| Contents |
| In discontinuous piecewise linear systems with two zones, it is shown that the existence of a focus in one zone is sufficient to get three nested limit cycles independently on the dynamics of the another linear zone. Starting from a situation with only one hyperbolic limit cycle, other two limit cycles are obtained by combining of a boundary focus bifurcation and a pseudo-Hopf bifurcation.
After some generic assumptions, and taking $\gamma_L\gamma_R< 0$, $m_L=i$, $a_L \le 0$ (a left focus), we show our results for $a_R < 0$ and $m_R\in\{i,0,1\}$ in the family of systems
$$
\mathbf{\dot{x}}=\left(
\begin{array}
{cr}
2\gamma_{\{L,R\}} & -1\\
\gamma_{\{L,R\}}^2 -m_{\{L,R\}}^2 & 0
\end{array}
\right) \mathbf{x}-\left(
\begin{array}
{c}%
-b_{\{L,R\}}\\
a_{\{L,R\}}
\end{array}
\right),
$$
where the subscripts $\{L,R\}$ indicate the left/right half planes. |
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