An Elegant Analytical Resolution of the Sprott-Zeraoulia Conjecture for Three-Dimensional Quadratic Differential Systems with Symmetric Jacobian Matrices

Abstract

The Sprott--Zeraoulia conjecture, formulated on the basis of numerical experiments, states that three-dimensional quadratic continuous-time systems with symmetric Jacobian matrices cannot exhibit chaotic behavior. We give an analytical resolution of this conjecture in terms of the principal precise notions of chaos used for continuous dynamical systems. The starting point is the classical, but decisive, fact that symmetry of the Jacobian on $\mathbb{R}^3$ is equivalent to a global gradient representation. Hence every system in the conjectured class can be written as $\dot{x}~=~\nabla V(x)$, where $V$ is a polynomial of degree at most three, and the conjecture becomes a problem about cubic polynomial gradient flows. For bounded trajectories, the monotonicity property of gradient systems first confines every omega-limit set to a critical level of $V$. When the equilibria on that level are isolated, convergence follows from the connectedness of the omega-limit set. In the general case, which may include curves or surfaces of equilibria, the \L{}ojasiewicz gradient inequality implies finite length and convergence of every bounded forward trajectory to a single equilibrium. Consequently, compact invariant sets support no nonstationary recurrence, positive-entropy measure, topological transitivity, Smale horseshoe, or Devaney, Auslander--Yorke, Li--Yorke, mean Li--Yorke, and distributional chaos. Solutions escaping to infinity are treated through the Poincar\'e compactification. Finally, we connect the gradient formulation with earlier Darboux-theoretic criteria: invariant algebraic surfaces satisfy $\langle\nabla V,\nabla f\rangle=Kf$, and an invariant affine plane with constant cofactor produces an exact tangential--normal decomposition of the potential and the flow.