Hopf spectral and multilinear normal-form data

Let $f(x,\lambda )$ be a real $C^3$ vector field on an open subset of ${\mathbb{R}}^d\times \mathbb{R}$, and suppose $f(0,0)=0$. Put $A=D_{x}f(0,0)$. Assume $A$ has a simple pair of eigenvalues $i\omega$ and $-i\omega$, where $\omega >0$, and no other eigenvalue on the imaginary axis. Let $A_{\mathbb{C}}$ be the complexification of $A$. Choose $q,p\in {\mathbb{C}}^d$ with

$Aq=i\omega q,A^{\ast }p=-i\omega p,q^{\ast }q=1,p^{\ast }q=1.$

Here $u^{\ast }v={\bar{u}}^{T}v$ is the standard Hermitian product. To obtain both normalizations, start with nonzero compatible eigenvectors $q_0,p_0$ and put $q=q_0\mathrm{/}\sqrt{q_0^{\ast }{q}_0}$. Simplicity gives $a:=p_0^{\ast }q\mathrm{\ne }0$ by For a simple eigenvalue, left and right eigenvectors pair nontrivially and may be normalized by $y^{\ast }x=1$. Set $p=p_0\mathrm{/}\bar{a}$; then $q^{\ast }q=1$ and $p^{\ast }q=a\mathrm{/}a=1$, while both eigenvector equations are preserved. Write $B=D_x^{2}f(0,0)$ and $C=D_x^{3}f(0,0)$ for the symmetric real multilinear derivatives, extended complex multilinearly. Since zero and $2i\omega$ are not eigenvalues of $A$, define

$G_{21}:=p^{\ast }\text{\hspace{0.17em}⁣}(C(q,q,\bar{q})-2B\text{\hspace{0.17em}⁣}(q,A^{-1}B(q,\bar{q}))+B\text{\hspace{0.17em}⁣}(\bar{q},(2i\omega I-A{)}^{-1}B(q,q)))$

and define the first Lyapunov coefficient in this normalization by

${\mathrm{\ell }}_1:=\frac{\mathrm{Re}{G}_{21}}{2\omega }\mathrm{.}$

If $q$ is rescaled to $cq$ and the unit-norm condition is dropped while $p^{\ast }q=1$ is preserved, then ${\mathrm{\ell }}_1$ rescales by $\mathrm{\mid }c{\mathrm{\mid }}^2$; under the declared condition $q^{\ast }q=1$, the remaining phase freedom leaves ${\mathrm{\ell }}_1$ unchanged. Its sign is the local cubic radial sign. The definition alone is not a bifurcation theorem.

For an equilibrium at arbitrary $(x_{\ast },{\lambda }_0)$, apply this definition to the translated field $\widehat{f}(y,\mu )=f(x_{\ast }+y,{\lambda }_0+\mu )$ at $(y,\mu )=(0,0)$. Thus $A=D_{x}f(x_{\ast },{\lambda }_0)$, $B=D_x^{2}f(x_{\ast },{\lambda }_0)$ and $C=D_x^{3}f(x_{\ast },{\lambda }_0)$, and $p,q,\omega$ refer to that same $A$. Translation preserves these spatial derivatives and their Euclidean normalization. If the equilibrium is continued as $\xi (\lambda )$, the further state change $y=x-\xi (\lambda )$ at each fixed parameter gives $\dot{y}=f(y+\xi (\lambda ),\lambda )$; because $\dot{\lambda }=0$, no extra time-derivative term appears. Spatial jets at the critical parameter are still the same $A,B,C$.