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Algebraic determination of limit cycles in a family of three-dimensional piecewise linear differential systems. (English) Zbl 1232.34047

The paper is concerned with the dynamics of a system

$\begin{array}{cc}\hfill \stackrel{˙}{𝐱}& =A𝐱-𝐛\phantom{\rule{2.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{S}_{-}\cup {{\Gamma }}_{-},\hfill \\ \hfill \stackrel{˙}{𝐱}& =B𝐱\phantom{\rule{2.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{{\Gamma }}_{-}\cup {S}_{0}\cup {{\Gamma }}_{+},\hfill \\ \hfill \stackrel{˙}{𝐱}& =A𝐱+𝐛\phantom{\rule{2.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{S}_{+}\cup {{\Gamma }}_{+},\hfill \end{array}$

where ${S}_{0}=\left\{\left(x,y,z\right)\in {ℝ}^{3}:-1 ${S}_{±}=\left\{\left(x,y,z\right)\in {ℝ}^{3}:±x>1\right\},$ and

${{\Gamma }}_{±}=\left\{\left(x,y,z\right)\in {ℝ}^{3}:x=±1\right\},$ $B=A+{\mathrm{𝐛𝐜}}^{\mathrm{T}},$ $A$ is a constant $3×3$ matrix, $𝐛,𝐜\in {ℝ}^{n}·$ This is a particular case of the Lur’e system in control theory. The authors develop algebraic tools for the study of symmetric periodic orbits using closing equations for symmetric periodic orbits. The principal result in the paper determines all the symmetric periodic orbits having two points in the plane ${{\Gamma }}_{+}$. Computations leading to the first harmonic bifurcation diagram are provided. In the final part of the paper, a particular case of the system under study, a dimensionless Chua oscillator is studied.

##### MSC:
 34C05 Location of integral curves, singular points, limit cycles (ODE) 34C25 Periodic solutions of ODE