We study the Becker--Döring bubblelator, a variant of the Becker--Döring coagulation-fragmentation system that models the growth of clusters by gain or loss of monomers. Motivated by models of gas evolution oscillators from physical chemistry, we incorporate the injection of monomers and depletion of large clusters. For a wide range of physical rates, the Becker--Döring system itself exhibits a dynamic phase transition as mass density increases past a critical value. We connect the Becker--Döring bubblelator to a transport equation coupled with an integrodifferential equation for the excess monomer density by formal asymptotics in the near-critical regime. For suitable injection/depletion rates, we argue that time-periodic solutions appear via a Hopf bifurcation. Numerics confirm that the generation and removal of large clusters can become desynchronized, leading to temporal oscillations associated with bursts of large-cluster nucleation.


  1. bubblelator
  2. oscillator
  3. time periodic solution
  4. growth process
  5. injection
  6. Hopf bifurcation

MSC codes

  1. 68Q25
  2. 68R10
  3. 68U05

Get full access to this article

View all available purchase options and get full access to this article.

Supplementary Material

PLEASE NOTE: These supplementary files have not been peer-reviewed.

Index of Supplementary Materials

Title of paper: Oscillations in a Becker-Döring model with injection and depletion

Authors: Andre Schlichting, Barbara Niethammer, Robert Pego, and Juan Lopez Velazquez

File: BD-SIAP-supp.pdf

Type: PDF

Contents: M.1 Time scale for the quasistationary approximation, SM.2 Numerical verification of transversality and non-resonance conditions, and SM.3 The case v« 1 and b≥ 0.

File: BD-SIAP-supp-video.mp4

Type: Video File

Contents: Dynamic simulation of the limit model in rescaled variables (3.2)-(3.4) with parameter: eta = 0.2, alpha = 1/3, and beta = 2/3.


