A nonlinear formulation of differential Galois theory was discussed here and here for three dimensional nonlinear systems (proof is on pages 6 – 10). For a two dimensional system, the following system of differential equations was considered: \begin{equation} \begin{split} \dot{x}_1&=X_1(x_1,x_2),\\ \dot{x}_2&=X_2(x_1,x_2), \end{split} \end{equation} such that $X_1,X_2$ are polynomial functions and $X_1(0,0)\neq0$. Let $K$ denote the differential field of rational functions, with constant field $\mathbb{C}$. The system is Liouvillian integrable iff the differential Galois group of $dx_2/dx_1=X_2(x_1,x_2)/X_1(x_1,x_2)$ over $K$ at $(0,0)$ is solvable. For the four dimensional system \begin{equation} \begin{split} \dot{x}&=X,\\ \dot{X}&=\sigma(Y-X-kx),\\ \dot{Y}&=-Y+rX-XZ,\\ \dot{Z}&=-bZ+XY. \end{split} \end{equation} does an equivalent formulation exist to determine whether Liouvillian integrability (or solvability by quadratures) holds where $k,r,\sigma\in\mathbb{R}^+$ and $b=1$? Any help would be much appreciated.

Note that when $k=0$, we have the closed system. \begin{equation} \begin{split} \dot{X}&=\sigma(Y-X),\\ \dot{Y}&=-Y+rX-XZ,\\ \dot{Z}&=-bZ+XY. \end{split} \end{equation}

As a corollary of Equation 6 and 12 in the second linked paper, the Lorenz-like system is at least partially integrable when \begin{equation} 4\sigma r=b^2\frac{(2n+1)^2}{4}-(\sigma-1)^2, \end{equation} for all $n\in\mathbb{Z}^+$.

The following theorem of Ishii may help for a complete integrability criterion. Assume that the system has a balance $\{{a},{p}\in\mathbb{Q}^4\}$ for which the Kovalevskaya matrix \begin{equation} K=JV_{a}-\mathop{\mathrm{diag}}(p), \end{equation} is semi-simple (such that $JV_{a}$ denotes the Jacobian of the system of ODEs at $a$). If the system is completely integrable, then the local general series are Puiseux series.

  • $\begingroup$ Are there any additional information about the constants $k$, $r$, $b$ and $\sigma$? $\endgroup$ Nov 28, 2021 at 2:55
  • $\begingroup$ Thanks for your reply. I am particularly interested in the system when $b=1$. Other than that, the constants assume positive real values $k,r,\sigma\in\mathbb{R}^+$. $\endgroup$
    – UNOwen
    Nov 28, 2021 at 3:23
  • $\begingroup$ We need to study the eigenvalues at equilibrium points. This is too difficult with all these undetermined constants. $\endgroup$ Nov 28, 2021 at 20:10
  • $\begingroup$ When $r>1+k-k/(\sigma+1)$, the system is asymptotically unstable (a real part of $\lambda$ is greater than zero for the linearisation). The only equilibrium occurs at $(0,0,0,0)$ and I am more interested in proving Liouvillian integrability (or if the system is unsolvable). $\endgroup$
    – UNOwen
    Nov 29, 2021 at 1:47
  • 1
    $\begingroup$ As you can see in theorem 5.2 of Goriely's "Integrability and nonintegrability of dynamical systems" (link: worldscientific.com/worldscibooks/10.1142/3846), the eigenvalues (more specific a resonance condition on them) play a big role in existence of first integrals. $\endgroup$ Nov 29, 2021 at 15:43


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