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) |
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