Associate Professor
Overview
Cecilia is an associate professor in the School of Mathematics and Physics at the University of Queensland. After completing undergraduate studies at Universidad de Guanajuato / CIMAT and PhD at the University of Maryland, College Park, she held research fellowships from the Pacific Institute for the Mathematical Sciences (PIMS) and the Australian Research Council (ARC). She has also held a Promoting Women Fellowship by UQ.
Cecilia is an expert in the field of random dynamical systems (RDS). Along with collaborators, she has developed a framework for the study of transport in RDS, relying on the so-called Lyapunov–Oseledets spectrum. Her key contributions include the development of tools and algorithms to (i) approximate coherent structures and Lyapunov exponents, (ii) establish limit laws and quantify fluctuations, (iii) develop a thermodynamic formalism and (iv) optimise mixing. Her work also includes significant advances on data assimilation, metastable and dynamical systems.
Cecilia has received significant research funding from the Australian Research Council, including a 2016 Discovery Early Career Researcher Award (DECRA) a 2018 ARC DP and a 2022 ARC DP as lead CI. She has led or co-led competitive applications for conference funding (20-60 participants), including a 2023 MATRIX Workshop, co-funded by the MATRIX-Simons Collaborative Fund, an Australian Mathematical Sciences Institute (AMSI) funded Mathsfest Workshop (ANU, 2016), a Banff International Research Station Workshop (Canada, 2015) and a BIRS-CMO Workshop (Mexico, 2018).
Cecilia has delivered over a hundred invited lectures, seminars and colloquia in almost twenty countries, including invited/keynote addresses at the ANZIAM 2023 annual conference, 2014 International Workshop Set Oriented Numerics (U Canterbury, NZ), 2017 Workshop Ergodic Theory, Algorithms & Rigorous Computations (U Warwick, UK), 2017 EMALCA (Latin-American & Caribbean Math School, Mexico) and participation at invitation-only workshops at AIM (USA), BIRS (Canada), Bernoulli Center (Switzerland), CMO (Mexico), CIRM (France), Centro De Giorgi (Italy), Lorentz Center (Netherlands) and MATRIX (Australia).
Cecilia's service roles include: MATRIX Scientific Committee (2019-), Australian Mathematical Society council (2018-2021) and Queensland representative at the ANZIAM Executive Committee (2019-2021).
Research Interests
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Dynamical Systems and Ergodic Theory
Cecilia’s research interests include dynamical systems, ergodic theory and related areas. Her recent work focuses on random dynamical systems, transfer operators, Lyapunov exponents and coherent structures.
Publications
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Journal Article: Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles
Gonzalez-Tokman, Cecilia and Quas, Anthony (2021). Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles. Journal of the European Mathematical Society, 23 (10), 3419-3457. doi: 10.4171/jems/1096
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Journal Article: A spectral approach for quenched limit theorems for random expanding dynamical systems
Dragičević, D., Froyland, G., González-Tokman, C. and Vaienti, S. (2018). A spectral approach for quenched limit theorems for random expanding dynamical systems. Communications in Mathematical Physics, 360 (3), 1121-1187. doi: 10.1007/s00220-017-3083-7
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Journal Article: Optimal mixing enhancement by local perturbation
Froyland, Gary, Gonzalez-Tokman, Cecilia and Watson, Thomas M. (2016). Optimal mixing enhancement by local perturbation. SIAM Review, 58 (3), 494-513. doi: 10.1137/15M1023221
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Journal Article: Stochastic stability of Lyapunov exponents and Oseledets splittings for semi-invertible matrix cocycles
Froyland, Gary, Gonzalez-Tokman, Cecilia and Quas, Anthony (2015). Stochastic stability of Lyapunov exponents and Oseledets splittings for semi-invertible matrix cocycles. Communications on Pure and Applied Mathematics, 68 (11), 2052-2081. doi: 10.1002/cpa.21569
Grants
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What predictions can I trust? Stability of chaotic random dynamical systems
(2022–2025) ARC Discovery Projects
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(2018–2021) University of New South Wales
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Coherent structures in chaotic dynamical systems
(2016–2020) ARC Discovery Early Career Researcher Award
Supervision
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Global stability of chaotic random dynamical systems
Doctor Philosophy
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Random Dynamical Systems, Transfer Operators and Stability
Doctor Philosophy
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(2021) Doctor Philosophy
Available Projects
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Dynamical Systems and Ergodic Theory
Topics available for student projects at PhD/Masters/Honours level include:
(i) Non-autonomous or random dynamical systems. These systems model the evolution of phenomena affected by external influences, such as deterministic forcing or stationary noise. Topics under investigation include Lyapunov exponents, multiplicative ergodic theory, statistical behaviour and stability.
