Existence of inverse Jacobi multipliers around Hopf points in \(\mathbb R^3\): emphasis on the center problem (Q424452)

It is considered the analytic three-dimensional system NEWLINE\[NEWLINE\begin{aligned} \dot x &=-y+ F_1(x,y,z),\\ \dot y &=x+ F_2(x,y,z),\\ \dot z &=\lambda z+ F_3(x,y,z),\end{aligned}\tag{1}NEWLINE\]NEWLINE where \(\lambda\in\mathbb{R}\setminus\{0\}\), \(F= (F_1, F_2, F_3): U\to\mathbb{R}^3\) is real analytic on the neighborhood of the origin \(U\subset\mathbb{R}^3\) with \(F(0)= 0\) and the Jacobian matrix satisfies \(DF(0)= 0\).NEWLINENEWLINE Theorem 1. The analytic system (1) has a center at the origin if and only if it admits a local analytic inverse Jacobi multiplier of the form \(V(x,y,z)= z+\cdots\) in a neighborhood of the origin in \(\mathbb{R}^3\). Moreover, when such \(V\) exists, the local analytic center manifold \(W^c\) is a subset of \(V^{-1}(0)\).NEWLINENEWLINE Theorem 2. Assume that the origin is a saddle focus for the analytic system (1). Then there exists a local \(C^\infty\) and non-flat inverse Jacobi multiplier of (1) having the expression \(V(x,y,z)= z(x^2+ y^2)^k+\cdots\) for some \(k\geq 2\). Moreover, there is a local \(C^\infty\) center manifold \(W^c\) such that \(W^c\subset V^{-1}(0)\).NEWLINENEWLINE Theorem 3. Consider the 3-parametric Lu family given by the quadratic system in \(\mathbb{R}^3\) given by NEWLINE\[NEWLINE\dot x= a(y- x),\quad \dot y= cy- xz,\quad \dot z=-bz+ xy,NEWLINE\]NEWLINE with parameters \(a,b,c\in\mathbb{R}^3\). The singularities \((\pm\sqrt{bc}, \pm\sqrt{bc},c)\) are centers if and only if \((a,b,c)\in L\), where the center variety \(L\) is the straight line \(L= \{(\alpha,\beta,\gamma)\in\mathbb{R}^3: \alpha\neq 0, \beta= 2\alpha, \gamma=\alpha\}\) of the parameter space. Moreover, when \((a,b,c)\in L\), \(V(x,y,z)= x^2- 2az\) is a global inverse Jacobi multiplier, \(\{V(x,y,z)= 0\}\) is a global center manifold for both singularities and the system reduced to the center manifold is Hamiltonian with Hamiltonian function \(H(x,y)= axy-{1\over 2} y^2-{1\over 8a} x^4\).