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differential equations using the results of the paper [9]. In this section, we will demonstrate numerically the formation of chaotic business cycles in the Kaldor-Kalecki model with time delay. 

Let us take into account the system, 

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Equation B$_{1}$ is the chaotic Van der Pol oscillator, which is used as the generator system in (5.15). Van der Pol type equations have played a role in economic modelling [40, 41, 58]. It is shown by Parlitz and Lauterborn [69] that equation B$_{1}$ is chaotic through period-doubling cascade. The process of period- doubling is described by Thompson and Stewart [82]. This implies that there are infinitely many unstable periodic solutions of B$_{1}$, all with different periods. Due to the absence of stability, any solution that starts near the periodic motions behaves irregularly . We will interpret the solution x ( t ) as an irregular productivity shock. 

System B$_{2}$ is the Kaldor-Kalecki model and it is the result of the perturbation of the model (3.10) of an aggregate economy with a productivity shock. We will observe numerically the appearance of a chaotic business cycle, and in particular, the entrainment by chaos of the limit cycle of system (3.10), in the next simulations. 

Let us take τ = 5 . 5 in B$_{2}$ so that the system possesses an orbitally stable limit cycle in the absence of perturbation [87]. We make use of the solution x ( t ) of B$_{1}$ with x (0) = 1 . 1008 , x $^{′}$(0) = − 1 . 5546 , and present in Figure 11 the solution of B$_{2}$ with the initial condition Y ( t ) = u ( t ) and K ( t ) = v ( t ) for t ∈ [ − τ, 0] , where u ( t ) = − 0 . 057 and v ( t ) = 0 . 063 are constant functions. Figure 11 reveals that the dynamics of B$_{2}$ exhibits chaotic business cycles. This result shows that our theory of chaotic business cycles can be extended to systems with time delay. 

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