x ′′ + 5( x 2 − 1) x ′ + x = 5 cos(2 . 467 t )   } B$_{1}$ Y ′ = 1 . 5 [tanh( Y ) − 0 . 25 K − (4 / 3) Y ] + 0 . 0045 x ( t )   K ′ = tanh( Y ( t − τ )) − 0 . 5 $_{K.}$   (5.15) 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  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. Figure 11: The appearance of chaotic business cycle in the Kaldor-Kalecki model B$_{2}$. −0.15 −0.1 −0.05 0 0.05 0.1 0.15 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 Y K 17