Publications

• D.J.W. Simpson and V. Avrutin.
  From two-dimensional continuous maps to one-dimensional discontinuous maps: a novel reduction explaining complex bifurcation structures in piecewise-linear families of maps.
  Submitted to Nonlinearity. arXiv
• I. Ghosh and D.J.W. Simpson.
  The VIVID Function for Numerically Continuing Periodic Orbits arising from Grazing Bifurcations of Hybrid Dynamical Systems.
  Submitted to J. Comput. Dyn. arXiv
• H. Tang, A. Champneys, and D.J.W. Simpson.
  Boundary Equilibrium Bifurcations Creating Multiple Limit Cycles in Impacting Hybrid Systems.
  SIAM J. Appl. Dyn. Syst., 24(4):2734-2766, 2025. arXiv
• P.A. Glendinning and D.J.W. Simpson.
  Extended Normal Forms for One-Dimensional Border-Collision Bifurcations.
  Nonlinearity, 38(10):105012, 2025. arXiv
• I. Ghosh and D.J.W. Simpson.
  Robust chaos in Rⁿ.
  Nonlinearity, 38(9):095013, 2025. arXiv
• D.J.W. Simpson.
  A Piecewise-Linear Fixed Point Theorem.
  Amer. Math. Month., 132(7):695-699, 2025. arXiv
• D.J.W. Simpson
  How to Compute Multi-Dimensional Stable and Unstable Manifolds of Piecewise-Linear Maps. In: Elaydi, S. and Gardini, L. and Tikjha, W. (eds). New Developments in Discrete Dynamical Systems, Difference Equations, and Applications, ICDEA 2023.
  Springer Proceedings in Mathematics & Statistics, 485:1-14, 2025. arXiv
• D.J.W. Simpson.
  Three Forms of Dimension Reduction for Border-Collision Bifurcations.
  Phys. Lett. A, 550:130603, 2025. arXiv
• D.J.W. Simpson.
  The Two-Dimensional Border-Collision Normal Form with a Zero Determinant.
  SIAM J. Appl. Dyn. Syst., 24(3): 2205-2245, 2025. arXiv
• D.J.W. Simpson.
  Nonsmooth Folds as Tipping Points.
  Chaos, 35(2):023125, 2025. arXiv
• I. Ghosh, R.I. McLachlan and D.J.W. Simpson.
  Robust Chaos in Orientation-Reversing and Non-Invertible Two-Dimensional Piecewise-Linear Maps.
  J. Nonlin. Sci., 35:16, 2025. arXiv
• I. Ghosh, R.I. McLachlan and D.J.W. Simpson.
  The Bifurcation Structure within Robust Chaos for Two-Dimensional Piecewise-Linear Maps.
  Commun. Nonlin. Sci. Numer. Simul., 134: 108025, 2024. arXiv
• D.J.W. Simpson.
  The Necessity of the Sausage-String Structure for Mode-Locking Regions of Piecewise-Linear Maps.
  Phys. D, 462: 134142, 2024. arXiv
• D.J.W. Simpson and P.A. Glendinning.
  Inclusion of Higher-Order Terms in the Border-Collision Normal Form: Persistence of Chaos and Applications to Power Converters.
  Phys. D, 462: 134131, 2024. arXiv
• P.A. Glendinning and D.J.W. Simpson.
  Differentiable Conjugacies for One-Dimensional Maps. In: Olaru, S., Cushing, J., Elaydi, S., Lozi, R. (eds). Difference Equations, Discrete Dynamical Systems and Applications, ICDEA 2022.
  Springer Proceedings in Mathematics & Statistics, 444:115-130, 2024. arXiv
• D.J.W. Simpson.
  Border-Collision Bifurcations from Stable Fixed Points to Any Number of Coexisting Chaotic Attractors.
  J. Difference Eq. Appl., 30(1): 90-110, 2024. arXiv
• D.J.W. Simpson.
  Detecting Invariant Expanding Cones for Generating Word Sets to Identify Chaos in Piecewise-Linear Maps.
  J. Difference Eq. Appl., 29:1094-1126, 2023. arXiv
• P.A. Glendinning and D.J.W. Simpson.
  Unstable Dimension Variability and Heterodimensional Cycles in the Border-Collision Normal Form.
  Phys. Rev. E., 108(2):L022202, 2023. arXiv
• H.O. Fatoyinbo and D.J.W. Simpson.
