STABILITY AND BIFURCATION IN A MODULATED BURGERS SYSTEM.

W. E. Olmstead; S. H. Davis

doi:10.1090/qam/644101

STABILITY AND BIFURCATION IN A MODULATED BURGERS SYSTEM.

W. E. Olmstead^*, S. H. Davis

^*Corresponding author for this work

Engineering Sciences and Applied Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude A( tau ) of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate theta ( tau ). Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in theta ( tau ). For each of these oscillatory-type modulations, it is found that A**2( tau ) has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.

Original language	English (US)
Pages (from-to)	467-477
Number of pages	11
Journal	Quarterly of Applied Mathematics
Volume	39
Issue number	4
DOIs	https://doi.org/10.1090/qam/644101
State	Published - 1982

ASJC Scopus subject areas

Applied Mathematics

Access to Document

10.1090/qam/644101

Fingerprint Dive into the research topics of 'STABILITY AND BIFURCATION IN A MODULATED BURGERS SYSTEM.'. Together they form a unique fingerprint.

Cite this

@article{25519c0853344c1188cdbcee633a3710,

title = "STABILITY AND BIFURCATION IN A MODULATED BURGERS SYSTEM.",

abstract = "The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude A( tau ) of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate theta ( tau ). Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in theta ( tau ). For each of these oscillatory-type modulations, it is found that A**2( tau ) has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.",

author = "Olmstead, {W. E.} and Davis, {S. H.}",

year = "1982",

doi = "10.1090/qam/644101",

language = "English (US)",

volume = "39",

pages = "467--477",

journal = "Quarterly of Applied Mathematics",

issn = "0033-569X",

publisher = "American Mathematical Society",

number = "4",

}

TY - JOUR

T1 - STABILITY AND BIFURCATION IN A MODULATED BURGERS SYSTEM.

AU - Olmstead, W. E.

AU - Davis, S. H.

PY - 1982

Y1 - 1982

N2 - The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude A( tau ) of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate theta ( tau ). Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in theta ( tau ). For each of these oscillatory-type modulations, it is found that A**2( tau ) has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.

AB - The stability of the null state for a nonlinear Burgers system is examined. The results include (i) an energy estimate for global stability for states involving arbitrary modulation in time, and (ii) an analysis of the bifurcation from the null state for slow modulations. For the slow modulations it is determined that the amplitude A( tau ) of the bifurcated disturbance velocity satisfies a Landau-type equation with time-dependent growth rate theta ( tau ). Particular attention is given to periodic and quasiperiodic modulations of the system, which lead to analogous behavior in theta ( tau ). For each of these oscillatory-type modulations, it is found that A**2( tau ) has the same long-time mean value as the unmodulated case, implying no alteration of the final mean kinetic energy. Applications to various fluid-dynamical phenomena are discussed.

UR - http://www.scopus.com/inward/record.url?scp=0019899832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0019899832&partnerID=8YFLogxK

U2 - 10.1090/qam/644101

DO - 10.1090/qam/644101

M3 - Article

AN - SCOPUS:0019899832

VL - 39

SP - 467

EP - 477

JO - Quarterly of Applied Mathematics

JF - Quarterly of Applied Mathematics

SN - 0033-569X

IS - 4

ER -

STABILITY AND BIFURCATION IN A MODULATED BURGERS SYSTEM.

Abstract

ASJC Scopus subject areas

Access to Document

Other files and links

Cite this