3.Minsky cycle as a Lotka-Volterra like system
Minsky himself did not formulate formal mathematical models that reflect his basic idea on the“financial instability hypothesis”.However, a lot of mathematical models that were inspired by Minsky's idea have been produced up to now since the seminal paper by Taylor and O'Connell（1985）was published.
In this section, we shall summarize the essence of the model that was presented by Asada（2001）as a typical example of such mathematical models.This approach interpret the“Minsky cycle”as a kind of Lotka-Volterra system of mathematical biology, which is based on the dynamic interaction of predator and prey.
The reduced form of a system of equations that was presented by Asada（2001）consists of the following two-dimensional system of differential equations.
where a dot over the symbol is time derivative, and the meanings of the symbols are as follows.
d=D/K=private debt-capital ratio.y=Y/K=income-capital ratio.D=stock of the real debt of the private firms.K=real capital stock that is owned by the private firms.Y=real national income.α=parameter that reflects the adjustment speed of the disequilibrium in the goods market.Variable y is used as a surrogate variable that reflects the degree of the utilization of capital stock and the rate of employment（1-rate of unemployment）in the labor market.In this simplified version of the model, the fixed price is assumed, and the government's economic activity and international transaction are abstracted from.
Furthermore, each partial derivative in this model becomes as follows because of some economic reasons（cf.Asada 2001）.
Eq.（1）（i）can be derived from the fact that the firms'investment expenditure that exceeds the firms'internal finance must be debt-financed.Eq.（1）（ii）expresses the“quantity adjustment”process in the goods market disequilibrium, which implies that the rate of utilization of the capital stock fluctuates according as the excess demand in the goods market per capital stock is positive or negative.In both equations（1）（i）and（1）（ii）, the following investment function is incorporated.
where 
of investment（rate of capital accumulation）, ρ=nominal rate of interest, πe=expected rate of price inflation, and ρ-πe=expected real rate of interest.Asada（2001）derived this type of investment function with debt effect from the expected profit maximization behavior of the firms by using the“principle of increasing risk”due to Kalecki（1937,1971）, which means that firms'risk increases as the investment expenditure increases.
In the simplified model in this section, the fixed price is assumed so that we have πe=0.Furthermore, in this section, we assume that the central bank always acts to keep the nominal rate of interestρto be constant, which means the lack of the active monetary policy.Therefore, πe and ρ do not enter into the system of equations（1）as variables.
In this model, the change of the private debt-capital ratio
becomes an increasing function of the income-capital ratio y（f12）＞0, because the increase of y induces the increase of the debt financing through the rise of the investment expenditure g.On the other hand, the change of the income-capital ratio
becomes a decreasing function of the debtcapital ratio d（f12）＜0, because the increase of d induces the decrease of the investment expenditure, which contributes to the reduction of the excess demand in the goods market through the decrease of the effective demand.These characteristics imply that we can consider the system of equations（1）as a“Lotka-Volterra”like system, in which the variable d plays the role of the predator and the variable y plays the role of the prey.
Asada（2001）proved that the cyclical fluctuations occur and in particular, the closed orbits around the equilibrium point exist at some range of the parameter valueαby means of the Hopf bifurcation theorem.Figure 1 illustrates a typical closed orbit of income and private debt that is produced in this system and the time trajectories of two variables corresponding to four phases of the closed orbit.The cyclical fluctuations of income and private debt in Figure 1 can be considered as a typical example of the“Minsky cycle”.
Figure 1 Emergence of Minsky cycle（Source:Asada,2001, p.81）
Figure 2 illustrates two types of bifurcation that can emerge in this system.Figure 2（a）is an example of the“subcritical”Hopf bifurcation. In this case, the closed orbit exists in the range of the parameter valueαat which the equilibrium point is locally stable and the closed orbit itself is unstable.Figure 2（b）is an example of the“supercritical”Hopf bifurcation.In this case, the closed orbit exists in the range of the parameter valueα at which the equilibrium point is locally unstable and the closed orbit itself is stable.
Figure 2 Subcritical Hopf bifurcation（a）and supercritical Hopf bifurcation（b）（Source:Asada,2001, p.83）
Both of the above mentioned two types of Hopf bifurcation are economically meaningful.The“subcritical”case corresponds to the“corridor stability”in the sense of Leijonfufvud（1973）, which means that this system is immune from small shocks that can contain the initial position inside the“corridor”（stable region）, but it is vulnerable to large shocks that leave the initial condition outside the“corridor”.In the“supercritical”case, the time trajectory of the variables that starts from the initial condition other than equilib-rium point converges to the limit cycle, and the cyclical fluctuations persist indefinitely.