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Volume 3, Issue 3
Hopf Bifurcation Analysis for a Delayed Business Cycle Model — The Equivalence of Multiple Time Scales Versus Center Manifold Reduction Methods

Xiaolin Zhen & Yuting Ding

DOI: 10.12150/jnma.2021.403

J. Nonl. Mod. Anal., 3 (2021), pp. 403-419.

Published online: 2022-06

[An open-access article; the PDF is free to any online user.]

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  • Abstract

In this paper, we study the Hopf bifurcation of a model with a second order term, which is the business cycle model with delay. Multiple time scales method, which is mainly used by the engineering researchers, and center manifold reduction method, which is mainly used by researchers from mathematical society, are used to derive the two types of normal forms near the Hopf critical point. A comparison between the two methods shows that the two normal forms are equivalent. Scholars can derive the normal form by choosing appropriate methods according to their actual demands. Moreover, bifurcation analysis and numerical simulations are given to verify the analytical predictions.

  • Keywords

Business cycle, Hopf bifurcation, Normal form, Multiple time scales, Center manifold reduction.

  • AMS Subject Headings

34K18, 37L10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JNMA-3-403, author = {Zhen , Xiaolin and Ding , Yuting}, title = {Hopf Bifurcation Analysis for a Delayed Business Cycle Model — The Equivalence of Multiple Time Scales Versus Center Manifold Reduction Methods}, journal = {Journal of Nonlinear Modeling and Analysis}, year = {2022}, volume = {3}, number = {3}, pages = {403--419}, abstract = {

In this paper, we study the Hopf bifurcation of a model with a second order term, which is the business cycle model with delay. Multiple time scales method, which is mainly used by the engineering researchers, and center manifold reduction method, which is mainly used by researchers from mathematical society, are used to derive the two types of normal forms near the Hopf critical point. A comparison between the two methods shows that the two normal forms are equivalent. Scholars can derive the normal form by choosing appropriate methods according to their actual demands. Moreover, bifurcation analysis and numerical simulations are given to verify the analytical predictions.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2021.403}, url = {http://global-sci.org/intro/article_detail/jnma/20670.html} }
TY - JOUR T1 - Hopf Bifurcation Analysis for a Delayed Business Cycle Model — The Equivalence of Multiple Time Scales Versus Center Manifold Reduction Methods AU - Zhen , Xiaolin AU - Ding , Yuting JO - Journal of Nonlinear Modeling and Analysis VL - 3 SP - 403 EP - 419 PY - 2022 DA - 2022/06 SN - 3 DO - http://doi.org/10.12150/jnma.2021.403 UR - https://global-sci.org/intro/article_detail/jnma/20670.html KW - Business cycle, Hopf bifurcation, Normal form, Multiple time scales, Center manifold reduction. AB -

In this paper, we study the Hopf bifurcation of a model with a second order term, which is the business cycle model with delay. Multiple time scales method, which is mainly used by the engineering researchers, and center manifold reduction method, which is mainly used by researchers from mathematical society, are used to derive the two types of normal forms near the Hopf critical point. A comparison between the two methods shows that the two normal forms are equivalent. Scholars can derive the normal form by choosing appropriate methods according to their actual demands. Moreover, bifurcation analysis and numerical simulations are given to verify the analytical predictions.

Zhen , Xiaolin and Ding , Yuting. (2022). Hopf Bifurcation Analysis for a Delayed Business Cycle Model — The Equivalence of Multiple Time Scales Versus Center Manifold Reduction Methods. Journal of Nonlinear Modeling and Analysis. 3 (3). 403-419. doi:10.12150/jnma.2021.403
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