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On the limit cycles of a class of piecewise linear differential systems in ${ℝ}^{4}$ with two zones. (English) Zbl 1251.37027

The authors study the maximum number of limit cycles of the 4-dimensional continuous piecewise linear vector field

${x}^{\text{'}}={A}_{0}x+\epsilon F\left(x\right),$

where $\epsilon$ is a small parameter, ${A}_{0}$ is an elliptic matrix with eigenvalues $+-i$ (in the Jordan normal form) and $F:{ℝ}^{4}\to {ℝ}^{4}$ is given by $F\left(x\right)=Ax+\phi \left({k}^{T}x\right)b$ for a matrix $A$, $k,b\in {ℝ}^{4}\setminus \left\{0\right\}$ and $\phi :ℝ\to ℝ$ is the piecewise linear function such that $\phi \left(s\right)=0$ for $s\in \left(\infty ,1\right)$ and $\phi \left(s\right)=ms$ for $s\in \left[1,\infty \right)$.

The main result is that the upper bound for the number of limit cycles of the above defined system is three, and there are systems having exactly three limit cycles. The proof is based on the non-smooth averaging theory.

##### MSC:
 37C10 Vector fields, flows, ordinary differential equations 34C23 Bifurcation (ODE) 34A36 Discontinuous equations
##### Keywords:
limit cycles; averaging theory; piecewise linear systems
##### References:
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