewise analytic functions , joint work with Ma³gorzata Moczurad, Journal of Complexity, 39 (2017), 69-93
8. Beyond the Melnikov method: a computer assisted approach with Maciej Capinski, Journal of Differential Equations, 262 (2017), 365-417
9. Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations with Jacek Cyranka , Journal of Differential Equations, 261(2016) 4648-4708
10. Existence of Periodic Solutions of the FitzHugh-Nagumo Equations for An Explicit Range of the Small Parameter, with Aleksander Czechowski, SIAM Journal of Applied Dynamical Systems, 15(2016), 1615–1655
11. Connecting orbits for a singular nonautonomous real Ginzburg-Landau type equation with Daniel Wilczak, SIAM Journal of Applied Dynamical Systems, Vol. 15 No. (1), 495-525 (2016)
12. New lower bound estimates for quadratures of bounded analytic functions , joint work with Ma³gorzata Moczurad and W³odzimierz Zwonek, Journal of Complexity, 34 (2016) 50-–67
13. Quasi-decidability of a Fragment of the Analytic First-order Theory of Real Numbers , with Peter Franek and Stefan Ratschan , Journal of Automated Reasoning, 57 (2016) 157-–185
14. Geometric proof for normally hyperbolic invariant manifolds with Maciej Capinski, Journal of Differential Equations, 259(2015) 6215--6286
15. Rigorous numerics for PDEs with indefinite tail: existence of a periodic solution of the Boussinesq equation with time-dependent forcing , with Aleksander Czechowski, Schedae Informaticae, 24 (2015), 143--158
16. Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof , with Jacek Cyranka , SIAM Journal of Applied Dynamical Systems, 14(2015) 787--821
17. A homoclinic orbit in a planar singular ODE- a computer assisted proof. , with R. Szczelina , SIAM Journal of Applied Dynamical Systems 12 (2013), 1541-–1565
18. Steady states bifurcations for theKuramoto-Sivashinsky equation - a computer assisted proof , a file with numerical data bifdata.txt , Journal of Computational Dynamics, 2 (2015), 95--142
19. Satisfiability of Systems of Equations of Real Analytic Functions is Quasi-decidable , with Peter Franek and Stefan Ratschan Mathematical Foundations of Computer Science 2011, Lecture Notes in Computer Science 6907 (2011) 315-326,
20. Transition Tori in the Planar Restricted Elliptic Three Body Problem , with Maciej Capinski , Nonlinearity, 24 (2011) 1395-1432
21. C^n-Lohner algorithm ,with D. Wilczak , Scheade Informaticae, Vol. 20, 9-46 (2011).
22. Cone Conditions and Covering Relations forNormally Hyperbolic Invariant Manifolds , with Maciej Capinski , Discrete and Continuous Dynamical Systems A , 30 (2011), 641-670
23. Rigorous Numerics for Dissipative PDEs III. An effectivealgorithm for rigorous integration of dissipativePDEs, a file with numerical data proofs.zip , the program and installation instruction , Topological Methods in Nonlinear Analysis, 36 (2010) 197–262
24. On stability of forcing relations for multidimensional perturbationsof interval maps with Ming-Chia Li , Fund. Math. 206 (2009) 241-251
25. Computer assisted proof of the existence of homoclinic tangency for the Henon map and for the forced-damped pendulum ,with D. Wilczak , SIAM Journal of Applied Dynamical Systems 8, Issue 4, pp. 1632-1663 (2009)
26. Covering relations, cone conditions and stable manifold theorem , J. of Diff. Equations 246 (2009) 1774--1819
27. Period doubling in the R\"ossler system - a computer assisted proof ,with D. Wilczak , Foundations of Computational Mathematics, 9 (2009) 611-649
28. A Lohner-type algorithm for control systems and ordinarydifferential inclusions , with Tomasz Kapela ,Discrete and Continuous Dynamical Systems B, 11 (2009), 365-385
29. Topological entropy for multidimensional perturbations of snap-backrepellers and one-dimensional maps ,with Ming-Chia Li and Ming-Jiea Lyu , Nonlinearity 21 (2008) 2555-2567
30. Rigorous verification of cocoon bifurcations in the Michelson system with H. Kokubu and D. Wilczak , Nonlinearity 20 (2007) 2147-2174
31. An interval method for finding fixed points and periodicorbits of infinite dimensional discrete dynamical systems, with Z. Galias , Int. J. of Bifurcation and Chaos 17,4261--4272 (2007)
32. Topological method for symmetric periodic orbitsfor maps with an reversing symmetry , with D. Wilczak ,Discrete and Continuous Dynamical Systems A, 17, 629--652 (2007)
33. Covering Relations and Non-autonomousPerturbations of ODEs , with Maciej Capinski, , Discrete and Continuous Dynamical Systems A, 14, 281--293 (2006)
34. Heteroclinic Connections between Periodic Orbits inPlanar Restricted Circular Three Body Problem - Part II , with D. Wilczak , Comm. Math. Phys. 259, 561-576 (2005),
35. Fixed point results based on the Wazewski method , with R. Srzednicki and K. Wojcik , in "Handbook of topological fixed point theory",Edited by: R.Brown, M. Furi, L. Górniewicz and B. Jiang , 903- 941,Kluwer 2004,
36. Topological horseshoes and delay differential equations , with K. Wojcik, Discrete and Continuous Dynamical Systems A, 12 (2005), 827--852
37. Covering relations for multidimensional dynamical systems II, with M. Gidea , J. of Diff. Equations 202(2004) 59--80
