10 Useful Transformations from Non-autonomous to Autonomous Systems
173
10.4 Conclusions
This chapter presented two alternative transformations from non-autonomous systems to autonomous in the special case of periodic forcing. The alternative transformations enabled the use of theorems derived for native autonomous systems which
otherwise, under the standard transformation, could not be utilized. The transformations presented are inspired by mathematical modeling which also suggests that
transformations for other special cases exist.
References
1. R. Albert, H.G. Othmer, The topology of the regulatory interactions predicts the expression
pattern of the segment polarity genes in drosophila melanogaster. J. Theor. Biol. 223(1), 1–18
(2003)
2. A. Ben-Tal, A study of symmetric forced oscillators. Ph. D. thesis, University of Auckland,
New Zealand (2001)
3. A. Ben-Tal, Simplified models for gas exchange in the human lungs. J. Theor. Biol. 238(2),
474–95 (2006)
4. A. Ben-Tal, I.G. Kevrekidis, Coarse-graining and simplification of the dynamics seen in bursting neurons. SIAM J. Appl. Dyn. Syst. 15(2), 1193–1226 (2016)
5. A. Ben-Tal, S.S. Shamailov, J.F.R. Paton, Central regulation of heart rate and the appearance
of respiratory sinus arrhythmia: new insights from mathematical modeling. Math. Biosci. 255,
71–82 (2014)
6. A. Ben-Tal, Y.J. Wang, M.C.A. Leite, The logic behind neural control of breathing pattern. Sci.
Rep. 9 (2019)
7. R.J. Butera, J. Rinzel, J.C. Smith, Models of respiratory rhythm generation in the pre-bötzinger
complex. I. bursting pacemaker neurons. J. Neurophys. 81, 382–397 (1999)
8. R.J. Butera, J. Rinzel, J.C. Smith, Models of respiratory rhythm generation in the pre-bötzinger
complex. II. populations of coupled pacemaker neurons. J. Neurophys. 81, 398–415 (1999)
9. D. Capecchi, S.R. Bishop, Periodic oscillations and attracting basins for a parametrically excited
pendulum. Dyn. Stab. Syst. 9(2), 123–143 (1994)
10. M.J. Clifford, S.R. Bishop, Inverted oscillations of a driven pendulum. Proc. R. Soc. A-Math.
Phys. Eng. Sci. 454(1979), 2811–2817 (1998)
11. J.H. Comroe, Physiology of Respiration, 2nd edn. (Year Book Medical Publishers, Inc., 1977)
12. C.A. Del Negro, G.D. Funk, J.L. Feldman, Breathing matters. Nat. Rev. Neurosci. 19(6), 351–
367 (2018)
13. L. Edelstein-Keshet, Mathematical models in biology. Soc. Ind. Appl. Math. (2005)
14. M. Elstad, E.L. O’Callaghan, A.J. Smith, A. Ben-Tal, R. Ramchandra, Cardiorespiratory interactions in humans and animals: rhythms for life. Am. J. Phys. Heart Circ. Phys. 315(1), H6–H17
(2018)
15. J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of
vector fields. Applied Mathematical Sciences, vol. 42, (Springer-Verlag, New York, 1996)
16. D. Jordan, P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists
and Engineers. (Oxford University Press, 2007)
17. S. Lenci, E. Pavlovskaia, G. Rega, M. Wiercigroch, Rotating solutions and stability of parametric pendulum by perturbation method. J. Sound Vib. 310(1–2), 243–259 (2008)
18. B.G. Lindsey, I.A. Rybak, J.C. Smith, Computational models and emergent properties of respiratory neural networks. Compr. Phys. 2, 1619–1670 (2012)
173
10.4 Conclusions
This chapter presented two alternative transformations from non-autonomous systems to autonomous in the special case of periodic forcing. The alternative transformations enabled the use of theorems derived for native autonomous systems which
otherwise, under the standard transformation, could not be utilized. The transformations presented are inspired by mathematical modeling which also suggests that
transformations for other special cases exist.
References
1. R. Albert, H.G. Othmer, The topology of the regulatory interactions predicts the expression
pattern of the segment polarity genes in drosophila melanogaster. J. Theor. Biol. 223(1), 1–18
(2003)
2. A. Ben-Tal, A study of symmetric forced oscillators. Ph. D. thesis, University of Auckland,
New Zealand (2001)
3. A. Ben-Tal, Simplified models for gas exchange in the human lungs. J. Theor. Biol. 238(2),
474–95 (2006)
4. A. Ben-Tal, I.G. Kevrekidis, Coarse-graining and simplification of the dynamics seen in bursting neurons. SIAM J. Appl. Dyn. Syst. 15(2), 1193–1226 (2016)
5. A. Ben-Tal, S.S. Shamailov, J.F.R. Paton, Central regulation of heart rate and the appearance
of respiratory sinus arrhythmia: new insights from mathematical modeling. Math. Biosci. 255,
71–82 (2014)
6. A. Ben-Tal, Y.J. Wang, M.C.A. Leite, The logic behind neural control of breathing pattern. Sci.
Rep. 9 (2019)
7. R.J. Butera, J. Rinzel, J.C. Smith, Models of respiratory rhythm generation in the pre-bötzinger
complex. I. bursting pacemaker neurons. J. Neurophys. 81, 382–397 (1999)
8. R.J. Butera, J. Rinzel, J.C. Smith, Models of respiratory rhythm generation in the pre-bötzinger
complex. II. populations of coupled pacemaker neurons. J. Neurophys. 81, 398–415 (1999)
9. D. Capecchi, S.R. Bishop, Periodic oscillations and attracting basins for a parametrically excited
pendulum. Dyn. Stab. Syst. 9(2), 123–143 (1994)
10. M.J. Clifford, S.R. Bishop, Inverted oscillations of a driven pendulum. Proc. R. Soc. A-Math.
Phys. Eng. Sci. 454(1979), 2811–2817 (1998)
11. J.H. Comroe, Physiology of Respiration, 2nd edn. (Year Book Medical Publishers, Inc., 1977)
12. C.A. Del Negro, G.D. Funk, J.L. Feldman, Breathing matters. Nat. Rev. Neurosci. 19(6), 351–
367 (2018)
13. L. Edelstein-Keshet, Mathematical models in biology. Soc. Ind. Appl. Math. (2005)
14. M. Elstad, E.L. O’Callaghan, A.J. Smith, A. Ben-Tal, R. Ramchandra, Cardiorespiratory interactions in humans and animals: rhythms for life. Am. J. Phys. Heart Circ. Phys. 315(1), H6–H17
(2018)
15. J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of
vector fields. Applied Mathematical Sciences, vol. 42, (Springer-Verlag, New York, 1996)
16. D. Jordan, P. Smith, Nonlinear Ordinary Differential Equations: An Introduction for Scientists
and Engineers. (Oxford University Press, 2007)
17. S. Lenci, E. Pavlovskaia, G. Rega, M. Wiercigroch, Rotating solutions and stability of parametric pendulum by perturbation method. J. Sound Vib. 310(1–2), 243–259 (2008)
18. B.G. Lindsey, I.A. Rybak, J.C. Smith, Computational models and emergent properties of respiratory neural networks. Compr. Phys. 2, 1619–1670 (2012)