Chris Bernhardt
 

Publications

  • Turing's Vision: The birth of computer science. MIT Press, May 2016
  • Powers of positive matrices, Mathematics Magazine. 91 (2018), no. 3, 218 - 227
  • Vertex maps on graphs -- Peron-Frobenius theory, Journal of Difference Equations and Applications, 22 (2016) No. 1, 55 - 66.
  • C. Bernhardt, P. C. Staecker, Generalizing the rotation interval to vertex maps on graphs, Trans. Amer. Math. Soc. 367 (2015), no. 6, 4235-4252. (Preprint: http://arxiv.org/abs/1209.4996)
  •  C. Bernhardt, Z. Gaslowitz, A. Johnson, W. Radil, Periods of orbits for maps on graphs     homotopic to the identity map, Journal of Difference Equations and Applications,  Vol.19, No.5 (2013), 758-771. (Preprint: arxiv.org/pdf/1108.2899.pdf )

  • Rotation matrices for vertex maps on graphs, Journal of Difference Equations and Applications, Vol. 18, No. 6, June 2012, 1033 - 1041. pdf
  •  Sharkovsky's theorem and maps on trees, Proceedings of the 14th International Conference on Difference Equations and Applications, (2009) 149 - 154
  • A Sharkovsky theorem for vertex maps on trees, Journal of Difference Equations and Applications, Vol. 17, No. 1, 103 - 113, January 2011 pdf
  • Evil twins alternate with odious twins, Math. Mag. 82 (2009), 57 - 62. pdf
  • A proof of the Cayley Hamilton Theorem, Math. Monthly, Vol. 116, No. 5, (2009), 456 - 457. pdf
  • Zero-entropy vertex maps on graphs, Journal of Difference Equations and Applications, Vol. 15, No. 1, 17 - 22, January 2009.
  •  The dynamics of non-hyperbolic toral automorphisms, J. Dyn. Syst. Geom. Theor. 5 (2007), no. 2, 153 - 164 (Joint work with student Sebastian Mineo.)
  •  Vertex maps for trees: algebra and periods of periodic orbits, Discrete Contin. Dyn. Syst, Vol. 14, No.3, pp. 399-408, March 2006.
  • A simple construction of a faithful, irreducible representation of $S_n$, J. Algebra Appl. 5 (2006), no. 2. 125-128
  • A discrete version of Sharkovsky's Theorem, Journal of Difference Equations and Applications, Vol. 10, No. 1, pp. 29-40, Jan 2004
  • Time-symmetric cycles, International Journal of Mathematics and Mathematical Sciences, Vol 2003, 26, pp 1683-1693, 2003
  • Unimodal time-symmetric dynamics, International Journal of Mathematics and Mathematical Sciences, Vol 2003, 14, pp 869-878, 2003
  • A proof of Sharkovsky's theorem, Journal of Difference Equations and Applications, Vol 9, 3-4, pp 373-379, 2003
  • Some Dynamical Properties Associated to the Difference Operator, College Math. J. 28 (1997), no. 1 20-26. (with T. Yuster).
  • On the Order of n-Modal Cycles, Trans. Amer. Math. Soc. 348 (1996) no. 9, 3827-3834.
  • Self-Similarity Maps for the Set of Unimodal Cycles, International Journal of Bifurcation and Chaos Appl. Sci Engrg.,  Vol. 5, 5, 1325-1330, 1995. (Re-printed in Thirty Years after Sharkovskii's Theorem: new perspectives (Murcia, 1994), World Sci. Ser. Nonlinear Sci. Ser. B Spec. Theme Issues Proc., 8, 1995.)
  • On the Existence of Fundamental Representatives in Maps of the Interval, Real Analysis Exchange, Vol. 19, 1993/1994, 608-611 (with E. Coven).
  • Evidence of Chaos in the S&P 500 Cash Index, Recent Advances in Futures and Options Research, JAI Press, Vol. 6, 1993, 179-192 (with R. Eldridge and I. Mulvey). (Reprinted in Chaos Theory and Nonlinear Dynamics, Irwin Professional Publishing.)
  • Comparing Periodic Orbits of Maps of the Interval, Trans. Amer. Math. Soc. Vol. 333, 2, 1992, 701-707 (with E. Coven, M. Misiurewicz and I. Mulvey).
  • A Polynomial Time Algorithm for the Forcing Relation on Cyclic Permutations, Contemporary Math, Vol. 135, 1992, 85-93 (with E. Coven).
  • The ordering on permutations induced by continuous maps of the  real line, Ergodic Theory and Dynamical Systems (1987), 7, 155-160
  • Simple permutations with a power of two, Ergodic Theory and Dynamical Systems (1984), 4, 179-186.
  • Periodic points and topological entropy of maps of the circle, Proceedings American Mathematical Society (2), 87, (1983), 516-518.
  • Rotation intervals of a class of endomorphisms of the circle, Proceedings London Mathematical Society  (1982), (3), 45, 258-280.
  • Periodic points of a class of endomorphisms of the circle, Journal London Mathematical Society, (2), 25 (1982), 539-550.
  • Periodic orbits of continuous mappings of the circle without fixed points, Ergodic Theory and Dynamical Systems (1981),1, 413-417.