On the definitions of bifurcation. (English) Zbl 0921.58042
Let \(f:\mathbb R^n\times\mathbb R\rightarrow\mathbb R^n\) be a \(C^1\)-function and \((x_0,\lambda_0)\) a zero of \(f\). The authors give two definitions for \((x_0,\lambda_0)\) to be a bifurcation point for zeroes of \(f\). The two definitions coincide if \(0\) is a simple eigenvalue of \(D_xf(x_0,\lambda_0)\) and if there are no other eigenvalues on the imaginary axis. There are examples where the two definitions differ.
Reviewer: Th.J.Bartsch (Gießen)
MSC:
| 37G99 | Local and nonlocal bifurcation theory for dynamical systems |
| 34C23 | Bifurcation theory for ordinary differential equations |
References:
| [1] | Chow S. N., Chap. 1 pp 2– (1982) |
| [2] | Wu J. K., J. Taiyuan University of Technology 24 pp 40– (1993) |
| [3] | Wu J. K., Mech. Practice 16 (1) pp 1– (1994) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
