A dynamical systems approach to modified point kinetics equations
DOI:
https://doi.org/10.31349/RevMexFis.72.061201Keywords:
Dynamical systems; Lyapunov stability; Jacobi stability; Kosambi-Cartan-Chern theory, reactor point-kineticsAbstract
We apply the formalism of dynamical systems and the Kosambi-Kartan-Chern theory to perform an extensive stability analysis of solutions to modified point kinetics equations, a set of two differential equations used in nuclear physics to describe the temporal behavior of a fission nuclear reactor. It is known that the stability of the system depends on reactivity, which appears in the equations as a parameter ρ, and the functional form of this parameter can describe different situations that may occur in a nuclear reactor, making the equations nonlinear or even nonautonomous in certain cases. We present a comparative study between linear stability and Jacobi stability, pointing out the usefulness of the geometrodynamical approach.
Downloads
References
A. F. Henry and E. Melkonian, Nuclear-Reactor Analysis, 1st ed. (The MIT Press, 1977), pp. 297-300
J. Duderstadt and L. Hamilton, Nuclear Reactor Analysis, 1st ed. (Wiley, 1991), pp. 237-239
T. W. Kerlin and B. R. Upadhyaya, Dynamics and Control of Nuclear Reactors, 1st ed. (Academic Press, 2019), pp. 21-25 DOI: https://doi.org/10.1016/B978-0-12-815261-4.00001-9
G. G. Theler and F. J. Bonetto, On the stability of the point reactor kinetics equations, Nucl. Eng. Des. 240 (2010) 1443, https://doi.org/10.1016/j.nucengdes.2010.03.007 DOI: https://doi.org/10.1016/j.nucengdes.2010.03.007
V. Pandey and S. Singh, Detailed bifurcation analysis with a simplified model for advance heavy water reactor system, Commun. Nonlinear Sci. Numer. Simul. 20 (2015) 186, https://doi.org/10.1016/j.nucengdes.2010.03.007 DOI: https://doi.org/10.1016/j.cnsns.2014.05.019
R. Vora, A. Chakraborty, and S. Singh, Bifurcation analysis of out-of-phase oscillations in boiling water reactors using multipoint neutron kinetics, Prog. Nucl. Energy 120 (2019), https://doi.org/10.1016/j.pnucene.2019.103218 DOI: https://doi.org/10.1016/j.pnucene.2019.103218
C. Z. Petersen et al., An analytical solution of the point kinetics equations with time variable reactivity by the decomposition method, Prog. Nucl. Energy, 53 (2011) 1091, https://doi.org/10.1016/j.pnucene.2011.01.001 DOI: https://doi.org/10.1016/j.pnucene.2011.01.001
A. Gandini and M. Salvatores, The Physics of Subcritical Multiplying Systems., J. Nucl. Sci. Tech. 39 (2002) 673, https://doi.org/10.1080/18811248.2002.9715249 DOI: https://doi.org/10.1080/18811248.2002.9715249
A. E. Aboanber and A. A. Nahla, A novel fractional technique for the modified point kinetics equations, J. Nucl. Sci. Tech. 24 (2016) 666, https://doi.org/10.1016/j.joems.2016.02.001 DOI: https://doi.org/10.1016/j.joems.2016.02.001
A. Nunes et al., A New Formulation to the Point Kinetics Equations Considering the Time Variation of the Neutron Currents, WJNST 05 (2015) 57, http://dx.doi.org/10.4236/wjnst.2015.51006 DOI: https://doi.org/10.4236/wjnst.2015.51006
P. Antonelli, S. Rutz, and S. Sabau, A Transient-State Analysis of Tyson’s Model for the Cell Division Cycle by Means of KCC-Theory, Open Syst. Inf. Dyn. 9 (2002) 223, https://doi.org/10.1023/A:1019752327311 DOI: https://doi.org/10.1023/A:1019752327311
C. Boehmer, T. Harko, and S. Sabau, Jacobi stability analysis of dynamical systems - applications in gravitation and cosmology, Adv. Theor. Math. Phys. 16 (2010) 1145, https://dx.doi.org/10.4310/ATMP.2012.v16.n4.a2 DOI: https://doi.org/10.4310/ATMP.2012.v16.n4.a2
C. Blaga, P. Blaga, and T. Harko, Jacobi and Lyapunov stability analysis of circular geodesics around a spherically symmetric dilaton black hole, Symmetry 15 (2023) 319, https://doi.org/10.3390/sym15020329 DOI: https://doi.org/10.3390/sym15020329
P. Singh et al., Stability of circular geodesics in equatorial plane of Kerr spacetime, Eur. Phys. J. Plus. 137 (2022), http://dx.doi.org/10.1140/epjp/s13360-022-02477-1 DOI: https://doi.org/10.1140/epjp/s13360-022-02477-1
Tiscareño, Tonatiuh, Rodríguez, Benito, and Chagoya, Javier, Stability analysis of geodesics in dynamical Chern-Simons black holes: a geometrical perspective, Eur. Phys. J. C 85 (2025) 439, https://doi.org/10.1140/epjc/s10052-025-14137-6 DOI: https://doi.org/10.1140/epjc/s10052-025-14137-6
C. Blaga and P. A. Blaga, Jacobi Stability of Circular Orbits Around Conformally InvariantWeyl Gravity Black Holes, Universe 12 (2026) 114, http://dx.doi.org/10.3390/universe12040114 DOI: https://doi.org/10.3390/universe12040114
M. W. Hirsch and S. Smale, Ecuaciones Diferenciales, Sistemas Dinámicos y Algebra lineal, 1st ed. (Alianza Editorial, España, 1983)
