A Period-Doubling Bubble in the Dynamics of Two Coupled Oscillators

  • J. C. Alexander
Part of the Lecture Notes in Biomathematics book series (LNBM, volume 66)


A period-doubling cascade in the bifurcation diagram of two Brusselators coupled by diffusion is continued to a particular parameter regime, where it is seen numerically to be associated with other bifurcation branches, and in particular, “decascades;” we call the resulting bifurcation effect a period-doubling bubble. Moreover the dynamics of the bubble formation can be described. The emphasis in this note in on describing the phenomenon, although the (strong) possibility of describing it analytically in terms of unfolding a singularity which comes from interactions of singularities of the single oscillators is discussed, as well as a discussion of possibly similar behavior in other coupled oscillators.


Hopf Bifurcation Bifurcation Diagram Unstable Manifold Homoclinic Orbit Couple Oscillator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.C. Alexander, “Spontaneous oscillations in two 2-component cells coupled by diffusion,” J. Math. Biol., submitted, 1984.Google Scholar
  2. [2]
    M. Ashkenazi & H.G. Othmer, “Spatial patterns in coupled bio-chemical oscillators,” J. Math. Biol. 5(1978), 305–350.MATHMathSciNetGoogle Scholar
  3. [3]
    M.J. Feigenbaum, “Universal behavior in nonlinear systems,” Los Alamos Sci. 1(1980), 4–29.MathSciNetGoogle Scholar
  4. [4]
    R.J. Field & R.M. Noyes, “Oscillations in chemical systems, IV. Limit cycle behavior in a model of a real chemical reaction,” J. Chem. Phys. 60(1974), 1877–1884.CrossRefGoogle Scholar
  5. [5]
    J. Guckenheimer & P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Appl. Math. Sci. #42, Springer-Verlag, 1983.Google Scholar
  6. [6]
    P. Holmes, “Unfolding a degenerate nonlinear oscillator: a codimension two bifurcation” in Nonlinear Dynamics, R.H.G. Helleman, ed., Ann. N.Y. Acad. Sci. #357(1980), 473–488.Google Scholar
  7. [7]
    R. Lefever & I. Prigogine, “Symmetry-breaking instabilities in dissipative systems II,” J. Chem. Phys. 48(1968), 1695–1700.CrossRefGoogle Scholar
  8. [8]
    G. Nicolis & I. Prigogine, Self-Organization in Nonequilibrium Systems, Wiley, 1977.MATHGoogle Scholar
  9. [9]
    R. O’Malley, Introduction to Singular Perturbations, Academic Press, 1974.MATHGoogle Scholar
  10. [10]
    I. Schreiber & M. Marek, “Strange attractors in coupled reaction-diffusion cells,” Physica 5D(1982), 258–272.MathSciNetGoogle Scholar
  11. [11]
    J.J. Tyson, “Analytic representation of oscillations, excitability, and travelling waves in a realistic model of the Belousov-Zhabotinskii reaction,” J. Chem. Phys. 66(1977), 905–915.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • J. C. Alexander
    • 1
  1. 1.Department of Mathematics and Institute for Physical Science and TechnologyUniversity of MarylandCollege ParkUSA

Personalised recommendations