  A Synopsis of the Non-Invertible, Two-Dimensional, Border-Collision Normal Form with Applications to Power Converters.
  Int. J. Bifurcation Chaos, 33(8):2330019, 2023. arXiv
• P.A. Glendinning and D.J.W. Simpson.
  Normal Forms, Differentiable Conjugacies and Elementary Bifurcations of Maps.
  SIAM J. Appl. Math., 83(2):816-836, 2023. arXiv
• I. Belykh, R. Kuske, M. Porfini and D.J.W. Simpson.
  Beyond the Bristol Book: Advances and Perspectives in Non-Smooth Dynamics and Applications. (an editorial)
  Chaos, 33(1):010402, 2023. preprint
  
• P.A. Glendinning and D.J.W. Simpson.
  Normal Forms for Saddle-Node Bifurcations: Takens’ Coefficient and Applications in Climate Models.
  Proc. R. Soc. A, 478:20220548, 2022. arXiv
• P.A. Glendinning and D.J.W. Simpson.
  Chaos in the Border-Collision Normal Form: A Computer-Assisted Proof Using Induced Maps and Invariant Expanding Cones.
  Appl. Math. Comput., 434:127357, 2022. arXiv
• I. Ghosh and D.J.W. Simpson.
  Renormalisation of the Two-Dimensional Border-Collision Normal Form.
  Int. J. Bifurcation Chaos, 32(12):2250181, 2022. arXiv
• H.O. Fatoyinbo, R.G. Brown, D.J.W. Simpson and B. van Brunt.
  Pattern Formation in a Spatially-Extended Model of Pacemaker Dynamics in Smooth Muscle Cells.
  Bull. Math. Biol., 84(8):86, 2022. arXiv
• D.J.W. Simpson.
  Dimension Reduction for Slow-Fast, Piecewise-Linear ODEs and Obstacles to a General Theory.
  Phys. D, 439:133368, 2022. arXiv
• S.S. Muni, R.I. McLachlan and D.J.W. Simpson.
  Unfolding Globally Resonant Homoclinic Tangencies.
  Discrete Contin. Dyn. Syst., 42(8):4013-4030, 2022. arXiv
• D.J.W. Simpson.
  Twenty Hopf-Like Bifurcations in Piecewise-Smooth Dynamical Systems.
  Phys. Rep., 970:1-80, 2022. arXiv
• I. Ghosh and D.J.W. Simpson.
  Robust Devaney Chaos in the Two-Dimensional Border-Collision Normal Form.
  Chaos, 32:043120, 2022. arXiv
• D.J.W. Simpson.
  On the Stability of Boundary Equilibria in Filippov Systems.
  Commun. Pure Appl. Anal., 20(9):3093-3111, 2021. arXiv
• S.S. Muni, R.I. McLachlan and D.J.W. Simpson.
  Homoclinic Tangencies with Infinitely Many Asymptotically Stable Single-Round Periodic Solutions.
  Discrete Contin. Dyn. Syst., 41(8):3629-3650, 2021. arXiv
• P.A. Glendinning and D.J.W. Simpson.
  A Constructive Approach to Robust Chaos using Invariant Manifolds and Expanding Cones.
  Discrete Contin. Dyn. Syst., 41(7):3367-3387, 2021. arXiv
• P.A. Glendinning and D.J.W. Simpson.
  Robust Chaos and the Continuity of Attractors.
  Trans. Math. Appl., 4(1):tnaa002, 2020. arXiv
• D.J.W. Simpson.
  Chaotic Attractors from Border-Collision Bifurcations: Stable Border Fixed Points and Determinant-Based Lyapunov Exponent Bounds.
  NZJM, 50:71-91, 2020. arXiv
• D.J.W. Simpson, V. Avrutin and S. Banerjee.
  The Nordmark Map and the Problem of Large-Amplitude Chaos in Impact Oscillators.
  Phys. Rev. E, 102:022211, 2020.
• D.J.W. Simpson.
  The Stability of Fixed Points on Switching Manifolds of Piecewise-Smooth Continuous Maps.
  J. Dyn. Diff. Equat., 32(3):1527-1552, 2020. arXiv
• H.O. Fatoyinbo, R.G. Brown, D.J.W. Simpson and B. van Brunt.
  Numerical Bifurcation Analysis of Pacemaker Dynamics in a Model of Smooth Muscle Cells.
  Bull. Math. Biol., 82(7):95, 2020. arXiv
• D.J.W. Simpson.
