Let's consider a family of smooth odd functions $\phi_v(u)\colon \mathbb{R}^2\to\mathbb{R}$, which ,looks like' a family of functions $f_y(x)=x^3-yx$ in the vicinity of $(0,0)$: $\phi_v(u)$ has no critical points for $v<0$; it has two non-degenerate critical points for $v>0$, and there is one degenerate critical point for $v=0$.
Does there exist a smooth change of variables $u$ and $v$ to $x(u,v)$ and $y(v)$ respectively that transforms $\phi(u,v)$ to the polynomial $x^3-yx$? More precisely, is following conjecture true?
Conjecture:
Let $\phi\colon [-u_0,u_0]\times [-v_0,v_0]\to\mathbb{R}$ be real analytic function such that
- $\phi'_u(0,0)=0$, $\phi''_{uu}(0,0)=0$, $\phi'''_{uuu}(0,0)\neq0$, $\phi''_{uv}(0,0)\neq 0$;
- for each $v\in[-v_0,0)$ there are precisely two solutions $u_{\pm}\in[-u_0,u_0]$ of the equation $\phi'_u(u,v)=0$ and $\phi''_{uu}(u_{\pm},v) \neq 0$;
- for each $v\in(0,v_0]$ there are no solutions of the equation $\phi'_u(u,v)=0$;
- $\phi(-u,v)=-\phi(u,v)$ for each $u$ and $v$.
Then there exist real analytic functions $x\colon [-u_0,u_0]\times[-v_0,v_0]\to\mathbb{R}$ and $y\colon [-v_0,v_0]\to\mathbb{R}$ such that $x'_u\neq 0$ and \begin{equation} \phi(u,v)=\frac{x(u,v)^3}{3} - y(v)x(u,v). \end{equation}
The first condition of Conjecture describes $(0,0)$ as a cusp point of the mapping $(u,v)\mapsto(\phi(u,v),v)$. This conjecture was proved only in some small vicinity of $(0,0)$ (see [1, Chapter VI, Lemma 2.3]). However, we could not find any theorems about changing variables on a large fixed vicinity of $(0,0)$. The specific function we are interested in: \begin{equation} \phi\colon [-\pi,\pi]\times[-\frac{1}{\sqrt2},1-\frac{1}{\sqrt2}]\to\mathbb{R},\qquad \phi(u,v)=u\cdot\left(v+\frac{1}{\sqrt{2}}\right) - \arcsin\frac{\sin u}{\sqrt{2}}. \end{equation}
References:
[1] Fedoryuk, M. V, The saddle-point method, Nauka, Moscow, 1977.
