# ON THE CREATION, GROWTH AND EXTINCTION OF OSCILLATORY SOLUTIONS FOR A SIMPLE POOLED CHEMICAL REACTION SCHEME.

Research output: Contribution to journal › Article

## Standard

**ON THE CREATION, GROWTH AND EXTINCTION OF OSCILLATORY SOLUTIONS FOR A SIMPLE POOLED CHEMICAL REACTION SCHEME.** / Merkin, J. H.; Needham, D. J.; Scott, S. K.

Research output: Contribution to journal › Article

## Harvard

*SIAM Journal on Applied Mathematics*, vol. 47, no. 5, pp. 1040-1060. https://doi.org/10.1137/0147068

## APA

*SIAM Journal on Applied Mathematics*,

*47*(5), 1040-1060. https://doi.org/10.1137/0147068

## Vancouver

## Author

## Bibtex

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## RIS

TY - JOUR

T1 - ON THE CREATION, GROWTH AND EXTINCTION OF OSCILLATORY SOLUTIONS FOR A SIMPLE POOLED CHEMICAL REACTION SCHEME.

AU - Merkin, J. H.

AU - Needham, D. J.

AU - Scott, S. K.

PY - 1987/1/1

Y1 - 1987/1/1

N2 - The equations which govern a simple pooled chemical reaction scheme are analyzed in detail in terms of a nondimensional parameter mu , which represents the amount of the pooled chemical originally present. It is shown that there is one finite equilibrium point, with a Hopf bifurcation occurring at mu equals 1. The phase plane at infinity is then examined and it is shown that there are equilibrium points at infinity at the positive ends of both axes, the nature of which are discussed. This enables the global phase portrait to be constructed for all positive mu . From this it emerges that the stable limit cycle created at mu equals 1 by a Hopf bifurcation is destroyed at mu //0 ( mu //0 less than 1) by an infinite period bifurcation, due to the formation of a heteroclinic orbit by the separatrices from the equilibrium points at infinity. The form of this heteroclinic orbit is then discussed, and it is shown that the value of mu //0 can be determined by simple numerical integration.

AB - The equations which govern a simple pooled chemical reaction scheme are analyzed in detail in terms of a nondimensional parameter mu , which represents the amount of the pooled chemical originally present. It is shown that there is one finite equilibrium point, with a Hopf bifurcation occurring at mu equals 1. The phase plane at infinity is then examined and it is shown that there are equilibrium points at infinity at the positive ends of both axes, the nature of which are discussed. This enables the global phase portrait to be constructed for all positive mu . From this it emerges that the stable limit cycle created at mu equals 1 by a Hopf bifurcation is destroyed at mu //0 ( mu //0 less than 1) by an infinite period bifurcation, due to the formation of a heteroclinic orbit by the separatrices from the equilibrium points at infinity. The form of this heteroclinic orbit is then discussed, and it is shown that the value of mu //0 can be determined by simple numerical integration.

UR - http://www.scopus.com/inward/record.url?scp=0023424681&partnerID=8YFLogxK

U2 - 10.1137/0147068

DO - 10.1137/0147068

M3 - Article

AN - SCOPUS:0023424681

VL - 47

SP - 1040

EP - 1060

JO - SIAM Journal on Applied Mathematics

JF - SIAM Journal on Applied Mathematics

SN - 0036-1399

IS - 5

ER -