Skip to main content

Theory and Modern Applications

Table 3 Existence and stability of the fixed point \(\pmb{y^{*}}\) of Poincaré map \(\pmb{{\mathcal{P}}(y_{i} ^{+})}\)

From: Global dynamics of a state-dependent feedback control system

Cases

\(\boldsymbol {A_{h}}\) and \(\boldsymbol {A_{h_{1}}}\)

Ï„

\(\boldsymbol {y^{*}}\)

Interval of \(\boldsymbol {y^{*}}\)

(SC123)

\(A_{h}\leq0\), ×

Ï„>0

EG

\(Y_{D}^{1}= [\tau, Y_{is}^{h}+\tau ]\)

\(A_{h}<0\), ×

τ = 0

EG

\(y^{*}=0\)

\(A_{h}=0\), ×

ENS

\(\forall y^{*}\in [0, Y_{is}^{h} ]\)

(SC11)

\(A_{h}> 0\), \(A_{h_{1}}\geq0\)

\(\frac{A_{h}}{p}<\tau\leq\tau_{2}^{h_{1}}\)

NE

 

\(\tau_{2}^{h_{1}}<\tau\)

ES

\((Y_{\max}^{h_{1}}, Y_{is}^{h_{1}}+\tau ]\)

τ = 0

EU

\(y^{*}=0\)

(SC12)

\(A_{h}\leq0\), \(A_{h_{1}}\geq0\)

\(\tau_{3}^{h_{1}}\leq\tau\leq\tau_{2}^{h_{1}}\)

NE

 

\(0<\tau<\tau_{3}^{h_{1}}\)

ES

\((0, Y_{\min}^{h_{1}} )\)

\(\tau>\tau_{2}^{h_{1}}\)

ES

\((Y_{\max}^{h_{1}}, Y_{is}^{h_{1}}+\tau ]\)

\(A_{h}<0\), ×

τ = 0

ES

\(y^{*}=0\)

\(A_{h}=0\), ×

ENS

\(\forall y^{*}\in [0, Y_{\min }^{h_{1}} )\)

(SC2)

\(A_{h}>0\), ×

\(\frac{A_{h}}{p}<\tau<\tau_{M}\)

NE

 

\(\tau_{M}\leq \tau\leq\tau_{2}\)

EU

\([Y_{\max}^{h}, \frac{b}{p}+\tau ]\)

\(\tau_{2}<\tau\)

ES

\([Y_{\max}^{h}, \frac{b}{p}+\tau ]\)

τ = 0

EU

\(y^{*}=0\)