Professor John Roberts

Professor John Roberts

Professor
Science
School of Mathematics & Statistics

ABOUT ME

 

Biography

My research is in dynamical systems (sometimes popularly termed "Chaos Theory"), which seeks to understand how systems change with time and how this evolution can be understood, classified and predicted. This area of research is an exciting interdisciplinary field which relates to, and uses ideas from pure and applied mathematics, physics and computer science. My current work focuses on two broad areas: the study of integrable systems (ordered dynamics based on rotations) and the study of arithmetic dynamics (a hybrid of dynamical systems with number theory). I work with my students, research associates and national and international collaborators.

Education

  • PhD in Applied Mathematics (University of Melbourne)
  • BSc Hons in Applied Mathematics (ANU)

 

RESEARCH

 

My Research Goals

  • Creating, characterising and understanding discrete integrable systems
  • Studying symmetry and time-reversal symmetry in dynamical systems
  • Studying arithmetic dynamical systems (dynamics over discrete spaces and number fields)

My Research in Detail

My research centres around two main (sometimes overlapping) topics in dynamical systems:

Integrable dynamics means “special”, loosely-speaking, and may refer to one or more “special” properties that the dynamics possesses, for example: ability to solve the dynamics; possession of enough constants of the motion or conservations laws; possession of so-called Lax pairs; slow growth of observables under the iterated dynamics; dynamics satisfying a consistency relation or having a lot of symmetry. These properties are non-generic in general dynamical systems. So systems that present with them, integrable systems, have been found to be rich sources of mathematical and physical beauty.

Integrable ordinary differential equations have been studied intensively since the time of Kepler and Newton, integrable partial differential equations since the discovery of solitons in the Korteweg-de Vries equation by Kruskal and collaborators. The question thus arises how to generalize this theory and its applications from the continuous case (integrable differential equations) to the discrete case (integrable difference equations), i.e. when all independent variables are discrete. Discrete integrable systems are currently the focus of many international research efforts, including my own.

Arithmetic dynamics is an exciting breakthrough science, merging various areas of mathematics (dynamics, number theory, algebra, geometry). This research area is rapidly expanding, has enormous potential for cross-fertilization, and is rich in important applications (cellular automata, pseudo-randomness and cryptography). Whereas the dynamics of maps over the continuum (of the real or complex numbers) has received enormous attention, the study of iterated maps over finite sets is relatively new and offers distinct challenges because some strictures from the continuum do not apply. Together with international collaborators, my research has used orbit statistics as a key to characterising and understanding the dynamics of birational maps. We have found universal distributions in the orbit statistics, related them to dynamical structure of the map, and then built probabilistic and combinatorial models to reproduce these distributions.

Current Student Projects (PhD and Honours)

I am very happy to supervise Honours projects in the broad area of dynamical systems. Recent projects include:

  • (Piecewise) Linear dynamics restricted to lattices
  • Discrete dynamical systems with non-constant parameters

Likewise, I would love to talk to (good!) prospective PhD students interested in discrete integrable systems and/or arithmetic dynamics. Thesis titles of my recent students include:

  • The arithmetic structure of discrete dynamical systems on the torus
  • Characterising singular curves in parametrised families of biquadratics
  • Algebraic aspects of integrability and reversibility in maps

Advice for prospective students

Dynamical systems is a wonderful area to work in. It affords an opportunity to do computer experimentation, look for patterns and test conjectures and, ultimately, to prove theorems that explain what you observe. Systems evolving with time are ubiquitous, from celestial mechanics [the historical origin of dynamical systems] to the stock market to mathematical biology to cryptosystems. Research in mathematics is becoming increasingly a cross-pollination of ideas and techniques from various mathematical subdisciplines, and nowhere is this more evident than in dynamical systems! Please contact me if you wish to know more.

 

TEACHING & OUTREACH

 

Courses I teach

MATH3201 Dynamical Systems and Chaos

Professional affiliations and service positions

Current: Member, EDI Committee, Australian Mathematical Society

Former: Deputy Head of School (2012-2020); School Research Director (2007-2020); Vice-President, Australian Mathematical Society (2016-2017); Chief Investigator of the ARC Centre of Excellence MASCOS

 

AWARDS & ACHIEVEMENTS

 

