First Results for Planar Continuous Systems with Three Zones

Abstract

In this chapter, we consider Liénard systems with three linear zones (2.5) satisfying (2.6) and (2.7), which we rewrite explicitly for ease of reading. Therefore, we are given the following system

$$\displaystyle \begin{aligned} \begin{array}{rcl} \dot{x}& = & F(x)-y,\\ \dot{y}& = & g(x)-a, \end{array} \end{aligned} $$

(4.1)

where

$$\displaystyle \begin{aligned} F(x)=\left\{\begin{array}{ll} t_{L}(x+1)-t_{C} & \text{ if }x < -1,\\ t_{C}x & \text{ if } |x| \leq 1,\\ t_{R}(x-1)+t_{C} & \text{ if }x > 1, \end{array}\right. \end{aligned} $$

(4.2)

and

$$\displaystyle \begin{aligned} g(x)=\left\{\begin{array}{ll} d_{L}(x+1)-d_{C} & \text{ if }x < -1,\\ d_{C}x & \text{ if }|x| \leq 1,\\ d_{R}(x-1)+d_{C} & \text{ if }x > 1, \end{array}\right. \end{aligned} $$

(4.3)

so that the phase plane is divided into three zones, possibly with different linear dynamics, namely the central zone $\mathcal {S}_C = \{(x,y)\in \mathbb {R}^2: -1< x < 1\}$, and the external zones $\mathcal {S}_{R} = \{(x,y)\in \mathbb {R}^2 : x > 1\}$, $\mathcal {S}_{L} = \{(x,y)\in \mathbb {R}^2 : x <-1\}$, separated by the straight lines $\Sigma _{1} = \{(x,y)\in \mathbb {R}^2: x = 1\}$ and $\Sigma _{-1} = \{(x,y)\in \mathbb {R}^2: x = -1\}$.