(ii) Theoretical and computational analysis of metastable and coherent structures in dynamical systems. Such structures encode important information of the long term evolution and transport phenomena in the underlying system. For example, they are useful to identify, analyse and quantify features of natural phenomena such as oceanic eddies and atmospheric vortices.
Publications
Featured Publications
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Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles
Gonzalez-Tokman, Cecilia and Quas, Anthony (2021). Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles. Journal of the European Mathematical Society, 23 (10), 3419-3457. doi: 10.4171/jems/1096
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A spectral approach for quenched limit theorems for random expanding dynamical systems
Dragičević, D., Froyland, G., González-Tokman, C. and Vaienti, S. (2018). A spectral approach for quenched limit theorems for random expanding dynamical systems. Communications in Mathematical Physics, 360 (3), 1121-1187. doi: 10.1007/s00220-017-3083-7
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Optimal mixing enhancement by local perturbation
Froyland, Gary, Gonzalez-Tokman, Cecilia and Watson, Thomas M. (2016). Optimal mixing enhancement by local perturbation. SIAM Review, 58 (3), 494-513. doi: 10.1137/15M1023221
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Froyland, Gary, Gonzalez-Tokman, Cecilia and Quas, Anthony (2015). Stochastic stability of Lyapunov exponents and Oseledets splittings for semi-invertible matrix cocycles. Communications on Pure and Applied Mathematics, 68 (11), 2052-2081. doi: 10.1002/cpa.21569
Book
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Mexican Mathematicians in the World: Trends and Recent Contributions
Galaz-García, Fernando, González-Tokman, Cecilia and Pardo Millán, Juan eds. (2021). Mexican Mathematicians in the World: Trends and Recent Contributions. Contemporary Mathematics, Providence, Rhode Island: American Mathematical Society. doi: 10.1090/conm/775
Book Chapter
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Galaz-García, Fernando, González-Tokman, Cecilia and Millán, Juan Carlos Pardo (2021). Preface. Mexican mathematicians in the world: trends and recent contributions. (pp. vii-viii) Providence, RI, United States: American Mathematical Society.
Journal Article
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Blachut, Chantelle, González-Tokman, Cecilia and Hernández-Dueñas, Gerardo (2023). A patch in time saves nine: methods for the identification of localised dynamical behaviour and lifespans of coherent structures. Journal of Nonlinear Science, 33 (4) 55. doi: 10.1007/s00332-023-09911-3
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Shifts in evolutionary balance of phenotypes under environmental changes
Kleshnina, Maria, McKerral, Jody C., González-Tokman, Cecilia, Filar, Jerzy A. and Mitchell, James G. (2022). Shifts in evolutionary balance of phenotypes under environmental changes. Royal Society Open Science, 9 (11) 220744, 1-17. doi: 10.1098/rsos.220744
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Equilibrium states for non-transitive random open and closed dynamical systems
Atnip, Jason, Froyland, Gary, Gonzalez-Tokman, Cecilia and Vaienti, Sandro (2022). Equilibrium states for non-transitive random open and closed dynamical systems. Ergodic Theory and Dynamical Systems, 43 (10) PII S0143385722000682, 3193-3215. doi: 10.1017/etds.2022.68
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Invariant measures for random expanding on average Saussol maps
Batayneh, Fawwaz and González-Tokman, Cecilia (2022). Invariant measures for random expanding on average Saussol maps. Stochastics and Dynamics, 22 (3) 2250015, 1-18. doi: 10.1142/S0219493722500150
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Lyapunov exponents for transfer operator cocycles of metastable maps: a quarantine approach
González-Tokman, C. and Quas, A. (2022). Lyapunov exponents for transfer operator cocycles of metastable maps: a quarantine approach. Transactions of the Moscow Mathematical Society, 82 (1), 65-76. doi: 10.1090/mosc/313
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On the number of invariant measures for random expanding maps in higher dimensions
Batayneh, Fawwaz and Gonzalez-Tokman, Cecilia (2021). On the number of invariant measures for random expanding maps in higher dimensions. Discrete and Continuous Dynamical Systems- Series A, 41 (12), 5887-5914. doi: 10.3934/dcds.2021100
