A new proof for the existence of topological horseshoe in a business cycle model

Complex chaotic dynamics is studies in the van der Pol oscillator model of business cycle forced by a sinusoidal function. The famous topological horseshoe theorem is applied to prove the existence of chaos from the mathematical viewpoint in this business cycle model. A rigorous proof for the existence of topological horseshoe is given. This technique combines the topological theory with the computer-assisted computations. A proper Poincaré section is first chosen to obtain the corresponding Poincaré map, which is proved to be semi-conjugate to the 2 -shift map. This result implies that the business cycle model has positive topological entropy, and thus is definitely chaotic.