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Research interests
- Dynamical systems and ergodic theory
- Maximum entropy optimisation methods
- Numerical effects in dynamics
- Mathematical physiology
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.
Publications
Preprints
- Christoper Bose & Rua Murray. First hyperbolic times for intermittent maps with unbounded derivative.
Preprint on arXiv.To appear in Dynamical systems: an international journal (2014)
- Christopher Bose, Cecilia Gonzalez-Tokman, Gary Froyland & Rua
Murray. Ulam's method for Lasota-Yorke maps with
holes. Preprint on
arXiv. To appear in SIAM Journal on Applied Dynamical Systems (2014)
- Christopher Bose & Rua Murray. Numerical approximation of conditionally invariant measures via Maximum
Entropy. Preprint on arXiv. To
appear in edited book: Ergodic theory, open dynamics, and coherent structures. Springer Proceedings in Mathematics & Statistics. Volume 70. ISBN 978-1-4939-0418-1
Papers
- Christopher Bose & Rua Murray. Polynomial decay of correlations in the
generalized baker's
transformation. Preprint on
arXiv. International Journal of Bifurcations and Chaos 23(8) (2013) 1350130
DOI: http://dx.doi.org/10.1142/S0218127413501307.
- Christopher Bose & Rua Murray. Maximum entropy estimates for
risk-neutral probability measures with non-strictly-convex
data. Journal of Optimisation Theory and Applications (2013) DOI 10.1007/s10957-013-0349-x Preprint version.
- Gary Froyland, Rua Murray & Ognjen Stancevic. Spectral degeneracy and
escape dynamics for intermittent maps with a hole. Nonlinearity 24 (2011) 2435-2463. http://dx.doi.org/10.1088/0951-7715/24/9/003.doi. Preprint: http://arxiv.org/abs/1012.2149
- Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete and continuous dynamical systems. 26(3):1007-1018 (2010). Preprint (302 kB PDF)
- Gary Froyland, Rua Murray & Dalia Terhesiu. Efficient computation of topological entropy, pressure, conformal measures, and equilibrium states in one dimension. Phys. Rev. E 76 (2007) 036702
- Christopher J. Bose & Rua Murray. Duality and the computation of approximately invariant measures for non-singular transformations. SIAM J. Optimization 18(2):691-709 (2007) Preprint (327kB PDF)
- Mirela Domijan, Rua Murray & James Sneyd. Dynamical probing of the mechanisms underlying calcium oscillations. J. Nonlinear Science. 16(5):483-506 (2006).
- Christopher J. Bose & Rua Murray. Dynamical conditions for convergence of a maximum entropy method for Frobenius-Perron operator equations. Appl. Math. Comp. 182(1):210-212 (2006).
- Christopher J. Bose & Rua Murray. Minimum 'energy' approximations to invariant measures for nonsingular transformations.Discrete and Continuous Dynamical Systems, 14(3):597-615 (2006).
- Rua Murray. Optimal partition choice for invariant measure approximation for one-dimensional maps. Nonlinearity 17(5):1623-1644 (2004).
- Giovanni Santoboni, Rua Murray & Steven Bishop. A simple use of the diffusion approximation for treating roundoff-induced problems in coupled maps with an invariant subset. Space-time chaos (Pamplona, 2000), 223--232, World Sci. Publishing, River Edge, NJ. (2001)
- Christopher J. Bose & Rua Murray. The exact rate of approximation in Ulam's method. Discrete and Continuous Dynamical Systems, 7(1):219-235 (2001).
- Giovanni Santoboni, Rua Murray & Steven Bishop. Round-off induced phenomena and diffusion processes: the 'premature' synchronisation of coupled maps. Phys. Lett. A, 271:358-367 (2000).
- Rua Murray. Existence, mixing and approximation of invariant densities for expanding maps on R r. Nonlinear Analysis TMA, 45(1):37-72 (2001). Preprint (362Kb PDF).
- Rua Murray. Approximation error for invariant density calculations. Discrete & Continuous Dynamical Systems, 4(3):535-557 (1998).
- M Keane, R Murray & L-S Young. Computing invariant measures for expanding circle maps. Nonlinearity, 11:27-46 (1998). Preprint version (331Kb PDF)
- R Murray. Invariant measures and stochastic discretisations of dynamical systems. Proceedings of 15th IMACS World Congress, Berlin, 1997. (144Kb PDF)
Other
- Rua Murray. Approximation of invariant measures for a class of maps with indifferent fixed points. University of Waikato, Mathematics Research Report Series II No. 106. (2005)
- Christopher J Bose & Rua Murray. Variational methods for the approximation of invariant measures. University of Waikato, Mathematics Research Report Series II No. 100. (2004)
- Rua Murray. Discrete approximation of invariant densities (2.6Mb PDF). PhD Thesis. University of Cambridge, August 1997.
- Rua Murray, Approximation error for invariant density calculations. Statistical Laboratory Research Report 1997-10 (1997).
- R D A Murray. Approximation and concrete error bounds for ACIM density
approximation for C 1+α expanding mappings.Statistical
Laboratory Research Report 1996-22 (1996).
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