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Thermodynamic Formalism for Random Weighted Covering Systems
Atnip, Jason, Froyland, Gary, González-Tokman, Cecilia and Vaienti, Sandro (2021). Thermodynamic Formalism for Random Weighted Covering Systems. Communications in Mathematical Physics, 386 (2), 819-902. doi: 10.1007/s00220-021-04156-1
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Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles
Gonzalez-Tokman, Cecilia and Quas, Anthony (2021). Stability and collapse of the Lyapunov spectrum for Perron-Frobenius operator cocycles. Journal of the European Mathematical Society, 23 (10), 3419-3457. doi: 10.4171/jems/1096
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A spectral approach for quenched limit theorems for random hyperbolic dynamical systems
Dragičević, D., Froyland, G., González-Tokman, C. and Vaienti, S. (2020). A spectral approach for quenched limit theorems for random hyperbolic dynamical systems. Transactions of the American Mathematical Society, 373 (1), 629-664. doi: 10.1090/tran/7943
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Blachut, Chantelle and González-Tokman, Cecilia (2020). A tale of two vortices: how numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems. Journal of Computational Dynamics, 7 (2), 369-399. doi: 10.3934/JCD.2020015
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Quenched stochastic stability for eventually expanding-on-average random interval map cocycles
Froyland, Gary, González-Tokman, Cecilia and Murray, R. U.A. (2019). Quenched stochastic stability for eventually expanding-on-average random interval map cocycles. Ergodic Theory and Dynamical Systems, 39 (10), 2769-2792. doi: 10.1017/etds.2017.143
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Hilbert space Lyapunov exponent stability
Froyland, Gary, Gonzalez-Tokman, Cecilia and Quas, Anthony (2019). Hilbert space Lyapunov exponent stability. Transactions of the American Mathematical Society, 372 (4), 2357-2388. doi: 10.1090/tran/7521
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A spectral approach for quenched limit theorems for random expanding dynamical systems
Dragičević, D., Froyland, G., González-Tokman, C. and Vaienti, S. (2018). A spectral approach for quenched limit theorems for random expanding dynamical systems. Communications in Mathematical Physics, 360 (3), 1121-1187. doi: 10.1007/s00220-017-3083-7
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Almost sure invariance principle for random piecewise expanding maps
Dragicevic, D., Froyland, G., Gonzalez-Tokman, C. and Vaienti, S. (2018). Almost sure invariance principle for random piecewise expanding maps. Nonlinearity, 31 (5), 2252-2280. doi: 10.1088/1361-6544/aaaf4b
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Optimal mixing enhancement by local perturbation
Froyland, Gary, Gonzalez-Tokman, Cecilia and Watson, Thomas M. (2016). Optimal mixing enhancement by local perturbation. SIAM Review, 58 (3), 494-513. doi: 10.1137/15M1023221
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Stability and approximation of invariant measures of Markov chains in random environments
Froyland, Gary and Gonzalez-Tokman, Cecilia (2016). Stability and approximation of invariant measures of Markov chains in random environments. Stochastics and Dynamics, 16 (1) 1650003, 1650003.1-1650003.23. doi: 10.1142/S0219493716500039
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A concise proof of the multiplicative ergodic theorem on banach spaces
González-Tokman, Cecilia and Quas, Anthony (2015). A concise proof of the multiplicative ergodic theorem on banach spaces. Journal of Modern Dynamics, 9 (01), 237-255. doi: 10.3934/jmd.2015.9.237
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Froyland, Gary, Gonzalez-Tokman, Cecilia and Quas, Anthony (2015). Stochastic stability of Lyapunov exponents and Oseledets splittings for semi-invertible matrix cocycles. Communications on Pure and Applied Mathematics, 68 (11), 2052-2081. doi: 10.1002/cpa.21569
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Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools
Froyland, Gary, Gonzalez-Tokman, Cecilia and Quas, Anthony (2014). Detecting isolated spectrum of transfer and Koopman operators with Fourier analytic tools. Journal of Computational Dynamics, 1 (2), 249-278. doi: 10.3934/jcd.2014.1.249
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A semi-invertible operator Oseledets theorem
