Andronov–Hopf and Neimark–Sacker bifurcations in time-delay differential equations and difference equations with applications to models for diseases and animal populations

Darlai, Rachadawan; Moore, Elvin J.; Koonprasert, Sanoe

doi:10.1186/s13662-020-02646-5

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 5 Nonlinear discrete HIV and ELM equations for versions with Neimark–Sacker bifurcations (values of ρ and η are given in Table 3)

From: Andronov–Hopf and Neimark–Sacker bifurcations in time-delay differential equations and difference equations with applications to models for diseases and animal populations

Version	HIV \(\psi = h_{m}\tau \)
dnn	\(y_{n+1}=y_{n}+\varepsilon \psi y_{n}-\delta \psi y_{n-m} -\varepsilon \psi (y_{n}+w^{*})^{2}\)
second order	\(y_{n+1} = y_{n} + \rho \psi y_{n}-\eta \psi y_{n-m}-2\varepsilon \psi y^{2}_{n} \)
nnd	\(y_{n+1}=y_{n}-\delta \psi y_{n}+\varepsilon \psi y_{n-m} -\varepsilon \psi (y_{n}+w^{})(y_{n-m}+w^{})\)
second order	\(y_{n+1} = y_{n}- \eta \psi y_{n-m} -\varepsilon \psi (y_{n}y_{n-m}+y_{n-m}y_{n})\)
dnd	\(y_{n+1}=y_{n}-\varepsilon \psi y_{n}-\delta \psi y_{n-m} -\varepsilon \psi (y_{n}+w^{})(y_{n-m}+w^{})\)
second order	\(y_{n+1}=y_{n} +\rho \psi y_{n}-\eta \psi y_{n-m}-\varepsilon \psi (y_{n}y_{n-m}+y_{n-m}y_{n})\)
ndd	\(y_{n+1}=y_{n}-\delta \psi y_{n}+\varepsilon \psi y_{n-m} -\varepsilon \psi (y_{n-m}+w^{*})^{2}\)
second order	\(y_{n+1}=y_{n} +\rho \psi y_{n}-\eta \psi y_{n-m} -2 \varepsilon \psi y_{n-m}^{2}\)
ddd	\(y_{n+1}=y_{n}+(\varepsilon -\delta ) \psi y_{n-m} -\varepsilon \psi (y_{n-m}+w^{*})^{2}\)
second order	\(y_{n+1} = y_{n}-\eta \psi y_{n-m} -2\varepsilon \psi y^{2}_{n-m} \)

Version	ELM \(\psi = h_{m}\tau \), Λ = β − r, \(\nu =\frac{\beta }{K}(\frac{\varLambda }{\beta })^{\frac{\gamma -1}{\gamma }}\), \(\xi =\frac{\beta }{K^{2}}(\frac{\varLambda }{\beta })^{\frac{\gamma -2}{\gamma }}\)
dnn	\(y_{n+1} = y_{n} +\beta \psi y_{n}-r\psi y_{n-m}-\alpha \psi \frac{\beta }{K^{\gamma }}(y_{n}+w^{*})^{\gamma +1}\)
third order	\(y_{n+1} = y_{n} +\rho \psi y_{n}-\eta \psi y_{n-m}- (\gamma ^{2}+\gamma )\nu \psi y_{n}^{2}- (\gamma ^{3}-\gamma )\xi \psi y_{n}^{3}\)
nnd	\(y_{n+1} = y_{n} +\psi \varLambda y_{n}-\alpha \psi \frac{\beta }{K^{\gamma }} (y_{n}+w^{})(y_{n-m}+w^{})^{\gamma }\)
third order	\(\begin{array}[t]{l} y_{n+1} = y_{n} -\eta \psi y_{n-m} -\nu \psi [(\gamma ^{2}-\gamma ) y_{n-m}^{2}+\gamma ( y_{n} y_{n-m}+y_{n-m} y_{n})] \\ \hphantom{y_{n+1} ={}}{}-\xi \psi [(\gamma ^{3}-3\gamma ^{2}+2\gamma )y_{n-m}^{3} \\ \hphantom{y_{n+1} ={}}{}+(\gamma ^{2}-\gamma )(y_{n} y_{n-m}^{2}+ y_{n-m} y_{n}y_{n-m}+ y_{n-m}^{2} y_{n})]\end{array}\)
dnd	\(\begin{array}[t]{l} y_{n+1} = y_{n} +\alpha \psi (1+\beta )y_{n}-r\psi y_{n-m} \\ \hphantom{y_{n+1} ={}}{}-\alpha \psi \frac{\beta }{K^{\gamma }}(y_{n}+w^{})(y_{n-m}+w^{})^{\gamma }\end{array}\)
third order	\(\begin{array}[t]{l} y_{n+1} = y_{n} + \rho \psi y_{n} -\eta \psi y_{n-m} \\ \hphantom{y_{n+1} ={}}{}-\nu \psi [(\gamma ^{2}-\gamma ) y_{n-m}^{2}+\gamma ( y_{n} y_{n-m}+y_{n-m} y_{n})] \\ \hphantom{y_{n+1} ={}}{}-\xi \psi [(\gamma ^{3}-3\gamma ^{2}+2\gamma )y_{n-m}^{3} \\ \hphantom{y_{n+1} ={}}{}+(\gamma ^{2}-\gamma )(y_{n} y_{n-m}^{2}+ y_{n-m} y_{n}y_{n-m}+ y_{n-m}^{2} y_{n})]\end{array}\)
ndd	\(y_{n+1} = y_{n} -r\psi y_{n}+\alpha \psi (1+\beta )y_{n-m}-\alpha \psi \frac{\beta }{K^{\gamma }}(y_{n-m}+w^{*})^{\gamma +1}\)
third order	\(\begin{array}[t]{l}y_{n+1} = y_{n} +\rho \psi y_{n}-\eta \psi y_{n-m} -(\gamma ^{2}+\gamma )\nu \psi y_{n-m}^{2} \\ \hphantom{y_{n+1} ={}}{}- (\gamma ^{3}-\gamma )\xi \psi y_{n-m}^{3}\end{array}\)
ddd	\(y_{n+1} = y_{n} +\psi \varLambda y_{n-m}-\alpha \psi \frac{\beta }{K^{\gamma }}(y_{n-m}+w^{*})^{\gamma +1}\)
third order	\(y_{n+1} = y_{n} -\eta \psi y_{n-m}-(\gamma ^{2}+\gamma )\nu \psi y_{n-m}^{2}- (\gamma ^{3}-\gamma )\xi \psi y_{n-m}^{3}\)

Back to article page