UC SPARK - University of Canterbury - New Zealand

Associate Professor Rua Murray

Mathematics and Statistics

Fields of Research

  • Dynamical systems and ergodic theory
  • Maximum entropy optimisation methods
  • Numerical effects in dynamics
  • Mathematical physiology

Researcher Summary

My core research is in the general area of dynamical systems. The subject is exciting because it is a meeting place for real applications, numerical computation and beautiful theorems. A lot of my work is on the probabilistic side of dynamical systems: this is called "ergodic theory", where one seeks to know about the statistical properties of complex systems. I have made significant contributions in "numerical ergodic theory": approximation of important dynamical objects on a computer, with confidence in the accuracy of the output, backed by mathematical rigour.   Since joining the University of Canterbury I have worked with collaborators on applications in ecology, energy, finance and physiology.

Subject Area: Disciplines

Research Groups


Research/Scholarly/Creative Works

Journal Articles
  • Froyland G., Gonzalez-Tokman C. and Murray RDA. (2018) Quenched stochastic stability for eventually expanding-on-average random interval map cocycles. Ergodic Theory and Dynamical Systems : 24 pages. http://dx.doi.org/10.1017/etds.2017.143.
  • Mohd MH., Murray R., Plank MJ. and Godsoe W. (2018) Effects of different dispersal patterns on the presence-absence of multiple species. Communications in Nonlinear Science and Numerical Simulation 56: 115-130. http://dx.doi.org/10.1016/j.cnsns.2017.07.029.
  • Davidson SM., Docherty PD. and Murray R. (2017) The dimensional reduction method for identification of parameters that trade-off due to similar model roles. Mathematical Biosciences 285: 119-127. http://dx.doi.org/10.1016/j.mbs.2017.01.003.
  • Mohd MH., Murray R., Plank MJ. and Godsoe W. (2017) Effects of biotic interactions and dispersal on the presence-absence of multiple species. Chaos, Solitons and Fractals 99: 185-194. http://dx.doi.org/10.1016/j.chaos.2017.04.012.
  • Gray RAL., Docherty PD., Fisk LM. and Murray R. (2016) A modified approach to objective surface generation within the Gauss-Newton parameter identification to ignore outlier data points. Biomedical Signal Processing and Control 30: 162-169. http://dx.doi.org/10.1016/j.bspc.2016.06.009.
  • Godsoe W., Murray R. and Plank MJ. (2015) Information on biotic interactions improves transferability of distribution models. American Naturalist 185(2): 281-290. http://dx.doi.org/10.1086/679440.
  • Godsoe W., Murray R. and Plank MJ. (2015) The effect of competition on species' distributions depends on coexistence, rather than scale alone. Ecography 38(11): 1071-1079. http://dx.doi.org/10.1111/ecog.01134.
  • Bose C. and Murray R. (2014) Maximum Entropy Estimates for Risk-Neutral Probability Measures with Non-Strictly-Convex Data. Journal of Optimisation Theory and Applications 161(1): 285-307. http://dx.doi.org/10.1007/s10957-013-0349-x.
  • Bose C., Froyland G., González-Tokman C. and Murray R. (2014) Ulam's Method for Lasota-Yorke Maps with Holes. SIAM Journal on Applied Dynamical Systems 13(2): 1010-1032. http://dx.doi.org/10.1137/130917533.
  • Bose CJ. and Murray R. (2007) Duality and the Computation of Approximate Invariant Densities for Nonsingular Transformations. SIAM Journal on Optimization 18(2): 691-709. http://dx.doi.org/10.1137/060658163.
  • Froyland G., Murray R. and Terhesiu D. (2007) Efficient computation of topological entropy, pressure, conformal measures, and equilibrium states in one dimension. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 76(3): 36702. http://dx.doi.org/10.1103/PhysRevE.76.036702.
  • Bose CJ. and Murray R. (2006) Dynamical conditions for convergence of a maximum entropy method for Frobenius-Perron operator equations. Applied Mathematics and Computation 182(1): 210-212. http://dx.doi.org/10.1016/j.amc.2006.01.089.
  • Bose CJ. and Murray R. (2006) Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete and Continuous Dynamical Systems 14(3): 597-615.