2012 Foreign Sabbatical Fellowship, Spanish Ministry of Science (Universitat Autonoma Barcelona)

2009 Isaac Newton Institute Visiting Fellow (Cambridge University)

2006 IAS Visiting Fellow (La Trobe University)

2006 AMSI/MASCOS Research Fellow

2004 UNSW John Yu Fellow

Phone
9385 7052
Location
School of Mathematics and Statistics University of New South Wales Sydney NSW 2052 The Red Centre Room 3065
  • Book Chapters | 2021
    Baake M; Coons M; Grimm U; Roberts JAG; Yassawi R, 2021, 'Aperiodic Order Meets Number Theory: Origin and Structure of the Field', in 2019-20 MATRIX Annals, Springer International Publishing, pp. 663 - 667, http://dx.doi.org/10.1007/978-3-030-62497-2_40
    Book Chapters | 1997
    Roberts JAG, 1997, 'Some Characterisations of Low-dimensional Dynamical Systems with Time-reversal Symmetry', in Control and Chaos, Birkhäuser Boston, pp. 106 - 133, http://dx.doi.org/10.1007/978-1-4612-2446-4_7
    Book Chapters | 1991
    Roberts JAG; Post T; Capel HW; Quispel GRW, 1991, 'Conservative Versus Reversible Dynamical Systems', in Solitons and Chaos, Springer Berlin Heidelberg, pp. 218 - 226, http://dx.doi.org/10.1007/978-3-642-84570-3_27
  • Journal articles | 2024
    Roberts JAG; Saito A; Vivaldi F, 2024, 'Critical curves of rotations', Indagationes Mathematicae, http://dx.doi.org/10.1016/j.indag.2024.02.001
    Journal articles | 2021
    Roberts JAG; Saito A; Vivaldi F, 2021, 'Critical curves of a piecewise linear map', Chaos, 31, pp. 073134, http://dx.doi.org/10.1063/5.0054334
    Journal articles | 2019
    Roberts JAG; Tran DT, 2019, 'Algebraic entropy of (integrable) lattice equations and their reductions', Nonlinearity, 32, pp. 622 - 653, http://dx.doi.org/10.1088/1361-6544/aaecda
    Journal articles | 2019
    Tran DT; Roberts JAG, 2019, 'Linear degree growth in lattice equations', Journal of Computational Dynamics, 6, pp. 449 - 467, http://dx.doi.org/10.3934/jcd.2019023
    Journal articles | 2018
    Baake M; Roberts JAG; Yassawi R, 2018, 'Reversing and extended symmetries of shift spaces', Discrete and Continuous Dynamical Systems- Series A, 38, pp. 835 - 866, http://dx.doi.org/10.3934/dcds.2018036
    Journal articles | 2018
    Evripidou CA; Quispel GRW; Roberts JAG, 2018, 'Poisson structures for difference equations', Journal of Physics A: Mathematical and Theoretical, 51, http://dx.doi.org/10.1088/1751-8121/aae746
    Journal articles | 2015
    Mahomed FM; Roberts JAG, 2015, 'Characterization of Hamiltonian symmetries and their first integrals', International Journal of Non-Linear Mechanics, 74, pp. 84 - 91, http://dx.doi.org/10.1016/j.ijnonlinmec.2015.04.005
    Journal articles | 2015
    Roberts JAG; Jogia D, 2015, 'Birational maps that send biquadratic curves to biquadratic curves', Journal of Physics A: Mathematical and Theoretical, 48, pp. 08ft02, http://dx.doi.org/10.1088/1751-8113/48/8/08FT02
    Journal articles | 2015
    Roberts JAG; Tran DT, 2015, 'Signatures over finite fields of growth properties for lattice equations', , http://dx.doi.org/10.1088/1751-8113/48/8/085201
    Journal articles | 2015
    Roberts JAG; Vivaldi F, 2015, 'Arithmetic exponents in piecewise-affine planar maps', , http://dx.doi.org/10.1016/j.physd.2015.01.007
    Journal articles | 2013
    Baake M; Neumarker N; Roberts JA, 2013, 'Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices', Discrete and Continuous Dynamical Systems, 33, pp. 527 - 553, http://dx.doi.org/10.3934/dcds.2013.33.527
    Journal articles | 2012
    Neumarker N; Roberts JA; Vivaldi F, 2012, 'Distribution of periodic orbits for the Casati-Prosen map on rational lattices', Physica D - Nonlinear Phenomena, 241, pp. 360 - 371, http://dx.doi.org/10.1016/j.physd.2011.10.006
    Journal articles | 2012