Gonzalez-Tokman, Cecilia and Quas, Anthony (2014). A semi-invertible operator Oseledets theorem. Ergodic Theory and Dynamical Systems, 34 (4), 1230-1272. doi: 10.1017/etds.2012.189
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Stability and approximation of random invariant densities for Lasota-Yorke map cocycles
Froyland, Gary, Gonzalez-Tokman, Cecilia and Quas, Anthony (2014). Stability and approximation of random invariant densities for Lasota-Yorke map cocycles. Nonlinearity, 27 (4), 647-660. doi: 10.1088/0951-7715/27/4/647
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Ulam's method for Lasota-Yorke maps with holes
Bose, Christopher, Froyland, Gary, Gonzalez-Tokman, Cecilia and Murray, Rua (2014). Ulam's method for Lasota-Yorke maps with holes. SIAM Journal on Applied Dynamical Systems, 13 (2), 1010-1032. doi: 10.1137/130917533
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Ensemble data assimilation for hyperbolic systems
Gonzalez-Tokman, Cecilia and Hunt, Brian R. (2013). Ensemble data assimilation for hyperbolic systems. Physica D: Nonlinear Phenomena, 243 (1), 128-142. doi: 10.1016/j.physd.2012.10.005
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Approximating invariant densities for metastable systems
Gonzalez Tokman, Cecilia, Hunt, Brian R. and Wright, Paul (2011). Approximating invariant densities for metastable systems. Ergodic Theory and Dynamical Systems, 31 (5), 1345-1361. doi: 10.1017/S0143385710000337
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Scaling laws for bubbling bifurcations
Gonzalez-Tokman, Cecilia and Hunt, Brian R. (2009). Scaling laws for bubbling bifurcations. Nonlinearity, 22 (11), 2607-2631. doi: 10.1088/0951-7715/22/11/002
Conference Publication
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González-Tokman, Cecilia (2018). Multiplicative ergodic theorems for transfer operators: towards the identification and analysis of coherent structures in non-autonomous dynamical systems. Second Meeting Matemáticos Mexicanos en el Mundo [Second meeting of Mexican Mathematicians Abroad], Guanajuato, Mexico, 15-19 December 2014. Guanajuato, Mexico: American Mathematical Society. doi: 10.1090/conm/709/14290
Grants (Administered at UQ)
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What predictions can I trust? Stability of chaotic random dynamical systems
(2022–2025) ARC Discovery Projects
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(2018–2021) University of New South Wales
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Coherent structures in chaotic dynamical systems
(2016–2020) ARC Discovery Early Career Researcher Award
PhD and MPhil Supervision
Current Supervision
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Global stability of chaotic random dynamical systems
Doctor Philosophy — Principal Advisor
Other advisors:
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Random Dynamical Systems, Transfer Operators and Stability
Doctor Philosophy — Principal Advisor
Other advisors:
Completed Supervision
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(2021) Doctor Philosophy — Principal Advisor
Other advisors:
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(2021) Doctor Philosophy — Principal Advisor
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Synchronisation of Nanomechanical Oscillators
(2017) Doctor Philosophy — Principal Advisor
Other advisors:
Possible Research Projects
Note for students: The possible research projects listed on this page may not be comprehensive or up to date. Always feel free to contact the staff for more information, and also with your own research ideas.
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Dynamical Systems and Ergodic Theory
Topics available for student projects at PhD/Masters/Honours level include:
(i) Non-autonomous or random dynamical systems. These systems model the evolution of phenomena affected by external influences, such as deterministic forcing or stationary noise. Topics under investigation include Lyapunov exponents, multiplicative ergodic theory, statistical behaviour and stability.
(ii) Theoretical and computational analysis of metastable and coherent structures in dynamical systems. Such structures encode important information of the long term evolution and transport phenomena in the underlying system. For example, they are useful to identify, analyse and quantify features of natural phenomena such as oceanic eddies and atmospheric vortices.