  • Bose C. and Murray R. (2001) The exact rate of approximation in Ulam's method. Discrete and Continuous Dynamical Systems 7(1): 219-235.
  • Santoboni G., Murray R. and Bishop S. (2000) Roundoff-induced phenomena and diffusion processes: the ``premature'' synchronisation of coupled maps. Physics Letters A 271 271(5-6): 358-367. http://dx.doi.org/10.1016/S0375-9601(00)00381-9.
  • Bose C. and Murray R. (2014) Numerical approximation of conditionally invariant measures via Maximum Entropy. In Bahsoun W; Bose C; Froyland G (Ed.), Ergodic theory, open dynamics, and coherent structures. Springer Proceedings in Mathematics & Statistics, Vol 70: 81-104. New York: Springer-Verlag. http://dx.doi.org/10.1007/978-1-4939-0419-8_5.
Conference Contributions - Published
  • Davidson SM., Docherty PD., Kretschmer J. and Murray R. (2017) The Novel Dimensional Reduction Method and Tikhonov Regularisation in Parameter Identification of Non-Linear Ill-Posed Problems. In IFAC-PapersOnLine 50(1): 5474-5479. http://dx.doi.org/10.1016/j.ifacol.2017.08.1085.
  • Fee CJ., Newberry F., Gordon A., Wilson PL., Moyers-Gonzalez M., Murray R., Huber T. and Dimartino S. (2016) 3D-printed chromatography columns for downstream capture of proteins in the presence of suspended cells. Phuket, Thailand: 2nd Bioprocess Asia Conference, 5-8 Dec 2016.
  • Bose C. and Murray R. (2014) Chapter 5 Numerical Approximation of Conditionally Invariant Measures via Maximum Entropy. In Springer Proceedings in Mathematics and Statistics 70: 81-104. http://dx.doi.org/10.1007/978-1-4939-0419-8__5.
  • Santoboni G., Murray R. and Bishop S. (2001) A simple use of the diffusion approximation for treating roundoff-induced problems in coupled maps with an invariant subset. Pamplona, Spain: Space-time chaos, 1 Jan 2000. : 223-232.
  • Murray R. (1997) Invariant measures and stochastic discretisations of dynamical systems. Berlin Germany: 15th IMACS World Congress on Scientific Computation, 1 Aug 1997.
Conference Contributions - Other
  • Fee CJ., Newberry F., Gordon A., Wilson P., Moyers-Gonzales M., Murray R., Huber T. and Dimartino S. (2016) Optimization of porous bed geometric features to maximize adsorption of proteins and passage of suspended solids in a 3D-printed adsorption column. Philadelphia, PA, USA: PREP Symposium 2016, 18-21 Jul 2016
  • Goodman M., David T., Docherty PD. and Murray R. (2016) Calcium Dynamics in Coupled Cellular Reaction Diffusion Equations. Queenstown, New Zealand: 34th Australasian Winter Conference on Brain Research, 27-31 Aug 2016
  • Goodman M., David T., Docherty PD. and Murray R. (2016) Homogenisation Theory Applied to Coupled Mammalian Cells. Christchurch, New Zealand: Health Research Society of Canterbury: Christchurch Hospitals' Grand Rounds series 2016, 27-27 May 2016
  • Goodman M., Murray R., Docherty PD. and David T. (2016) Homogenisation Theory with Coupled Cellular Reaction Diffusion Equations. Boston, MA< USA: SIAM Conference on the Life Sciences 2016, 11-14 Jul 2016
  • Asuncion J., Krumdieck S., Rendall S., Page S. and Murray R. (2013) Geographic Energy Adaptive Potential of Farmers' Market System as Compared with Conventional Supermarket System. Washington DC, USA: Transportation Research Board 92nd Annual Meeting, 13-17 Jan 2013
  • Asuncion J., Page S., Murray R. and Krumdieck S. (2012) Analysis of the truck trip generation characteristics of supermarkets and convenience stores. Rotorua, New Zealand: IPENZ Transportation Group Conference, 18-21 Mar 2012
  • Asuncion J., Rendall S., Murray R. and Krumdieck S. (2012) New Zealand intermodal freight network and the potential for mode shifting. Rotorua, New Zealand: IPENZ Transportation Group Conference, 18-21 Mar 2012
Reports Other
  • Ackerley E. and Murray R. (2013) Progress on Improving Student Success in First-Year Mathematics at the University of Canterbury. The CULMS Newsletter 8: 7-10. [Print and Online].
  • Murray R. (2006) Geon Ho Choe, Computational Ergodic Theory. Algorithms and Computation in Mathematics 13, 2005; 453pp. Newsletter of the New Zealand Mathematical Society 96: 28-29. Wellington: New Zealand Mathematical Society Inc.. http://ifs.massey.ac.nz/outreach/mathnews/NZMS96/news96.pdf. [Book Review].