  Unfolding Codimension-Two Subsumed Homoclinic Connections in Two-Dimensional Piecewise-Linear Maps.
  Int. J. Bifurcation Chaos, 30(3):2030006, 2020. arXiv
• D.J.W. Simpson.
  Hopf-Like Boundary Equilibrium Bifurcations involving Two Foci in Filippov Systems.
  J. Diff. Eq., 267(11):6133-6151, 2019. arXiv
• M.E. Roberts, C. Kueh, E. Greenbank, D. Clarke, S. van Hove, D.J.W. Simpson, A. Williams and J. Williams.
  Modelling the Mechanical Action of a Front Loading Washing Machine. Proceedings of the 2017 Mathematics and Statistics in Industry NZ Study Group, MINZ-2017.
  ANZIAM J., 59: M30-M62, 2019.
  
• H.A. Al Fran, D.J.W. Simpson and C.P. Tuffley.
  Characterisation and Classification of Signatures of Spanning Trees of the n-Cube.
  Australas. J. Combin., 75(3):259-295, 2019. arXiv
• D.J.W. Simpson.
  A General Framework for Boundary Equilibrium Bifurcations of Filippov Systems.
  Chaos, 28(10):103114, 2018. arXiv
• D.J.W. Simpson.
  A Compendium of Hopf-Like Bifurcations in Piecewise-Smooth Dynamical Systems.
  Phys. Lett. A, 382(35):2439-2444, 2018. arXiv
• M.R. Jeffrey, G. Kafanas and D.J.W. Simpson.
  Jitter in Dynamical Systems with Intersecting Discontinuity Surfaces.
  Int. J. Bifurcation Chaos, 28(6):1830020, 2018. arXiv
• D.J.W. Simpson.
  The Structure of Mode-Locking Regions of Piecewise-Linear Continuous Maps: II. Skew Sawtooth Maps.
  Nonlinearity, 31(5):1905-1939, 2018. arXiv
• D.J.W. Simpson and R. Kuske.
  The Influence of Localised Randomness on Regular Grazing Bifurcations with Applications to Impacting Dynamics.
  J. Vib. Contr., 24(2):407-426, 2018. arXiv
• D.J.W. Simpson.
  Grazing-Sliding Bifurcations Creating Infinitely Many Attractors.
  Int. J. Bifurcation Chaos, 27(12):1730042, 2017. arXiv
• D.J.W. Simpson.
  Open Problems on Border-Collision Bifurcations. In: Colombo A., Jeffrey M., Lázaro J., Olm J. (eds). Proceedings from the workshop: Advances in Nonsmooth Dynamics, CRM, Barcelona, Feb-Apr, 2016.
  Extended Abstracts Spring 2016. 8:163-166, 2017.
  
• D.J.W. Simpson and C.P. Tuffley.
  Subsumed Homoclinic Connections and Infinitely Many Coexisting Attractors in Piecewise-Linear Maps.
  Int. J. Bifurcation Chaos, 27(2):1730010, 2017. arXiv
• D.J.W. Simpson.
  The Structure of Mode-Locking Regions of Piecewise-Linear Continuous Maps: I. Nearby Mode-Locking Regions and Shrinking Points.
  Nonlinearity, 30(1):382-444, 2017. arXiv
• D.J.W. Simpson.
  The Instantaneous Local Transition of a Stable Equilibrium to a Chaotic Attractor in Piecewise-Smooth Systems of Differential Equations.
  Phys. Lett. A, 380(38):3067-3072, 2016. arXiv
• D.J.W. Simpson.
  Unfolding Homoclinic Connections formed by Corner Intersections in Piecewise-Smooth Maps.
  Chaos, 26:073105, 2016. arXiv
• D.J.W. Simpson.
  Border-Collision Bifurcations in R^N.
  SIAM Rev., 58(2):177-226, 2016. preprint
• D.J.W. Simpson and M.R. Jeffrey.
  Fast Phase Randomisation via Two-Folds.
  Proc. R. Soc. A, 472(2186):20150782, 2016. (open access)
• D.J.W. Simpson and R. Kuske.
  Stochastic Perturbations of Periodic Orbits with Sliding.
  J. Nonlin. Sci., 25(4):967-1014, 2015. arXiv
• D.J.W. Simpson and R. Kuske.
  The Positive Occupation Time of Brownian Motion with Two-Valued Drift and Asymptotic Dynamics of Sliding Motion with Noise.