E. H. M. Badger and I. G. C. Dryden, The formation of gum particles in coal-gas, Trans. Faraday Soc., 35 (1939), pp. 607--628, https://doi.org/10.1039/tf9393500607.
J. M. Ball, J. Carr, and O. Penrose, The Becker-Döring cluster equations: Basic properties and asymptotic behaviour of solutions, Comm. Math. Phys., 104 (1986), pp. 657--692, http://projecteuclid.org/euclid.cmp/1104115173.
R. C. Ball, C. Connaughton, P. P. Jones, R. Rajesh, and O. Zaboronski, Collective oscillations in irreversible coagulation driven by monomer inputs and large-cluster outputs, Phys. Rev. Lett., 109 (2012), 168304, https://doi.org/10.1103/physrevlett.109.168304.
K. Bar-Eli and R. M. Noyes, Gas-evolution oscillators. \textrm10. A model based on a delay equation, J. Phys. Chem., 96 (1992), pp. 7664--7670, https://doi.org/10.1021/j100198a034.
R. Becker and W. Döring, Kinetische Behandlung der Keimbildung in übersättigten Dämpfen., Ann. Phys., 24 (1935), pp. 719--752, https://doi.org/10.1002/andp.19354160806.
B. P. Belousov, A periodic acting reaction and its mechanism, in Collection of Short Papers on Radiation Medicine, Medgiz, Moscow, 1959, pp. 145--152.
J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, Julia: A fresh approach to numerical computing, SIAM Rev., 59 (2017), pp. 65--98, https://doi.org/10.1137/141000671.
P. G. Bowers and R. M. Noyes, Chemical oscillations and instabilities. 51. Gas evolution oscillators. 1. Some new experimental examples, J. Amer. Chem. Soc., 105 (1983), pp. 2572--2574, https://doi.org/10.1021/ja00347a010.
N. V. Brilliantov, W. Otieno, S. A. Matveev, A. P. Smirnov, E. E. Tyrtyshnikov, and P. L. Krapivsky, Steady oscillations in aggregation-fragmentation processes, Phys. Rev. E, 98 (2018), 012109, https://doi.org/10.1103/physreve.98.012109.
S. S. Budzinskiy, S. A. Matveev, and P. L. Krapivsky, Hopf bifurcation in addition-shattering kinetics, Phys. Rev. E, 103 (2021), L040101, https://doi.org/10.1103/PhysRevE.103.L040101.
J. Carr, D. B. Duncan, and C. H. Walshaw, Numerical approximation of a metastable system, IMA J. Numer. Anal., 15 (1995), pp. 505--521, https://doi.org/10.1093/imanum/15.4.505.
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations, Appl. Math. Sci., Springer, New York, 1995, https://doi.org/10.1007/978-1-4612-4206-2.
M. Doumic, K. Fellner, M. Mezache, and H. Rezaei, A bi-monomeric, nonlinear Becker\textendashDöring-type system to capture oscillatory aggregation kinetics in prion dynamics, J. Theoret. Biol., 480 (2019), pp. 241--261, https://doi.org/10.1016/j.jtbi.2019.08.007.
Y. Farjoun and J. C. Neu, Exhaustion of nucleation in a closed system, Phys. Rev. E, 78 (2008), 051402, https://doi.org/10.1103/physreve.78.051402.
W. Feller, An Introduction to Probability Theory and Its Applications. Vol. I, 3rd ed., John Wiley and Sons, New York, 1968.
R. J. Field and R. M. Noyes, Oscillations in chemical systems. \textrmIV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), pp. 1877--1884, https://doi.org/10.1063/1.1681288.
S. K. Friedlander, Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, 2nd ed., Topics Chem. Engrg., Oxford University Press, Oxford, 2000.
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Appl. Math. Sci. 99, Springer, New York, 1993, https://doi.org/10.1007/978-1-4612-4342-7.
E. Hingant and R. Yvinec, Deterministic and stochastic Becker\textendashDöring equations: Past and recent mathematical developments, in Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology, D. Holcman, ed., Springer, Cham, 2017, pp. 175--204, https://doi.org/10.1007/978-3-319-62627-7_9.
S. G. Johnson, QuadGK.jl: Gauss--Kronrod integration in Julia, 2013, https://github.com/JuliaMath/QuadGK.jl.
A. J. Lotka, Contribution to the theory of periodic reactions, J. Phys. Chem., 14 (1910), pp. 271--274, https://doi.org/10.1021/j150111a004.
A. J. Lotka, Undamped oscillations derived from the law of mass action, J. Amer. Chem. Soc., 42 (1920), pp. 1595--1599, https://doi.org/10.1021/ja01453a010.
S. A. Matveev, P. L. Krapivsky, A. P. Smirnov, E. E. Tyrtyshnikov, and N. V. Brilliantov, Oscillations in aggregation-shattering processes, Phys. Rev. Lett., 119 (2017), 260601, https://doi.org/10.1103/physrevlett.119.260601.
R. McGraw and J. H. Saunders, A condensation feedback mechanism for oscillatory nucleation and growth, Aerosol Sci. Technol., 3 (1984), pp. 367--380, https://doi.org/10.1080/02786828408959025.
J. S. Morgan, XXX.\textemdashThe periodic evolution of carbon monoxide, J. Chem. Soc. Trans., 109 (1916), pp. 274--283, https://doi.org/10.1039/ct9160900274.
B. Niethammer, On the evolution of large clusters in the Becker-Döring model, J. Nonlinear Sci., 13 (2003), pp. 115--155, https://doi.org/10.1007/s00332-002-0535-8.
R. L. Pego and J. J. L. Velázquez, Temporal oscillations in Becker-Döring equations with atomization, Nonlinearity, 33 (2020), pp. 1812--1846, https://doi.org/10.1088/1361-6544/ab6815.
O. Penrose, Metastable states for the Becker-Döring cluster equations, Comm. Math. Phys., 124 (1989), pp. 515--541, http://projecteuclid.org/euclid.cmp/1104179294.
O. Penrose, The Becker-Döring equations at large times and their connection with the LSW theory of coarsening, J. Stat. Phys., 89 (1997), pp. 305--320, https://doi.org/10.1007/BF02770767.
O. Penrose, J. Lebowitz, J. Marro, M. Kalos, and J. Tobochnik, Kinetics of a first-order phase transition: computer simulations and theory, J. Stat. Phys., 34 (1984), pp. 399--426, https://doi.org/10.1007/BF01018552.
S. E. Pratsinis, S. K. Friedlander, and A. J. Pearlstein, Aerosol reactor theory: Stability and dynamics of a continuous stirred tank aerosol reactor, AIChE J., 32 (1986), pp. 177--185, https://doi.org/10.1002/aic.690320202.
I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. II, J. Chem. Phys., 48 (1968), pp. 1695--1700, https://doi.org/10.1063/1.1668896.
A. Schlichting, Macroscopic limit of the Becker-Döring equation via gradient flows, ESAIM Control Optim. Calc. Var., 25 (2019), 22, https://doi.org/10.1051/cocv/2018011.
M. Slemrod, The Becker-Döring equations, in Modeling in Applied Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, 2000, pp. 149--171.
K. W. Smith and R. M. Noyes, Gas evolution oscillators. 3. A computational model of the Morgan reaction, J. Phys. Chem., 87 (1983), pp. 1520--1524, https://doi.org/10.1021/j100232a014.
K. W. Smith, R. M. Noyes, and P. G. Bowers, Chemical oscillations and instabilities. Part 52. Gas evolution oscillators. 2. A reexamination of formic acid dehydration, J. Phys. Chem., 87 (1983), pp. 1514--1519, https://doi.org/10.1021/j100232a013.
J. J. Tyson, The Belousov-Zhabotinskii Reaction, Springer, Berlin, 1976, https://doi.org/10.1007/978-3-642-93046-1.
V. Volterra, Variazioni efluttuazioni del numero di individui in specie animali conviventi, Mem. Reale. Accad. Naz. Lincei, 2 (1926), pp. 31--113.
Z. Yuan, P. Ruoff, and R. M. Noyes, Chemical oscillations and instabilities. Part 66. Gas evolution oscillators. 7. A quantitative modeling test for the Morgan reaction, J. Phys. Chem., 89 (1985), pp. 5726--5732, https://doi.org/10.1021/j100272a031.
A. Zhabotinsky, Periodic course of the oxidation of malonic acid oxidation in a solution, Biofizika, 9 (1964), pp. 306--311.

Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1194 - 1219
ISSN (online): 1095-712X


Submitted: 16 February 2021
Accepted: 9 March 2022
Published online: 21 July 2022


  1. bubblelator
  2. oscillator
  3. time periodic solution
  4. growth process
  5. injection
  6. Hopf bifurcation

MSC codes

  1. 68Q25
  2. 68R10
  3. 68U05



Funding Information

Hausdorff Research Institute
Deutsche Forschungsgemeinschaft https://doi.org/10.13039/501100001659 : 390685587, 390685813, 211504053
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1812609, DMS-2106534

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.