Reflection-symmetric cubic double-zero normal form ($\dot{q}=p,\dot{p}=-\lambda q+\kappa p+\sigma q^3-2\sigma q^{2}p$)

The sign $\sigma =\pm 1$ distinguishes an inverted quartic potential from a double-well potential. The origin has determinant $\lambda$ and trace $\kappa$. Nonzero equilibria obey $q^2=\lambda /\sigma$, with determinant $-2\lambda$ and trace $\kappa -2\lambda$. Its energy $p^2/2+\lambda q^2/2-\sigma q^4/4$ has derivative $(\kappa -2\sigma q^2)p^2$. The origin's Hopf bifurcation is supercritical for $\sigma =1$ and subcritical for $\sigma =-1$. In the latter case the nonzero equilibria have supercritical Hopf bifurcations at $\kappa =2\lambda <0$.

// Target: dynamical-systems.bigb

For a reflection-symmetric vector field $\dot{u}=v+f_{30}{u}^3+f_{21}{u}^{2}v+\cdots$, $\dot{v}=g_{30}{u}^3+g_{21}{u}^{2}v+\cdots$, introduce $x=u$, $y=\dot{u}$. Differentiating gives $\dot{x}=y$ and a cubic acceleration with $x^{2}y$ coefficient $b=g_{21}+3f_{30}$. The contribution $3f_{30}$ comes from differentiating the first equation's cubic term. Further cubic near-identity transformations can remove nonresonant terms but cannot turn $b=0$ into a nonzero coefficient by a nonsingular scaling. This is a useful check before assigning a generic reflection-symmetric Bogdanov–Takens bifurcation to a calculated Taylor expansion.

For a vector field with linear part $\dot{u}=v$, $\dot{v}=0$ and cubic terms $a_i{u}^3+b_i{u}^{2}v+c_{i}u{v}^2+d_i{v}^3$, write $(u,v)=(x,y)+(h_1,h_2)$ with homogeneous cubic $h_i$. The transformed cubic field is $f_3+Ah-DhA(x,y{)}^T$, where $A(x,y{)}^T=(y,0{)}^T$. Taking the coefficients of $h_2$ to be $(-a_1,c_2/2,d_2,0)$ and those of $h_1$ to be $((b_1+c_2/2)/3,(c_1+d_2)/2,d_1,0)$ eliminates every cubic term in $\dot{x}$ and the $x{y}^2,y^3$ terms in $\dot{y}$. The remaining normal form is $\dot{x}=y$, $\dot{y}=P{x}^3+Q{x}^{2}y$ through cubic order. The invariant damping coefficient contains the contribution $3a_1$ from differentiating the first equation.

Near a nilpotent origin with $\sigma =1$, $\mu =0$, the weighted scaling $u_{\mathrm{old}}={\epsilon }^{2}u$, $v_{\mathrm{old}}=\epsilon v$, $\tau =\epsilon t$, ${\mu }_{\mathrm{old}}={\epsilon }^2\mu$, $\sigma =1+{\epsilon }^{2}s$ makes the leading cubic confinement flow $u^{\prime }=-sv+v^3/2$, $v^{\prime }=2u$. Direct differentiation proves conservation of $H=u^2+s{v}^2/2-v^4/8$. For $s>0$ the origin is a centre and $(0,\pm \sqrt{2s})$ are saddles. Their heteroclinic connections have $H=s^2/2$ and $u=\pm (2s-v^2)/(2\sqrt{2})$ for $|v|\le \sqrt{2s}$. Positive lower energy levels surround the centre; higher levels escape the potential well.