  This paper presents a new canonical duality methodology for solving general
nonlinear dynamical systems. Instead of the conventional iterative methods, the
discretized nonlinear system is first formulated as a global optimization
problem via the least squares method. The canonical duality theory shows that
this nonconvex minimization problem can be solved deterministically in
polynomial time if a global optimality condition is satisfied. The so-called
pseudo-chaos produced by Runge-Kutta type of linear iterations are mainly due
to the intrinsic numerical error accumulations. Otherwise, the global
optimization problem could be NP-hard and the nonlinear system can be really
chaotic.
  A conjecture is proposed, which reveals the connection between chaos in
nonlinear dynamics and NP-hardness in computer science. The methodology and the
conjecture are verified by applications to the well-known logistic equation, a
forced memristive circuit and the Lorenz system. Computational results show
that the canonical duality theory can be used to identify chaotic systems and
to obtain realistic global optimal solutions in nonlinear dynamical systems.
