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Article Contents

# How interest rate influences a business cycle model

• * Corresponding author: Jingli Ren

This research is supported by the National Natural Science Foundation of China (11771407), and the Innovative Research Team of Science and Technology in Henan Province (17IRTSTHN007)

• We study the effect of interest rate on phenomenon of business cycle in a Kaldor-Kalecki model. From the information of the People's Bank of China and the Federal Reserve System, we know the interest rate is not a constant but with remarkable periodic volatility. Therefore, we consider periodically forced interest rate in the model and study its dynamics. It is found that, both limit cycle through Hopf bifurcation in unforced system and periodic solutions generated by period doubling bifurcation or resonance in periodically forced system, can lead to cyclical economic fluctuations. Our analysis reveals that the cyclical fluctuation of interest rate is one of a key formation mechanism of business cycle, which agrees well with the pure monetary theory on business cycle. Moreover, this fluctuation can cause chaos in a business cycle system.

Mathematics Subject Classification: Primary: 34C23, 37G15; Secondary: 37D45.

 Citation:

• Figure 1.  S-shape of $I(Y)$

Figure 2.  Fluctuations of interest rate. (a) Interest rate of FED from September 1986 to February 2007; (b) Interest rate of PBC from July 1998 to October 2015

Figure 3.  Phase portraits of a fold bifurcation of system (5) for $\alpha = 2$, $\beta = 0.6$, $\gamma = 0.283368$, $\delta = 0.3$, $m = 2.294$, $n = 4.612$, $C_{0} = 0.2$. (a) $d = 0.6$. (b) $d = 0.750017$. (c) $d = 0.8$

Figure 4.  Bifurcation diagram of system (5) in $d-\gamma$ plane for $\alpha = 3$, $\beta = 0.6$, $\delta = 0.3$, $m = 3$, $n = 5.6$, $C_{0} = 0.6$

Figure 5.  Phase portrait of Hopf bifurcation of system (5) for $\alpha = 3$, $\beta = 0.6$, $d = 0.195$, $\delta = 0.3$, $m = 3$, $n = 5.6$, $C_{0} = 0.6$. (a) Supercritical Hopf bifurcation for $\gamma = 1.175$. (b) Subcritical Hopf bifurcation for $\gamma = 0.66$

Figure 6.  Time series of limit cycles corresponding to the cases in Fig. 5. (a) Stable limit cycle generated by supercritical Hopf bifurcation for $\gamma = 1.175$. (b) Stable limit cycle in system (5) for case $\gamma = 0.66$. (c) Unstable limit cycle generated by subcritical Hopf bifurcation for $\gamma = 0.66$

Figure 7.  (b) Bifurcation diagrams of the forced system in with $\alpha = 3, \beta = 0.6, \gamma = 0.4, \delta = 0.3, m = 3, n = 5.6, C_{0} = 0.2$. (c) partial enlargements of (b)

Figure 8.  (a) Bifurcation diagrams of the forced system in with $\alpha = 3, \beta = 0.6, \gamma = 0.7, \delta = 0.3, m = 3.6, n = 10.58, C_{0} = 0.2$. (b) partial enlargements of (a)

Figure 9.  (a) Bifurcation diagrams of the forced system in $(\epsilon - d)$ plane with $\alpha = 5, \beta = 0.6, \gamma = 0.85, \delta = 0.3, m = 3.211, n = 3.368, C_{0} = 1$

Figure 10.  (a) Bifurcation diagrams of the forced system in $(\epsilon - d)$ plane with $\alpha = 5, \beta = 0.6, \gamma = 0.8, \delta = 0.3, m = 3.211, n = 3.368, C_{0} = 1$

Figure 11.  Phase portrait of different solutions. (a) A stable period-two orbit for $d = 0.3, \epsilon = 0.8$. (b), (c) Poincaré section and Time series of the stable period-two orbit. (d) A stable period-four orbit for $\ d = 0.18, \epsilon = 0.8$. (e), (f) Poincaré section and Time series of the stable period-four orbit. (g) Phase portrait of torus for $d = 0.2, \epsilon = 0.1$. (h), (i) Poincaré section and Time series of the torus

Figure 12.  Phase portrait of chaotic attractor. (a) Chaotic attractor through cascade of period doublings for $d = 0.25, \epsilon = 0.5$. (b), (c) Corresponding Poincaré section and Time series of (a). (d) Chaotic attractor through torus destruction for $d = 0.2, \epsilon = 0.2$. (e), (f) Corresponding Poincaré section and Time series of (d)

Figure 13.  (a) Spectrum of largest Lyapunov exponents for $\epsilon = 0.5$. (b) Spectrum of largest Lyapunov exponents for $\epsilon = 0.2$

Figure 14.  (a) Bifurcation diagram in $(Y-d)$ plane for $\epsilon = 0.5$. (b) Bifurcation diagram in $(Y-d)$ plane for $\epsilon = 0.2$

Figure 15.  (a) A stable period-three orbit for $\gamma = 0.85,d = 0.45, \epsilon = 0.2$. (b), (c) Poincaré section and Time series of the stable period-three orbit. (d) A stable period-five orbit for $\gamma = 0.8,d = 0.235, \epsilon = 0.7$. (e), (f) Poincaré section and Time series of the stable period-five orbit

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