Theory and Modern Applications
Table 2 The relations among the key parameters ( i.e. θ , \(\pmb{V_{L}}\) , and τ ), the signs of \(\pmb{A_{h_{1}}}\) and \(\pmb{A_{h}}\) and the domains of the Poincaré map \(\pmb{{\mathcal{P}}(y_{i}^{+})}\)
From: Global dynamics of a state-dependent feedback control system
Cases | \(\boldsymbol {V_{L}}\) | \(\boldsymbol {\theta_{1}V_{L}}\) | \(\boldsymbol {A_{h}}\) and \(\boldsymbol {A_{h_{1}}}\) | \(\boldsymbol {{\mathcal{P}}(y_{i}^{+})}\) |
|---|---|---|---|---|
(C1) | \(V_{L}< x_{\min}^{h_{2}}\) | \(x_{3}^{*}\leq\theta_{1}V_{L}\leq x_{\min}\) | \(A_{h}\leq0\), \(A_{h_{1}}\geq 0\) | \(y_{i}^{+} \in Y_{D}^{h_{1}}\) |
\(x_{\min}< \theta_{1}V_{L}< x_{\mathrm{mid}}\) | \(A_{h}> 0\), \(A_{h_{1}}\geq0\) | |||
\(x_{\mathrm{mid}}\leq\theta_{1}V_{L}\leq x_{1}^{*}\) | \(A_{h}\leq0\), \(A_{h_{1}}\geq0\) | |||
\(\theta_{1}V_{L}< x_{3}^{*}\) | \(A_{h}\leq0\), × | \(y_{i}^{+} \in Y_{D}^{1}\) | ||
\(x_{1}^{*}<\theta_{1}V_{L}\) | ||||
\(x_{\min}^{h_{2}}\leq V_{L}\) | \(x_{3}^{*}\leq \theta_{1}V_{L}\leq x_{1}^{*}\) | \(A_{h}\leq0\), \(A_{h_{1}}\geq0\) | \(y_{i}^{+} \in Y_{D}^{h_{1}}\) | |
\(\theta_{1}V_{L}< x_{3}^{*}\) | \(A_{h}\leq0\), × | \(y_{i}^{+} \in Y_{D}^{1}\) | ||
\(x_{1}^{*}<\theta_{1}V_{L}\) | ||||
(C2) |  | \(x_{4}^{*}<\theta_{1}V_{L}\) | \(A_{h}>0\), × | \(y_{i}^{+} \in Y_{D}^{h}\) |
\(\theta_{1}V_{L}\leq x_{4}^{*}\) | \(A_{h}\leq0\), × | \(y_{i}^{+} \in Y_{D}^{1}\) | ||
(C3) |  |  | \(A_{h}\leq0\), × | \(y_{i}^{+} \in Y_{D}^{1}\) |