    Ormerod JT; Quispel GR; Roberts JA, 2012, 'Book Review- Discrete Integrable Systems: QRT Mappings and Elliptic Surfaces', Discrete Integrable Systems: QRT Mappings and Elliptic Surfaces
    Journal articles | 2009
    Roberts JA; Vivaldi F, 2009, 'A combinatorial model for reversible rational maps over finite fields', Nonlinearity, 22, pp. 1965 - 1982
    Journal articles | 2008
    Baake M; Roberts JA; Weiss A, 2008, 'Periodic orbits of linear endomorphisms on the 2-torus and its lattices', Nonlinearity, 21, pp. 2427 - 2446
    Journal articles | 2008
    Pettigrew JA; Roberts JA, 2008, 'Characterizing singular curves in parametrized families of biquadratics', JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 41, pp. 115203-1 - 115203-28
    Journal articles | 2006
    Baake M; Roberts JA, 2006, 'The structure of reversing symmetry groups', Bulletin of the Australian Mathematical Society, 73, pp. 445 - 459
    Journal articles | 2006
    Jogia DM; Roberts JA; Vivaldi F, 2006, 'An algebraic geometric approach to integrable maps of the plane', Journal of Physics A - Mathematical and General, 39, pp. 1133 - 1149
    Journal articles | 2006
    Roberts JA; Quispel GR, 2006, 'Creating and relating three-dimensional integrable maps', Journal of Physics A - Mathematical and General, 39, pp. L605 - L615
    Journal articles | 2005
    Baake M; Roberts JA, 2005, 'Symmetries and reversing symmetries of polynomial automorphisms of the plane', Nonlinearity, 18, pp. 791 - 816
    Journal articles | 2005
    Quispel GR; Capel HW; Roberts JA, 2005, 'Duality for discrete integrable systems', Journal of Physics A - Mathematical and General, 38, pp. 3965 - 3980
    Journal articles | 2005
    Roberts JA; Vivaldi F, 2005, 'Signature of time-reversal symmetry in polynomial automorphisms over finite fields', Nonlinearity, 18, pp. 2171 - 2192
    Journal articles | 2003
    Iatrou A; Roberts JA, 2003, 'Integrable mappings of the plane preserving biquadratic invariant curves III', Physica A - Statistical Mechanics and Its Applications, 326, pp. 400 - 411
    Journal articles | 2003
    Roberts JA; Baake M, 2003, 'Symmetries and reversing symmetries of area-preserving polynomial mappings in generalised standard form', Physica A - Statistical Mechanics and Its Applications, 317, pp. 95 - 112
    Journal articles | 2003
    Roberts JA; Jogia DM; Vivaldi F, 2003, 'The Hasse-Weil bound and integrability detection in rational maps', Journal of Nonlinear Mathematical Physics, 10, pp. 166 - 180
    Journal articles | 2003
    Roberts JA; Vivaldi F, 2003, 'Arithmetical method to detect integrability in maps', Physical Review Letters, 90
    Journal articles | 2003
    Roberts JAG; Jogia D; Vivaldi F, 2003, 'The Hasse-Weil Bound and Integrability Detection in Rational Maps', Journal of Nonlinear Mathematical Physics, 10, pp. 166 - 166, http://dx.doi.org/10.2991/jnmp.2003.10.s2.15
    Journal articles | 2003
    Roberts JAG; Vivaldi F, 2003, 'Erratum: Arithmetical method to detect integrability in maps (Phys. Rev. Lett. (2003) 90 (034102))', Physical Review Letters, 90
    Journal articles | 2003
    Roberts JAG; Vivaldi F, 2003, 'Publisher’s Note: Arithmetical Method to Detect Integrability in Maps [Phys. Rev. Lett. 90, 034102 (2003)]', Physical Review Letters, 90, http://dx.doi.org/10.1103/PhysRevLett.90.079902
    Journal articles | 2002
    Iatrou A; Roberts JA, 2002, 'Integrable mappings of the plane preserving biquadratic invariant curves II', Nonlinearity, 15, pp. 459 - 489
    Journal articles | 2002
    Iatrou A; Roberts JAG, 2002, 'Integrable mappings of the plane preserving biquadratic invariant curves II (vol 15, pg 459, 2002)', NONLINEARITY, 15, pp. 955 - 955, https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:000175697700028&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=891bb5ab6ba270e68a29b250adbe88d1
    Journal articles | 2002
