In this work, we present a monoparametric family of piecewise linear systems to generate multiscroll attractors through a polynomial family defined by path curves that connect to the roots. The idea is to define path curves where the roots of a polynomial can take values by determining an initial and a final polynomial. As a consequence, structural stability and bifurcation of the system can be obtained. Structural stability is obtained by preserving the same stability of the initial and final polynomials. However, the system bifurcates by changing the stability of the final polynomial with respect to the initial polynomial. The aim is achieved by the design of a piecewise linear controller that is applied to affine linear systems. Our results are mathematically proved and numerical examples are also provided to illustrate the approach.

中文翻译：

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通过多项式的路径连接集生成滚动吸引子的分段线性系统的单参数族

在这项工作中，我们提出了一个单参数的分段线性系统族，通过连接到根的路径曲线定义的多项式族来生成多滚动吸引子。这个想法是定义路径曲线，其中多项式的根可以通过确定初始多项式和最终多项式来取值。因此，可以获得系统的结构稳定性和分叉。通过保持初始多项式和最终多项式的相同稳定性来获得结构稳定性。然而，系统通过改变最终多项式相对于初始多项式的稳定性而分岔。该目标是通过设计应用于仿射线性系统的分段线性控制器来实现的。我们的结果在数学上得到了证明，并且还提供了数值示例来说明该方法。