Neimark-Sacker Bifurcations Of Numerical Discretization For Van Der Pol Equation With Two Delays

If the solution of a differential dynamical system depends not only the current state, butalso the states some times or some period ago, then we call it to be the delay differentialdynamical system, which is described by delay differential equations. Because of delaydifferential dynamical system is more complicated than ordinary differential dynamical system,there are theoretical and practicable meanings for researches on delay differential dynamicalsystem.For the aspect of qualitative analysis, complicated dynamical behavior will occur in delaydynamical system because of the initial condition is not a initial value but a function. Be-sides, there are lots of difficulties to obtain the analytic solution since the complexity of delaydifferential equations, so it is necessary to carry out a numerical research.In the parameter dependent nonlinear dynamical system, when the parameters varied,changes may be occured in the qualitative structure of the solution for certain parameter values.This is the primary question for bifurcation theory on dynamical systems. Hopf bifurcationis a kind of bifurcation phenomenons be cared specially, which describes of the solution ofa system from equilibrium to periodic. For the aspect of numerical analysis for dynamicalsystem, we have sufficient reasons to require the numerical method preserve the dynamicalbehaviors of the original dynamical systems. For example, if there will be a Hopf bifurcationoccured at parameter valueμ=μ^* for a dynamical system, we naturally expect that therewill be a Hopf bifurcation occur nearμ=μ^* in the corresponding numerical method.This paper considers the van der Pol equation with two delaysThe linearized system of (1) at E₀ = (0, 0) iswhere a= f′(0) The solution structure of delay differential system (1) will changes whenÏ„₁ andÏ„₂changed. Therefore,Ï„₁ andÏ„₂ are same important for equation (5). In the paper, itby fixingÏ„₂ and takingÏ„₁ as a bifurcation pararmeter, obtain the following conclusionTheorem There existsÏ„₁ =Ï„₁⁰ such that equationsω²=aωsinÏ„₁ω+cos(Ï„₁+Ï„₂)ωaωcosÏ„₁ω-sin(Ï„₁+Ï„2)ω=0hold, and Re[dλ／dÏ„₁]|_{λ=iω0Ï„1=Ï„1⁰}≠0then for system (1), a Hopf-type bifurcation occurs at E₀=(0, 0) asÏ„₁ passes throughÏ„₁⁰.To solve the system (1) by using of Euler method, where we use linear interpolation toapproximate x(t-Ï„₁) and y(t-Ï„₂), we obtain the discretization of system (1)In this paper, by the comparison and the analysis of the matrix form between delaydifferential system and discretization directly, We can obtain the characteristic structure ofdiscretization (6). We prove thatTheorem There will be a Neimark-Sacker bifurcation occurs at parameter valueÏ„₁^h=Ï„₁⁰+ O(h) for the Euler discretization (6) of the van Her Pol equation (1) with two delays asthe stepsize h sufficiently small, whereÏ„₁⁰ is the Hopf bifurcation parameter value of equation(1).This theorem is the main conclusion of this paper, which show that, for the Euler dis-cretization of van der Pol equation with two delays, there will be a Neimark-Sacker bifurcationoccurs at the parameter valueÏ„₁^h=Ï„₁⁰+O(h), provided that the Van der Pol equation un-dergoes a Hopf bifurcation at parameter parameter valueÏ„₁=Ï„₁⁰, which mean that the Eulerdiscretization preserve the Hopf bifurcation of the van der Pol equation with two delays whenwe take one of the delay as a parameter.