38. Covering relations for multidimensional dynamical systems I, with M. Gidea , J. of Diff. Equations, 202(2004) 32--58
39. Rigorous numerics for dissipative PartialDifferential Equations II.Periodic orbit for the Kuramoto-Sivashinsky PDE - a computerassisted proof , Foundations of Computational Mathematics, 4 (2004), 157--185
40. On smooth dependence on intial conditions for dissipative PDEs, an ODE-type approach , J. of Diff. Equations, 195/2 (2003), 271--283
41. The existence of simple choreographies for N-body problem - a computer assisted proof , with Tomasz Kapela , a C++-program used in this paper can be downloaded here , Nonlinearity, vol. 16 (2003), 1899-1918
42. Periodic, homoclinic and heteroclinic orbits for H\'{e}non Heiles Hamiltoniannear the critical energy level , with Gianni Arioli, Nonlinearity, vol. 16, No. 5 (2003), 1833--1852
43. Trapping regions and ODE-type proof ofan existence and uniqueness for Navier-Stokes equations with periodic boundaryconditions on the plane, Univ. Iag. Acta Math. 41(2003) 89-113
44. Attracting fixed points for the Kuramoto-Sivashinsky equation , SIAM Journal on Applied Dynamical Systems,(2002) Volume 1, Number 2 pp. 215-235
45. Heteroclinic Connections between Periodic Orbits inPlanar Restricted Circular Three Body Problem - A ComputerAssisted Proof , with D. Wilczak ,Comm. Math. Phys. 234 (2003) 1, 37-75,
46. C^1-Lohner algorithm , Foundations of Computational Mathematics, (2002) 2:429-465
47. Isolating segments, fixed point index and symbolic dynamics III. Applications. with K. Wojcik J. Diff. Eq. Vol. 183, No. 1, (2002), pp. 262-278
48. Abundance of homoclinic and heteroclinic orbitsand rigorous bounds for the topological entropy for the Henon map, with Z. Galias , Nonlinearity, 14 (2001), 909-932
49. Rigorous Numerics for Partial DifferentialEquations: the Kuramoto-Sivashinsky equation., with K. Mischaikow ,Foundations of Computational Mathematics, (2001) 1:255-288
50. Topological entropy for multidimensional perturbations of one dimensional maps, with M. Misiurewicz Int. J. of Bifurcation and Chaos, Vo. 11, No.5 (2001), 1443-1446
51. Isolating segments, fixed point index andsymbolic dynamics II. Homoclinic solutions. , with K. Wojcik, J. of Diff. Equations, 172, 189--211 (2001)
52. Symbolic dynamics for the H\'enon--Heiles hamiltonian on the critical energy level , with Gianni Arioli, Journal of Diff. Equations, 171, 173-202 (2001)
53. On periodic points for systems of weakly coupled 1-dim maps,Nonlinear Analysis TMA, Vol 46/7, 1039-1062 (2001)
54. Set arithmetics and the enclosing problem in dynamics , with Marian Mrozek , Ann. Pol. Mat. (2000), 237--259
55. On existence of infinitely many homoclinic solutions. with K. Wojcik , Monatshetfe fur Mathematik, 130, 155-160 (2000)
56. Isolating segments, fixed point index and symbolic dynamics with K. Wojcik, Journal of Diff. Equations, 161, 245--288, (2000)
57. Multidimensional perturbations of one-dimensional maps and stability of Sharkovskii ordering, Int. J. of Bifurcations and Chaos, Vol. 9, No. 9, (1999), 1867-1876
58. Sharkovskii's Theorem for multidimensional perturbations ofone-dimensional maps II Topological Methods in Nonlinear Analysis (1999),14,169-182
59. Sharkovskii's Theorem for multidimensional perturbations ofone-dimensional maps, Erg. Th. & Dyn. Systems, (1999), 19, 1655--1684
60. Computer assisted proof of chaos in the Lorenz system, Z. Galias Physica D, 115(1998), 165-188
61. On the discrete Conley index in the invariant subspace, with K. Wojcik and A. Szymczak, Topology and Applications, 87(1998) ,105-116,
62. Computer assisted proof of the horseshoe dynamics in the H\'enon map, Random \& Computational Dynamics, Vol. 5, No.1, 1997, 1--19
63. Computer assisted proof of chaos in the H\'enon map and in the R\"ossler equations, Nonlinearity, 1997, Vol. 10, No. 1, 243--252
64. An existence theorem for perturbation of nonlinear functionalboundary value problems for ODE's, Rivista di Matematica Pura e Applicata , 19, 1996 (101--105)
65. Fixed point index for iterations, topological horseshoe andchaos, Topological Methods in Nonlinear Analysis, 1996, Vol. 8, No. 1, 169--177
66. On a Matched Pair of Lie Groups for the $\kappa$-Poincar\'e in2-Dimensions, with A. Sitarz, Mathematical Physics Electronic Journal 1996, Vol. 2, No. 5, pp. 8
67. On dynamical System with an Integral Invariant on the Torus, Univ. Iag. Acta Math. , 1995, XXXII, 157--174

## Conference reports

1. How to show an existence of homoclinic trajectories using topological tools ? , with K. Wojcik , Equadiff'99 , 246-248, Editors: B. Fiedler, K. Gr\"oger, J. Sprekels. 2000, World Scientific, Singapore, New Jersey, London, Hong-Kong
2. Symbolic dynamics for the R\"ossler folded towel map,Conley Index Theory, Warszawa 1999, Banach Center Publulications Vol. 47,253--258
3. Chaos in the Lorenz equations for classical parameter values.A computer assisted proof , with Z. Galias Univ. Iag. Acta Math. , 1998, XXXVI,
4. Rigorous verification of chaos in the R\"ossler equations, Scientific Computing and Validated Numerics, G. Alefeld, A. Frommer and B. Lang editors, Akademie Verlag, Berlin, 1996, 287--292