Conclusion

Abstract

In this monograph, we adopted a forced oscillator model of nonlinear economic cycles as a prototype to model the fundamental dynamical behaviors of a complex economic system, which exhibits multistability (coexistence of attractors), multiscale (power-law dependence on frequency), and coexistence of regularity and irregularity (order and chaos). It is important to point out that although we have selected the van der Pol model for its mathematical simplicity and its wide interest in economics, in view of the universal mathematical properties of nonlinear dynamical systems, the dynamical characteristics investigated in this simple model is actually applicable to other more sophisticated economic scenarios. We succeeded in characterizing the anatomy of a complex economic system by classifying its structure and dynamics. In terms of the system structure, our analysis shows that a complex economic system is composed of a hierarchy of stable and unstable structures, namely, stable and unstable manifolds of a fixed point in the state space and in the Poincaré section, stable and unstable periodic orbits, stable and unstable manifolds of a chaotic saddle, stable (periodic) and unstable (chaotic) attractors. In particular, we showed that unstable periodic orbits are the building blocks of chaotic saddles and chaotic attractors; moreover, chaotic saddles are embedded in a chaotic attractor and are responsible for the transient motion preceding the convergence to an attractor (periodic or chaotic).