1. II.A Business cycle Model A forced van der Pol oscillator model of business cycle was chosen as a prototype model to study the complex economic dynamics. (8) When, (8) is the famous van der Pol equation. For VDP equation, the origin in the phase space is the only equilibrium solution, which is an unstable fixed point if. In the case that the exogenous forcing is not equal to zero, (8) can evolve into a periodic attractor or an aperiodic solution when we vary the control parameters and.

2. II.A Business cycle Model Chaotic dynamics has been numerically studied [12]. [12] A. C.-L Chian, F. A. Borotto, E. L. Rempel, and C. Rogers, Attractor merging crisis in chaotic business cycles, Chaos, Solitons and Fractals, 2005, 24: 869-875. A rigorous mathematical proof for the existence of chaos is needed.

3. ☺ Our work: A rigorous proof for existence of chaos from mathematical point of view is given. II.A Business cycle Model

4. For the convenience of discussion in the following, we rewrite (8) as follows which are four coupled first-order differential equations. (9)

5. II.A Business cycle Model Limit cycle and Chaos A rigorous proof for the existence of topological horseshoe will be given. Fig. 16 (a) A limit cycle of period-1 in the state space for. (b) A chaotic attractor in the state space for. (a) (b)

6. II.A Business cycle Model Fig. 17. The attractor of business cycle model with and the Poincaré section. Firstly, we consider the plane which is shown in Fig. 17.

7. II.A Business cycle Model On this Poincaré section, after many trial numerical simulations we choose a proper cross section with its four vertices being and will study the corresponding Poincaré map This Poincaré map is defined as follows: for each point, is chosen to be the sixth return intersection point with under the flow of business cycle model with initial condition.

8. II.A Business cycle Model Theorem 3. The Poincaré map corresponding to the cross-sections has the property that there exists a closed invariant set for which is semi-conjugate to the 2-shift map. Therefore, This implies that for the parameter, the business cycle system (8) has positive topological entropy. Fig.18. The cross section and its image under the sixth return Poincaré map with. The drawing that the arrow points at is a magnification of a part of the original diagram.

9. II.A Business cycle Model Fig. 19 subset a and its image Proof. According to the Topological horseshoe Theorem, to prove this statement, we must find two mutually disjointed subsets of, such that a -connected family with respect to them is existed. After many attempts, the first subset is denoted by with and be its left and right edge, respectively, as shown in Fig. 19. Under the sixth return Poincaré map, the subset is mapped to its image with mapped to and mapped to. It is easy to see that is on the left side of the edge, and is on the right side of the edge. In this case, we say that the image lies wholly across the quadrangle with respect to and.

10. II.A Business cycle Model Fig. 20 subset b and its image The second subset is shown in Fig. 20, with and be its left and right edge, respectively. Just like the situation for subset, the subset is mapped to under the Poincaré map, is mapped to which is on the left side of the edge, and is mapped to which is on the right side of the edge. Thus the image lies wholly across the quadrangle with respect to and as well.

11. II.A Business cycle Model Conclusions: Given a rigorous computer-assisted proof for the existence of chaos in this business cycle model

12. Let enter into the magic chaos world

13. 

14. Thanks for your coming