One dimensional centre manifolds, part 3

I now want to generalize the discussion relating the case that the leading order term in the dynamics on a centre manifold is quadratic and the generic fold bifurcation to a related case where one more derivative has been taken. So the aim is to relate the case that the leading order term in the dynamics on the centre manifold is cubic and the generic cusp bifurcation. To do this we express $h_{xxx}$ in terms of derivatives of $f^1$ . In fact we will only write out those terms explicitly which do not always vanish at the bifurcation point. The result is $h_{xxx}(0,0,0)=f^1_{xxx}(0,0,0)+3\psi_{xx}(0,0)f^1_{xy}(0,0,0)$ .
In this way it is possible to express the condition for the non-vanishing of the cubic term in the dynamics on the centre manifold by $f^1_{xxx}(0,0,0)+3\psi_{xx}(0,0)f^1_{xy}(0,0,0)\ne 0$ . Note, however, that the expression $f^1_{xy}(0,0,0)$ is not intrinsic to the centre manifold. Because of the vanishing of the first derivatives the second order derivatives $\psi_{xx}(0,0)$ and $f^1_{xy}(0,0)$ define tensors in the full dimension of the dynamical system. Hence we can write $\psi_{xx}(0,0)f^1_{xy}(0,0,0)$ invariantly in the form $\psi^i_{,jk}(0,0)f^m_{,il}(0,0,0)p_mq^jq^kq^l$ . It remains to compute $\psi^i_{,jk}(0,0)$ . The invariance of the centre manifold implies that $f^i_{,j}\psi^j=f^i_{,jk}q^jq^k (x_1)^2+\ldots$ . Hence $f^i_{,l}(0,0,0)\psi^l_{,jk}(0,0)q^jq^k=f^i_{,jk}(0,0,0)q^jq^k$ . This equation has a unique solution for $\psi^l_{,jk}(0,0)q^jq^k$ . In this way one of the conditions for a generic cusp bifurcation can be expressed in an invariant way. This corresponds to (8.128) in the book of Kuznetsov. In trying to understand these things better I was confused by equation (5.28) in the book which should be almost the same as (8.128) but in fact contains a typo. There is a repetition of $B(q,$ . For a generic cusp bifurcation we need two parameters $\alpha_1$ and $\alpha_2$ . The genericity condition involving derivatives with respect to parameters can easily translated into an invariant form. The result is
$p_if^i_{\alpha_1}(0,0,0)p_jf^j_{,k\alpha_2}(0,0,0)q^k-p_if^i_{\alpha_2}(0,0,0)p_jf^j_{,k\alpha_1}(0,0,0)q^k=0.$ Formulae similar to those just given were used in a paper I wrote with Juliette Hell some years ago (Nonlin. Analysis: RWA. 24, 175). Looking back at that I have the feeling that we must have been doing some sleepwalking when writing that paper. Or maybe I was sleepwalking and Juliette was paying attention. In any case, the final result was correct.

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