Generalized Lorenz-Type Systems

Exact general solutions of the Lorenz system of differential equations with arbitrary parameters are unknown. We first present exact particular solutions of this system for special values of the parameters, in terms of Jacobian elliptic functions, but no sensitive dependence on the initial conditions leading to chaotic behavior. is exhibited by these solutions. We next consider two generalizations of the Lorenz system involving quadratic and cubic terms and obtain their exact general solutions in terms of Bessel and modified Bessel functions, exhibiting sensitive dependence on initial conditions for certain parameter ranges. Finally, a generalized system involving arbitrary powers is reduced to the general Duffing equation with damping, which can be solved exactly when some of the parameters are interconnected. The exact solutions presented, showing precisely how chaotic behavior occurs, are useful for testing conjectures about such differential systems, as well as testing the validity and accuracy of approximate computer solutions, for which numerical errors may grow exponentially fast