system admits an asymptotically stable equilibrium. The term 0 . 6 y$_{1}$ ( t ) describes the effect exports from Country 2 to Country 1 , modelled as a function of the income of Country 1 , y$_{1}$ ( t ) , have on the rate of change in the income of Country 2 . The function ν$_{2}$ ( t, ζ ) reflects productivity shocks in Country 2 , which is a binary variable. The higher value of ν$_{2}$ stands for faster productivity growth, and the lower value for slower productivity growth, which leads to slower output growth. 

Since the periodic motions that are embedded in the chaotic attractor of system A$_{2}$ with µ = 3 . 8 and the function (4.13) have incommensurate periods, one can confirm using the results of [8] that system A$_{3}$ is chaotic with infinitely many quasi-periodic solutions in the basis. This will be shown through simulations in Figure 9. 

System A$_{4}$ describes the aggregate economy of Country 3 . It is obtained by perturbing system (3.8) with the solutions of A$_{2}$. It is a replicator with respect to system A$_{2}$, while the term 2( y$_{1}$ ( t ) + 0 . 5) is the input. This term represents the effect of exports from Country 3 to Country 1 , modelled as a function of income in Country 1 , y 1( t ) , on the rate of growth of income in Country 3 . 

In the absence of perturbations, A$_{4}$ possesses an orbitally stable limit cycle [72]. Theorem 2.1 implies that system A$_{4}$ admits chaotic business cycles, provided that the value of the parameter µ = 3 . 8 is used in system A$_{2}$. Since the orbitally stable cycle of system (3.8) occurs through a Hopf bifurcation, one can talk about the bifurcation of the cyclic chaos . 

System A$_{5}$ models the dynamics of Country 4 . It is constructed by perturbing the Kaldor model (3.6) with the solutions of A$_{4}$, or in economic terms, by perturbing the aggregate economy with exports from Country 4 to Country 3 , which are a fraction of the income of Country 3 , Y ( t ) . The eigenvalues of the associated linear system are − 1 / 5 and − 1 / 10 . In the absence of the perturbation term 0 . 01 Y ( t ) , the system possesses an asymptotically stable equilibrium, for sufficiently small values of a$_{3}$. We will make use of system A$_{5}$ to demonstrate the attraction of chaotic business cycles . 

## 4.2 Simulations 

In this part of the paper, we will demonstrate numerically the chaotic behavior in system (4.11). In what follows, we will use a$_{1}$ = 3 × 10 − $^{6}$, a$_{2}$ = 10 − $^{6}$, a$_{3}$ = 5 × 10 − $^{6}$, µ = 3 . 8 and κ$_{0}$ = 0 . 63 . 

Let us start with system A$_{2}$. Setting the initial data y$_{1}$ ( t$_{0}$ ) = 0 . 12 , k$_{1}$ ( t$_{0}$ ) = 0 . 08 , where t$_{0}$ = 0 . 63 , we graph in Figure 2 the y$_{1}$ coordinate of system A$_{2}$. It is seen in the figure that system A$_{2}$ behaves chaotically. 

To show the extension of chaos by system A$_{3}$, we use the solution in Figure 2 as perturbation in system A$_{3}$ and present in Figure 3 the time series of the y$_{2}$ coordinate of $_{A$_{3}$}$. The initial data y$_{2}$ ( t$_{0}$ ) = 0 . 95 , k$_{2}$ ( t$_{0}$ ) = 0 . 38 , where t$_{0}$ = 0 . 63 , is used in the simulation. Figure 3 reveals that the chaos of system A$_{2}$ is extended such that the system A$_{3}$ also possesses chaos. In order to confirm the extension of chaos once more, we depict in Figure 4 the projection of the trajectory of the coupled Kaldor system A$_{2}$ − A$_{3}$, 

