108
M. Lucas et al.
series recordings for identification of such stability mechanisms in open systems,
building upon the initial work of [29, 30].
In conclusion, we have studied oscillatory systems that are explicitly driven by
time-variable external influences; such models aim at explicitly taking into account
the thermodynamic openness that is crucial for living systems to stay alive. Modelling
this openness deterministically and explicitly may be key to advancing our understanding of the underlying mechanisms at play in the stability of living systems, such
as the brain [6], and even beyond, e.g. in cosmology [36].
Acknowledgements We would like to thank Joe Rowland Adams for useful feedback on the
manuscript. We also would like to thank three anonymous referees for their useful comments which
have helped to improve the manuscript. This study has been supported by an EPSRC Doctoral
Prize Fellowship, the DFG grant CRC 701, the EPSRC grant EP/M006298/1, and the European
Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement No 642563.
References
1. D. Aeyels, Stability of nonautonomous systems by Liapunov’s direct method. Banach Cent.
Publ. 32(1), 9–17 (1995)
2. V. Anagnostopoulou, T. Jaeger, Nonautonomous saddle-node bifurcations: random and deterministic forcing. J. Differ. Equations 253(2), 379–399 (2012)
3. V.A. Antonov, Modeling of processes of cyclic evolution type. Synchronization by a random
signal. Vestn. Leningr. Univ Mat Mekh Astron. (vyp. 2), 67–76 (1984)
4. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)
5. P. Ashwin, S. Wieczorek, R. Vitolo, P. Cox, Tipping points in open systems: bifurcation, noiseinduced and rate-dependent examples in the climate system. Phil. Trans. R Soc. A 370(1962),
1166–1184 (2012)
6. J.D. Barrow, Conjecture about the general cosmological solution of Einstein’s equations. Phys.
Rev. D 102(2), 024017 (2020)
7. P. Boškoski, D. Iatsenko, G. Lancaster, S. McCormack, J. Newman, G.V. Policharla, V. Ticcinelli, T. Stankovski, A. Stefanovska, PyMODA v0.1.0 (2020)
8. M. Braˇ ciˇ c Lotriˇ c, A. Stefanovska, Synchronization and modulation in the human cardiorespiratory system. Phys. A 283(3–4), 451–461 (2000)
9. A. Carter, Classical and Statistical Thermodynamics (Prentice Hall, Raymond f. Boyer Library
Collection, 2001)
10. P.T. Clemson, A. Stefanovska, Discerning non-autonomous dynamics. Phys. Rep. 542(4), 297–
368 (2014)
11. B. de Saedeleer, M. Crucifix, S. Wieczorek, Is the astronomical forcing a reliable and unique
pacemaker for climate? A conceptual model study. Clim. Dyn. 40(1–2), 273–294 (2013)
12. M. Faggian, F. Ginelli, F. Rosas, Z. Levnaji´ c, Synchronization in time-varying random networks
with vanishing connectivity. Sci. Rep. 9(1), 10207 (2019)
13. P. Gandhi, E. Knobloch, C. Beaume, Dynamics of phase slips in systems with time-periodic
modulation. Phys. Rev. E 92(6), 062914 (2015)
14. C. Gardiner, Stochastic Methods, vol. 4 (Springer, Berlin, 2009)
15. P. Gaspard, Cycles, randomness, and transport from chaotic dynamics to stochastic processes.
Chaos 25(9), 097606 (2015)
16. M. Ghil, The wind-driven ocean circulation: applying dynamical systems theory to a climate
problem. Discrete Contin. Dyn. Syst. A 37(1), 189–228 (2017)
M. Lucas et al.
series recordings for identification of such stability mechanisms in open systems,
building upon the initial work of [29, 30].
In conclusion, we have studied oscillatory systems that are explicitly driven by
time-variable external influences; such models aim at explicitly taking into account
the thermodynamic openness that is crucial for living systems to stay alive. Modelling
this openness deterministically and explicitly may be key to advancing our understanding of the underlying mechanisms at play in the stability of living systems, such
as the brain [6], and even beyond, e.g. in cosmology [36].
Acknowledgements We would like to thank Joe Rowland Adams for useful feedback on the
manuscript. We also would like to thank three anonymous referees for their useful comments which
have helped to improve the manuscript. This study has been supported by an EPSRC Doctoral
Prize Fellowship, the DFG grant CRC 701, the EPSRC grant EP/M006298/1, and the European
Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
grant agreement No 642563.
References
1. D. Aeyels, Stability of nonautonomous systems by Liapunov’s direct method. Banach Cent.
Publ. 32(1), 9–17 (1995)
2. V. Anagnostopoulou, T. Jaeger, Nonautonomous saddle-node bifurcations: random and deterministic forcing. J. Differ. Equations 253(2), 379–399 (2012)
3. V.A. Antonov, Modeling of processes of cyclic evolution type. Synchronization by a random
signal. Vestn. Leningr. Univ Mat Mekh Astron. (vyp. 2), 67–76 (1984)
4. A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, C. Zhou, Synchronization in complex networks. Phys. Rep. 469(3), 93–153 (2008)
5. P. Ashwin, S. Wieczorek, R. Vitolo, P. Cox, Tipping points in open systems: bifurcation, noiseinduced and rate-dependent examples in the climate system. Phil. Trans. R Soc. A 370(1962),
1166–1184 (2012)
6. J.D. Barrow, Conjecture about the general cosmological solution of Einstein’s equations. Phys.
Rev. D 102(2), 024017 (2020)
7. P. Boškoski, D. Iatsenko, G. Lancaster, S. McCormack, J. Newman, G.V. Policharla, V. Ticcinelli, T. Stankovski, A. Stefanovska, PyMODA v0.1.0 (2020)
8. M. Braˇ ciˇ c Lotriˇ c, A. Stefanovska, Synchronization and modulation in the human cardiorespiratory system. Phys. A 283(3–4), 451–461 (2000)
9. A. Carter, Classical and Statistical Thermodynamics (Prentice Hall, Raymond f. Boyer Library
Collection, 2001)
10. P.T. Clemson, A. Stefanovska, Discerning non-autonomous dynamics. Phys. Rep. 542(4), 297–
368 (2014)
11. B. de Saedeleer, M. Crucifix, S. Wieczorek, Is the astronomical forcing a reliable and unique
pacemaker for climate? A conceptual model study. Clim. Dyn. 40(1–2), 273–294 (2013)
12. M. Faggian, F. Ginelli, F. Rosas, Z. Levnaji´ c, Synchronization in time-varying random networks
with vanishing connectivity. Sci. Rep. 9(1), 10207 (2019)
13. P. Gandhi, E. Knobloch, C. Beaume, Dynamics of phase slips in systems with time-periodic
modulation. Phys. Rev. E 92(6), 062914 (2015)
14. C. Gardiner, Stochastic Methods, vol. 4 (Springer, Berlin, 2009)
15. P. Gaspard, Cycles, randomness, and transport from chaotic dynamics to stochastic processes.
Chaos 25(9), 097606 (2015)
16. M. Ghil, The wind-driven ocean circulation: applying dynamical systems theory to a climate
problem. Discrete Contin. Dyn. Syst. A 37(1), 189–228 (2017)