Professor John A G Roberts

Office: Room 3065, Red Centre
Phone: +61 2 9385 7052
Email:: Jag.Roberts@unsw.edu.au
Mail:

Photo credit: The Imagination Agency

Deputy Head, School of Mathematics and Statistics, UNSW

Chief Investigator of the ARC Centre of Excellence MASCOS

Vice-President, Australian Mathematical Society

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Research Interests and Selected Papers

Nonlinear dynamical systems, with particular interests in the following topics (and their interrelation):

Integrable dynamical systems

G.R.W. Quispel, J.A.G. Roberts and C.J. Thompson, Integrable mappings and soliton equations II, Physica D 34 (1989) 183-192.
A. Iatrou and J.A.G. Roberts, Integrable mappings of the plane preserving biquadratic invariant curves II, Nonlinearity 15 (2002) 459-489. PDF version
J.A.G. Roberts, A. Iatrou and G.R.W. Quispel, Interchanging parameters and integrals in dynamical systems: the mapping case. J. Phys. A: Math. Gen. 35 (2002) 2309-2325. PDF version
J.A.G. Roberts and F. Vivaldi, Arithmetical method to detect integrability in maps, Phys. Rev. Lett. 90 (2003) 034102. PDF version
G.R.W. Quispel, H.W. Capel and J.A.G. Roberts, Duality for discrete integrable systems, J. Phys. A: Math. Gen. 38 (2005) 3965-3980. PDF version
D. Jogia, J.A.G. Roberts and F. Vivaldi, An algebraic geometric approach to integrable maps of the plane, J. Phys. A: Math. Gen.39 (2006) 1133-1149. PDF version
J.A.G. Roberts and G.R.W. Quispel, Creating and relating 3-dimensional integrable maps, J. Phys. A: Math. Gen. 39 (2006) L605-L615. PDF version
J. Pettigrew and J.A.G. Roberts, Characterizing singular curves in parametrized families of biquadratics, J. Phys. A: Math. Theor. 41 (2008) 115203. PDF version

Symmetry and Time-reversal symmetry

J.A.G. Roberts and G.R.W. Quispel, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems, Phys. Rep. 216 (1992) 63-177.
J.A.G. Roberts and J.S.W. Lamb, Self-similarity of period-doubling branching in 3D reversible mappings, Physica D 82 (1995) 317-332.PDF version
J.S.W. Lamb and J.A.G. Roberts, Time-reversal symmetry in dynamical systems: a survey, Physica D 112 (1998) 1-39.PDF version
M.Baake and J.A.G. Roberts, Symmetries and reversing symmetries of polynomial automorphisms of the plane, Nonlinearity 18 (2005) 791-816. PDF version
J.A.G. Roberts and F. Vivaldi, Signature of time-reversal symmetry in polynomial automorphisms over finite fields, Nonlinearity 18 (2005) 2171-2192. PDF version
M.Baake and J.A.G. Roberts, The structure of reversing symmetry groups, Bull. Austral. Math. Soc. 73 (2006) 445-459. PDF version

Dynamical systems arising from quasiperiodic physical phenomena

J.A.G. Roberts and M. Baake, Trace maps as 3D reversible dynamical systems with an invariant, J. Stat. Phys. 74 (1994) 829-888. PDF version
J.A.G. Roberts, Escaping orbits in trace maps, Physica A 228 (1996) 295-325. PDF version
M. Baake and J.A.G. Roberts, Reversing symmetry group of GL(2,Z) and PGL(2,Z) matrices with connections to cat maps and trace maps, J. Phys. A: Math. Gen. 30 (1997) 1549-1573. PDF version
M.Baake and J.A.G. Roberts, Symmetries and reversing symmetries of toral automorphisms, Nonlinearity 14 (2001) R1-24. PDF version

