2021•Unpublished venueRequires access
The Hopf Bifurcation
Robert L. Devaney
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Abstract
As in the case of one-dimensional maps, the lack of hyperbolicity is usually a signal for the occurrence of bifurcations. In one-dimensional systems, these occur when the eigenvalue at a periodic point is either + 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429280801/4a0ae1af-5c20-413c-b1ed-996514b22423/content/math33_1.tif"/> (usually the saddle node bifurcation) or − 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429280801/4a0ae1af-5c20-413c-b1ed-996514b22423/content/math33_2.tif"/> (usually the period-doubling bifurcation). In complex dynamics, different bifurcations occur when the derivative is complex and lies on the unit circle. For higher dimensional systems, these types of bifurcations also occur, but there are other possible bifurcations of periodic points as well. The most typical of these is the Hopf bifurcation, which we will describe in this section.