    Roberts JA; Iatrou A; Quispel GR, 2002, 'Interchanging parameters and integrals in dynamical systems: the mapping case', Journal of Physics A - Mathematical and General, 35, pp. 2309 - 2325
    Journal articles | 2001
    Baake M; Roberts JAG, 2001, 'Symmetries and reversing symmetries of toral automorphisms', Nonlinearity, 14, http://dx.doi.org/10.1088/0951-7715/14/4/201
    Journal articles | 2001
    Iatrou A; Roberts JAG, 2001, 'Integrable mappings of the plane preserving biquadratic invariant curves', Journal of Physics A: Mathematical and General, 34, pp. 6617 - 6636, http://dx.doi.org/10.1088/0305-4470/34/34/308
    Journal articles | 2001
    Pettigrew JA; Roberts JA; Vivaldi F, 2001, 'Complexity of regular invertible p-adic motions', Chaos, 11, pp. 849 - 857
    Journal articles | 1998
    Lamb JSW; Roberts JAG, 1998, 'Time-reversal symmetry in dynamical systems: A survey', Physica D: Nonlinear Phenomena, 112, pp. 1 - 39, http://dx.doi.org/10.1016/S0167-2789(97)00199-1
    Journal articles | 1997
    Baake M; Roberts JAG, 1997, 'Reversing symmetry group of Gl(2, ℤ) and PGl(2, ℤ) matrices with connections to cat maps and trace maps', Journal of Physics A: Mathematical and General, 30, pp. 1549 - 1573, http://dx.doi.org/10.1088/0305-4470/30/5/020
    Journal articles | 1996
    Roberts JAG, 1996, 'Escaping orbits in trace maps', Physica A: Statistical Mechanics and its Applications, 228, pp. 295 - 325, http://dx.doi.org/10.1016/0378-4371(95)00428-9
    Journal articles | 1995
    Roberts JAG; Lamb JSW, 1995, 'Self-similarity of period-doubling branching in 3-D reversible mappings', Physica D: Nonlinear Phenomena, 82, pp. 317 - 332, http://dx.doi.org/10.1016/0167-2789(94)00229-J
    Journal articles | 1994
    Roberts JAG; Baake M, 1994, 'Trace maps as 3D reversible dynamical systems with an invariant', Journal of Statistical Physics, 74, pp. 829 - 888, http://dx.doi.org/10.1007/BF02188581
    Journal articles | 1993
    Lamb JSW; Roberts JAG; Capel HW, 1993, 'Conditions for local (reversing) symmetries in dynamical systems', Physica A: Statistical Mechanics and its Applications, 197, pp. 379 - 422, http://dx.doi.org/10.1016/0378-4371(93)90592-R
    Journal articles | 1992
    Roberts JAG; Capel HW, 1992, 'Area preserving mappings that are not reversible', Physics Letters A, 162, pp. 243 - 248, http://dx.doi.org/10.1016/0375-9601(92)90441-N
    Journal articles | 1992
    Roberts JAG; Quispel GRW, 1992, 'Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems', Physics Reports, 216, pp. 63 - 177, http://dx.doi.org/10.1016/0370-1573(92)90163-T
    Journal articles | 1990
    Roberts JAG, 1990, 'Order and chaos in reversible dynamical systems', Bulletin of the Australian Mathematical Society, 42, pp. 527 - 528, http://dx.doi.org/10.1017/s0004972700028707
    Journal articles | 1989
    Quispel GRW; Roberts JAG; Thompson CJ, 1989, 'Integrable mappings and soliton equations II', Physica D: Nonlinear Phenomena, 34, pp. 183 - 192, http://dx.doi.org/10.1016/0167-2789(89)90233-9
    Journal articles | 1989
    Quispel GRW; Roberts JAG, 1989, 'Conservative and dissipative behaviour in reversible dynamical systems', Physics Letters A, 135, pp. 337 - 342, http://dx.doi.org/10.1016/0375-9601(89)90004-2
    Journal articles | 1988
    Quispel GRW; Roberts JAG; Thompson CJ, 1988, 'Integrable mappings and soliton equations', Physics Letters A, 126, pp. 419 - 421, http://dx.doi.org/10.1016/0375-9601(88)90803-1
    Journal articles | 1988
    Quispel GRW; Roberts JAG, 1988, 'Reversible mappings of the plane', Physics Letters A, 132, pp. 161 - 163, http://dx.doi.org/10.1016/0375-9601(88)90274-5
    Journal articles | 1988
    Roberts JAG; Thompson CJ, 1988, 'Dynamics of the classical heisenberg spin chain', Journal of Physics A: Mathematical and General, 21, pp. 1769 - 1780, http://dx.doi.org/10.1088/0305-4470/21/8/013
