Dynamical properties of chemical systems near Hopf bifurcation points.

In this paper, we numerically investigate local properties of dynamical systems close to a Hopf bifurcation instability. We focus on chemical systems and present an approach based on the theory of normal forms for determining numerical estimates of the limit cycle that branches off at the Hopf bifurcation point. For several numerically ill-conditioned examples taken from chemical kinetics, we compare our results with those obtained by using traditional approaches where an approximation of the limit cycle is restricted to the center subspace spanned by critical eigenvectors, and show that inclusion of higher-order terms in the normal form expansion of the limit cycle provides a significant improvement of the limit cycle estimates. This result also provides an accurate initial estimate for subsequent numerical continuation of the limit cycle. (c) 2000 American Institute of Physics.

[1]  K. Showalter,et al.  NORMAL MODES FOR CHEMICAL REACTIONS FROM TIME SERIES ANALYSIS , 1999 .

[2]  Michael T. Heath,et al.  Scientific Computing , 2018 .

[3]  Baltazar D. Aguda,et al.  Periodic-chaotic sequences in a detailed mechanism of the peroxidase-oxidase reaction , 1991 .

[4]  A. Zhabotinsky.  A history of chemical oscillations and waves. , 1991, Chaos.

[5]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[6]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[7]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[8]  D Thomas,et al.  An experimental enzyme-membrane oscillator. , 1973, Biochimica et biophysica acta.

[9]  M. Ipsen,et al.  Systematic derivation of amplitude equations and normal forms for dynamical systems. , 1998, Chaos.

[10]  Milan Kubíček,et al.  Book-Review - Computational Methods in Bifurcation Theory and Dissipative Structures , 1983 .

[11]  I. Schreiber,et al.  Chaotic patterns in a coupled oscillator-excitator biochemical cell system. , 1999, Chaos.

[12]  G. Sell,et al.  The Hopf Bifurcation and Its Applications , 1976 .

[13]  I. Epstein,et al.  Glycolytic pH oscillations in a flow reactor. , 1994, Biophysical chemistry.

[14]  Floris Takens,et al.  Singularities of vector fields , 1974 .

[15]  B. A. Scott,et al.  Magnetic susceptibility and nuclear magnetic resonance studies of transition-metal monophosphides. , 1968, The Journal of chemical physics.

[16]  Yoshiki Kuramoto,et al.  On the Formation of Dissipative Structures in Reaction-Diffusion Systems Reductive Perturbation Approach , 1975 .

[17]  A. Zhabotinsky,et al.  Concentration Wave Propagation in Two-dimensional Liquid-phase Self-oscillating System , 1970, Nature.

[18]  Philip Holmes,et al.  The limited effectiveness of normal forms: a critical review and extension of local bifurcation studies of the Brusselator PDE , 1997 .

[19]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[20]  L. Olsen,et al.  Chaos in an enzyme reaction , 1977, Nature.

[21]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[22]  G. Iooss,et al.  Elementary stability and bifurcation theory , 1980 .

[23]  rensen,et al.  Quenching of chemical oscillations: General theory , 1990 .

[24]  M. Holodniok,et al.  New algorithms for the evaluation of complex bifurcation points in ordinary differential equations. a comparative numerical study , 1984 .

[25]  N. Zabusky,et al.  Chemical Oscillations in a Membrane , 1973, Nature.

[26]  Irving R. Epstein,et al.  A General Model for pH Oscillators , 1991 .