Algebraic dynamics

J. Pettigrew, J.A.G. Roberts and F. Vivaldi, Complexity of regular invertible p-adic motions, Chaos 11 (2001) 849-857. PDF version
J.A.G. Roberts and F. Vivaldi, Arithmetical method to detect integrability in maps, Phys. Rev. Lett. 90 (2003) 034102. PDF version
J.A.G. Roberts, D. Jogia and F. Vivaldi, The Hasse-Weil bound and integrability detection in rational maps, J. Nonlinear Math. Phys. 10 (2003) 166-180. PDF version
J.A.G. Roberts and F. Vivaldi, Signature of time-reversal symmetry in polynomial automorphisms over finite fields, Nonlinearity 18 (2005) 2171-2192. PDF version
M.Baake, J.A.G. Roberts and A. Weiss, Periodic orbits of linear endomorphisms on the 2-torus and its lattices, Nonlinearity 21 (2008) 2427-2446. PDF version
J.A.G. Roberts and F. Vivaldi, A combinatorial model for reversible rational maps over finite fields, Nonlinearity 22 (2009) 1965-1982. PDF version
J.A.G. Roberts, Order and symmetry in birational difference equations and their signatures over finite phase spaces, Proceedings of the Workshop Future Directions in Difference Equations, Conference Collection 69, Univ. Vigo, Serv. Publ., Vigo, (2011) pp. 213-221. PDF version
N.Neumaerker, J.A.G. Roberts and F. Vivaldi, Distribution of periodic orbits for the Casati-Prosen map on rational lattices, Physica D 241 (2012) 360-371. PDF version
M.Baake, N.Neumaerker and J.A.G. Roberts, Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices, Discrete and Continuous Dynamical Systems 33 (2013) 527-553. PDF version

Research Associates

Current and recent research associates:

Dinh Tran [integrable systems], 2016-17
Alina Ostafe [arithmetic dynamics] [UNSW VC Postdoc], 10/2013-
Dinh Tran [integrable systems], 2011-13
Natascha Neumaerker [arithmetic dynamics], 2009-10
Danesh Jogia [integrable systems], 2008

Graduate Students

I very much enjoy supervising (good!) postgraduate students - formulating ideas and projects and seeing them through to fruition in papers and theses. Please feel free to contact me if your research interests match mine.

Current and recent students:

Tim Siu, Arithmetic dynamics and integrable systems, PhD, 2014-
Natascha Neumaerker, The arithmetic structure of discrete dynamical systems on the torus, PhD, 2008-12 (PhD granted by University of Bielefeld; co-supervision with Prof M Baake)
Jim Pettigrew, Characterising singular curves in parametrised families of biquadratics, PhD, 2004-9
Danesh Jogia, Algebraic aspects of integrability and reversibility in maps, PhD, 2003-7
Rahmi Rusin, A Study of Limit Cycles and Quadratic Systems, MSc (minor thesis), 2002-3.
Apostolos Iatrou, Integrable Mappings of the Plane preserving biquadratic invariant curves, PhD (La Trobe University), 2000-3.
Jim Pettigrew, Invertible p-adic dynamics of polynomial maps, MSc (La Trobe University), 1999-2001.

Selected Recent Publications/Talks

J.A.G. Roberts and D.T. Tran, Towards some exact results for the (vanishing) algebraic entropy of (integrable) lattice equations, preprint (2014) PDF version
J.A.G. Roberts and D.T. Tran, Signatures over finite fields of growth properties for lattice equations, J. Phys. A: Math. Theor. 48 (2015) 085201 (20pp) PDF version
J.A.G. Roberts and D.Jogia, Birational maps that send biquadratic curves to biquadratic curves, J. Phys. A: Math. Theor. 48 (2015) 08FT02 (13 pp) PDF version
J.A.G. Roberts and F. Vivaldi, Arithmetic exponents in piecewise-affine planar maps, Physica D 298-299 (2015) 1-12. PDF version
F.M. Mahomed and J.A.G. Roberts, Characterization of Hamiltonian symmetries and their first integrals, Int. J. Non-Linear Mechanics 74 (2015) 84-91. PDF version
M.Baake, J.A.G. Roberts and R. Yassawi, Reversing and extended symmetries of shift spaces, preprint (2016) PDF version
J.A.G. Roberts and D.T. Tran, Algebraic entropy of (integrable) lattice equations and their reductions, preprint (2017) PDF version
D.T. Tran and J.A.G. Roberts Linear degree growth in lattice equations, preprint (2017) PDF version