L. Perko, Differential equations and dynamical systems, 3rd ed. (Springer Verlag, New York, 2001) DOI: https://doi.org/10.1007/978-1-4613-0003-8
E. A. Coddington and N. Levinston, Theory of Ordinary Differential equations, 1st ed. (Dover publications, United states of America, 1987) DOI: https://doi.org/10.1007/BFb0067079
F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, 1st ed. (Springer Verlag, Berlin, 1990) DOI: https://doi.org/10.1007/978-3-642-97149-5_1
P. E. Kloeden and R. Martin, Nonautonomous Dynamical Systems (American Mathematical Society, Providence, Rhode Island, 2011) DOI: https://doi.org/10.1090/surv/176
V. Marinca and N. Herisanu, Nonlinear Dynamical Systems in Engineering: Some Approximate Approaches (Springer Verlag, 2011) DOI: https://doi.org/10.1007/978-3-642-22735-6
T. Caraballo and X. Han, Aplied Nonautonomous and Random Dynamical Systems Aplied Dymanical Systems (Springer Verlag, Switzerland, 2016) DOI: https://doi.org/10.1007/978-3-319-49247-6
T. Harko, P. Pantaragphong, and S. V. Sabau, Kosambi-CartanChern (KCC) theory for higher-order dynamical systems, Int. J. Geom. Methods Mod. Phys. 13 (2016) 1650014, https://doi.org/10.1142/S0219887816500146 DOI: https://doi.org/10.1142/S0219887816500146
W. Krabs, A General Predator-Prey Model, Math. Comput. Model. Dyn. Syst. 9 (2003) 387, http://dx.doi.org/10.1076/mcmd.9.4.387.27896 DOI: https://doi.org/10.1076/mcmd.9.4.387.27896
M. Lines and A. Medio, Nonlinear Dynamical Systems in Economics (Springer-Verlag Wien, 2005) DOI: https://doi.org/10.1007/3-211-38043-4
K. Rateitschak and O. Wolkenhauer, Thresholds in transient dynamics of signal transduction pathways, J. Theor. Biol. 264 (2010) 334, https://doi.org/10.1016/j.jtbi.2010.02.001 DOI: https://doi.org/10.1016/j.jtbi.2010.02.001
M. Crucifix and J. Rougier, On the use of simple dynamical systems for climate predictions, Eur. Phys. J. Spec. 174 (2009) 11, https://doi.org/10.1140/epjst/e2009-01087-5 DOI: https://doi.org/10.1140/epjst/e2009-01087-5
P. E. Kloeden and C. Patzsche, Nonautonomous Dynamical Systems in the Life Sciences, 1st ed. (Springer Verlag, Switzerland, 2013) DOI: https://doi.org/10.1007/978-3-319-03080-7
M. Fels, The Equivalence Problem for Systems of Second- Order Ordinary Differential Equations, Proc. Lond. Math. Soc. 71 (1995) 221, https://doi.org/10.1112/plms/s3-71.1.221 DOI: https://doi.org/10.1112/plms/s3-71.1.221
S. Roy, A. Ray, and A. R. Chowdhury, KosambiCartan- Chern perspective on chaos: Unveiling hidden attractors in nonlinear autonomous systems, Phys. Rev. E 109 (2024) 044205, https://link.aps.org/doi/10.1103/PhysRevE.109.044205 DOI: https://doi.org/10.1103/PhysRevE.109.044205
F. Wang et al., Jacobi Stability Analysis and the Onset of Chaos in a Two-Degree-of-Freedom Mechanical System, Int. J. Bifurc. Chaos Appl. Sci. Eng. 31 (2021) 2150075, https://doi.org/10.1142/S0218127421500759 DOI: https://doi.org/10.1142/S0218127421500759
Q. Liu and X. Zhang, Jacobi Stability Analysis of Liu System: Detecting Chaos, Mathematics 12 (2024), https://doi.org/10.3390/math12131981 DOI: https://doi.org/10.3390/math12131981
T. Harko et al., Jacobi stability analysis of the Lorenz system, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550081, https://doi.org/10.1142/S0219887815500814 DOI: https://doi.org/10.1142/S0219887815500814
Y. Chen and Z. Yin, The Jacobi Stability of a Lorenz-Type Multistable Hyperchaotic System with a Curve of Equilibria, Int. J. Bifurc. Chaos Appl. Sci. Eng. 29 (2019) 1950062, https://doi.org/10.1142/S0218127419500627 DOI: https://doi.org/10.1142/S0218127419500627
M. K. Gupta and C. K. Yadav, Jacobi Stability Analysis of Rössler System, Int. J. Geom. Methods. Mod. Phys. 27 (2017) 1750056, https://doi.org/10.1142/S0218127417500560 DOI: https://doi.org/10.1142/S0218127417500560
H. Abolghasem, Liapunov Stability Versus Jacobi Stability, J. Dyn. Syst. Geom. Theor. 10 (2012) 13, https://doi.org/10.1080/1726037X.2012.10698604 DOI: https://doi.org/10.1080/1726037X.2012.10698604
S. Sabau, Some remarks on Jacobi stability, Nonlinear Anal. Theory Methods Appl. 63 (2005) 143, https://doi.org/10.1016/j.na.2005.02.061 DOI: https://doi.org/10.1016/j.na.2005.02.061
Y. Cai et al., A numerical solution to the nonlinear point kinetics equations using Magnus expansion, Ann. Nucl. Energy. 89 (2016) 84, https://doi.org/10.1016/j.anucene.2015.11.021 DOI: https://doi.org/10.1016/j.anucene.2015.11.021
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2026 Tonatiuh Manuel Tiscareño Sanchez, S. L. Hinojosa Ruiz, S. Tototzintle Huitle
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Authors retain copyright and grant the Revista Mexicana de Física right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