  Stoch. Dyn., 14(4):1450010, 2014. arXiv
• D.J.W. Simpson and R. Kuske.
  Stochastically Perturbed Sliding Motion in Piecewise-Smooth Systems.
  Discrete Contin. Dyn. Syst. Ser. B, 19(9):2889-2913, 2014. arXiv
• D.J.W. Simpson.
  On the Relative Coexistence of Fixed Points and Period-Two Solutions near Border-Collision Bifurcations.
  Appl. Math. Lett., 38:162-167, 2014. arXiv
• D.J.W. Simpson.
  Scaling Laws for Large Numbers of Coexisting Attracting Periodic Solutions in the Border-Collision Normal Form.
  Int. J. Bifurcation Chaos, 24(9):1450118, 2014. arXiv
• D.J.W. Simpson.
  Sequences of Periodic Solutions and Infinitely Many Coexisting Attractors in the Border-Collision Normal Form.
  Int. J. Bifurcation Chaos, 24(6):1430018, 2014. arXiv
• M.R. Jeffrey and D.J.W. Simpson.
  Non-Filippov Dynamics Arising from the Smoothing of Nonsmooth Systems, and its Robustness to Noise.
  Nonlinear Dyn., 76(2):1395-1410, 2014. arXiv
• D.J.W. Simpson.
  On Resolving Singularities of Piecewise-Smooth Discontinuous Vector Fields via Small Perturbations.
  Discrete Contin. Dyn. Syst., 34(9):3803-3830, 2014. arXiv
• D.J.W. Simpson, S.J. Hogan and R. Kuske.
  Stochastic Regular Grazing Bifurcations.
  SIAM J. Appl. Dyn. Syst., 12(2), 533-559, 2013. arXiv
• D.J.W. Simpson and J.D. Meiss.
  Aspects of Bifurcation Theory for Piecewise-Smooth, Continuous Systems.
  Phys. D, 241(22):1861-1868, 2012. arXiv
• D.J.W. Simpson, R. Kuske and Y.-X. Li.
  Dynamics of Simple Balancing Models with Time-Delayed Switching Feedback Control.
  J. Nonlin. Sci., 22(2):135-167, 2012. arXiv
• D.J.W. Simpson and R. Kuske.
  Mixed-Mode Oscillations in a Stochastic Piecewise-Linear System.
  Phys. D, 240:1189-1198, 2011. arXiv
• D.J.W. Simpson.
  Bifurcations in Piecewise-Smooth Continuous Systems.
  World Scientific, Singapore, 2010.
  
• D.J.W. Simpson and J.D. Meiss.
  Resonance near Border-Collision Bifurcations in Piecewise-Smooth, Continuous Maps.
  Nonlinearity, 23(12):3091-3118, 2010. arXiv
• D.J.W. Simpson and J.D. Meiss.
  Simultaneous Border-Collision and Period-Doubling Bifurcations.
  Chaos, 19(3):033146, 2009. arXiv
• D.J.W. Simpson and J.D. Meiss.
  Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps.
  Nonlinearity, 22(5):1123-1144, 2009. arXiv
• D.J.W. Simpson, D.S. Kompala and J.D. Meiss.
  Discontinuity Induced Bifurcations in a Model of Saccharomyces cerevisiae.
  Math. Biosci., 218(1):40-49, 2009. arXiv
• D.J.W. Simpson and J.D. Meiss.
  Unfolding a Codimension-Two Discontinuous Andronov-Hopf Bifurcation.
  Chaos, 18(3):033125, 2008. arXiv
• D.J.W. Simpson and J.D. Meiss.
  Neimark-Sacker Bifurcations in Planar, Piecewise-Smooth, Continuous Maps.
  SIAM J. Appl. Dyn. Syst., 7(3):795-824, 2008.
• B. Marts, D.J.W. Simpson, A. Hagberg and A.L. Lin.
  Period Doubling in a Periodically Forced Belousov-Zhabotinsky Reaction.
  Phys. Rev. E, 76(2):026213, 2007.
• D.J.W. Simpson and J.D. Meiss.
  Andronov-Hopf Bifurcations in Planar, Piecewise-Smooth, Continuous Flows.
  Phys. Lett. A, 371(3):213-220, 2007. arXiv
• D.J.W. Simpson, V. Kirk and J. Sneyd.
  Complex Oscillations and Waves of Calcium in Pancreatic Acinar Cells.
  Phys. D, 200:303-324, 2005.