Nonchaotic Behavior in Quadratic Three-Dimensional Differential Systems with a Symmetric Jacobian Matrix

Abstract

In this paper, we give an algebraic criterion to determine the nonchaotic behavior for polynomial differential systems defined in ℝ3 and, using this result, we give a partial positive answer for the conjecture about the nonchaotic dynamical behavior of quadratic three-dimensional differential systems having a symmetric Jacobian matrix. The algebraic criterion presented here is proved using some ideas from the Darboux theory of integrability, such as the existence of invariant algebraic surfaces and Darboux invariants, and is quite general, hence it can be used to study the nonchaotic behavior of other types of differential systems defined in ℝ3, including polynomial differential systems of any degree having (or not having) a symmetric Jacobian matrix.