# On the solvability of a nonlinear differential system

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: $$$$\begin{split} \dot{x}_1&=X_1(x_1,x_2),\\ \dot{x}_2&=X_2(x_1,x_2), \end{split}$$$$ such that $$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)$$ is solvable. For the four dimensional system $$$$\begin{split} \dot{x}&=X,\\ \dot{X}&=\sigma(Y-X-kx),\\ \dot{Y}&=-Y+rX-XZ,\\ \dot{Z}&=-bZ+XY. \end{split}$$$$ does an equivalent formulation exist to determine whether Liouvillian integrability (or solvability by quadratures) holds where $$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. $$$$\begin{split} \dot{X}&=\sigma(Y-X),\\ \dot{Y}&=-Y+rX-XZ,\\ \dot{Z}&=-bZ+XY. \end{split}$$$$

As a corollary of Equation 6 and 12 in the second linked paper, the Lorenz-like system is at least partially integrable when $$$$4\sigma r=b^2\frac{(2n+1)^2}{4}-(\sigma-1)^2,$$$$ for all $$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 $$$$K=JV_{a}-\mathop{\mathrm{diag}}(p),$$$$ is semi-simple (such that $$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.

• Are there any additional information about the constants $k$, $r$, $b$ and $\sigma$? Nov 28 at 2:55
• Thanks for your reply. I am particularly interested in the system when $b=1$. Other than that, the constants assume positive real values $k,r,\sigma\in\mathbb{R}^+$. Nov 28 at 3:23
• We need to study the eigenvalues at equilibrium points. This is too difficult with all these undetermined constants. Nov 28 at 20:10
• 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). Nov 29 at 1:47
• 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. Nov 29 at 15:43