Numerical Hopf bifurcation of linear multistep methods for a class of delay differential equations

Normal forms of non-resonance and weak resonance double Hopf bifurcation in the retarded functional differential equations and applications

“…Many scholars have extended and improved the method of the dynamical behavior of ODE [8,19,28,29,41] to study the dynamical behavior of DDE, and they have obtained a lot of very good results, such as stability and Hopf bifurcation [1,5,14,15,26,30], synchronization [24], Bogdanov-Takens bifurcation [6,17], zero-Hopf bifurcation [11,25,33,36,37], triple zero bifurcation [10], and double Hopf bifurcation [2][3][4]12,13,18,[21][22][23]34,35,38,40,42,45]. Some researchers also have studied the direction of numerical Hopf bifurcation and stability of bifurcating invariant curve for the delay differential equations by using multistep method and Runge-Kutta method [20,31,32].…”

Stability analysis in a second-order differential equation with delays

“…For numerical analysis of Hopf bifurcation for delay differential equations, more and more authors like to compute it by the normal form with which Wulf and Ford did much work [11,18] for a general delay, differential equations and many authors applied the technology to a lot of models [8,19]. In our papers [14][15][16], we discussed in detail the numerical Hopf bifurcation for more general delay differential equations with the various numerical methods. But, the above papers only involves a single delay as bifurcation parameters.…”

Numerical bifurcation of a delayed diffusive food‐limited model with Dirichlet boundary condition

“…It is a natural requirement of an adequate numerical method that it possesses the discrete equivalents of the qualitative properties the continuous system satisfies. In , the dynamics of numerical discrete difference equations can inherit those of the original differential equations.. The essence of the method of lines is semi‐discretization, where a PFDE in spatial and temporal variables is discretized in the spatial variables only.…”

“…It is a natural requirement of an adequate numerical method that it possesses the discrete equivalents of the qualitative properties the continuous system satisfies. In [10][11][12][13][14][15][16], the dynamics of numerical discrete difference equations can inherit those of the original differential equations. .…”