Global Optimal Trajectory in Chaos and NP-Hardness

TL;DR

This paper introduces a canonical duality approach to solve nonlinear dynamical systems, linking chaos with NP-hardness, and demonstrates its effectiveness on classical chaotic models.

Contribution

It proposes a novel canonical duality methodology that can determine global optima in nonlinear systems and connects chaos with NP-hardness in computational complexity.

Findings

The method can identify chaotic systems accurately.

Global solutions are obtained efficiently for certain nonlinear systems.

The approach reveals a link between chaos and NP-hardness.

Abstract

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.