  • Working Papers | 2021
    Roberts JAG; Saito A; Vivaldi F, 2021, Critical curves of a piecewise linear map, http://dx.doi.org, http://dx.doi.org/10.1063/5.0054334
    Working Papers | 2017
    Roberts JAG; Tran DT, 2017, Algebraic entropy of (integrable) lattice equations and their reductions, http://dx.doi.org, http://web.maths.unsw.edu.au/~jagr/RT17a.pdf
    Working Papers | 2017
    Tran DT; Roberts JAG, 2017, Linear degree growth in lattice equations, http://dx.doi.org, https://arxiv.org/abs/1702.08295
    Working Papers | 2017
    2017, Reversing and extended symmetries of shift spaces, http://dx.doi.org10.3934/dcds.2018036
  • Preprints | 2023
    Roberts JAG; Saito A; Vivaldi F, 2023, Critical curves of rotations, , http://arxiv.org/abs/2303.17955v1
    Preprints | 2021
    Roberts JAG; Saito A; Vivaldi F, 2021, Critical curves of a piecewise linear map, , http://dx.doi.org/10.48550/arxiv.2104.07197
    Preprints | 2018
    Evripidou CA; Quispel GRW; Roberts JAG, 2018, Poisson structures for difference equations, , http://dx.doi.org/10.48550/arxiv.1806.07233
    Preprints | 2014
    Roberts JAG; Vivaldi F, 2014, Growth of heights in piecewise-affine planar maps, , http://dx.doi.org/10.48550/arxiv.1408.4264
    Preprints | 2012
    Baake M; Neumaerker N; Roberts JAG, 2012, Orbit structure and (reversing) symmetries of toral endomorphisms on rational lattices, , http://dx.doi.org/10.48550/arxiv.1205.1003
    Conference Papers | 2011
    Roberts JA, 2011, 'Order and symmetry in birational difference equations and their signatures over finite phase spaces'
    Preprints | 2009
    Roberts JAG; Vivaldi F, 2009, A combinatorial model for reversible rational maps over finite fields, , http://dx.doi.org/10.48550/arxiv.0905.4135
    Preprints | 2008
    Baake M; Roberts JAG; Weiss A, 2008, Periodic orbits of linear endomorphisms on the 2-torus and its lattices, , http://dx.doi.org/10.48550/arxiv.0808.3489
    Preprints | 2006
    Baake M; Roberts JAG, 2006, The structure of reversing symmetry groups, , http://dx.doi.org/10.48550/arxiv.math/0605296
    Preprints | 2005
    Baake M; Roberts JAG, 2005, Symmetries and reversing symmetries of polynomial automorphisms of the plane, , http://dx.doi.org/10.48550/arxiv.math/0501151
    Preprints | 2002
    Roberts JAG; Baake M, 2002, Symmetries and reversing symmetries of area-preserving polynomial mappings in generalised standard form, , http://dx.doi.org/10.48550/arxiv.math/0206096
    Preprints | 2000
    Baake M; Roberts JAG, 2000, Symmetries and reversing symmetries of toral automorphisms, , http://dx.doi.org/10.48550/arxiv.math/0006092
    Preprints | 1999
    Baake M; Roberts JAG, 1999, Symmetries and reversing symmetries of trace maps, , http://arxiv.org/abs/math/9901124v1
    Conference Papers | 1995
    Wagner H; Baake M; Roberts JAG, 1995, 'Trace maps and dynamical systems', in Janot C; Mosseri R (ed.), PROCEEDINGS OF THE 5TH INTERNATIONAL CONFERENCE ON QUASICRYSTALS, WORLD SCIENTIFIC PUBL CO PTE LTD, FRANCE, AVIGNON, pp. 104 - 107, presented at 5th International Conference on Quasicrystals, FRANCE, AVIGNON, 22 May 1995 - 26 May 1995, https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1995BH70Q00022&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=891bb5ab6ba270e68a29b250adbe88d1
    Conference Papers | 1993
    ROBERTS JAG; BAAKE M, 1993, 'THE DYNAMICS OF TRACE MAPS', in Seimenis J (ed.), HAMILTONIAN MECHANICS, PLENUM PRESS DIV PLENUM PUBLISHING CORP, POLAND, TORUN, pp. 275 - 285, presented at NATO Advanced Research Workshop on Hamiltonian Mechanics - Integrability and Chaotic behaviour, POLAND, TORUN, 28 June 1993 - 02 July 1993, https://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&KeyUT=WOS:A1994BC43P00026&DestLinkType=FullRecord&DestApp=ALL_WOS&UsrCustomerID=891bb5ab6ba270e68a29b250adbe88d1