We reconsider the multiplier–accelerator model of business cycles, first introduced by Samuelson and then modified by many authors. The original simple model, besides damped oscillations, also leads to divergent oscillations. To avoid this, we introduce two different types of governmental expenditures leading a two-dimensional continuous piecewise linear map that can generate sustained oscillations (attracting cycles). The map is defined by three different linear functions in three different partitions of the phase plane, and this peculiarity influences the overall dynamics of the system. We show that, similar to the classical Samuelson model, there is a unique feasible equilibrium as well as converging oscillations. However, close to the center bifurcation value the attracting equilibrium coexists with attracting cycles of different periods, which lose stability via a center bifurcation simultaneously with the equilibrium. Moreover, we show that attracting cycles of particular type also exist when the equilibrium becomes an unstable focus. For several families of attracting cycles, by introducing the symbolic representation, we obtain boundaries of the related periodicity regions, associated with border collision bifurcations.

我们重新考虑了商业周期的乘数-加速器模型，该模型由Samuelson首次提出，然后被许多作者修改。原始的简单模型，除了阻尼振荡，还导致发散振荡。为避免这种情况，我们引入了两种不同类型的政府支出，它们导致了二维连续的分段线性映射，从而可能产生持续的振荡（吸引周期）。该图由相平面的三个不同分区中的三个不同线性函数定义，该特性影响系统的整体动力学。我们证明，与经典的Samuelson模型类似，存在唯一的可行平衡以及收敛振动。但是，接近中心分叉值时，吸引平衡与不同时期的吸引周期共存，通过中心分叉与平衡同时失去稳定性。此外，我们表明，当平衡成为不稳定的焦点时，也会存在特定类型的吸引循环。对于几个吸引循环族，通过引入符号表示，我们获得了与边界碰撞分叉相关的相关周期性区域的边界。