| If the evolution of differential dynamical system depends not only the current state, but also the states some times or some blocks past, then we call it as delay differential dynamical system, which is described by delay differential equations. Because of delay differential dynamical system is more profound than ordinary dynamical systems, there are deep theoretical and practicable meanings for researches on delay differential dynamical system.For the aspect of qualitative analysis, complicated dynamical behavior will occur for delay dynamical system because of the initial condition is not a initial value but a initial function. Besides, there are lots of difficulties to obtain the analytic solution since the complexity of delay differential equation, so it is very necessary to have a numerical analysis research.The elemental difference between delay and ordinary differential equation is that the introduction of delay may infect the dy-namical behavior essentially, so it is very important to investigate the dynamical behavior using delay as a parameter, meanwhile, it is a new challenge for numerical simulation.Hopf bifurcation is a kind of bifurcation phenomenons be cared specially, which describe the evolution of the solution for a system from equilibrium state to periodic state. For the aspect of numerical analysis for dynamical system, we have sufficient reasons to require the numerical method preserve the dynamical behaviors of the original dynamical systems. For example, if there is an Hopf bifurcation at parameter value \i = \i* for a dynamical system, we naturally expect that an Hopf bifurcation occured near \i — fi* in the corresponding numerical method.In this dissertation, we take the sunflower equation and its forward Euler method as an example, proved that when the equation has a Hopf bifurcation at some parameter value, then the numerical method also has a Neimark-Sacker bifurcation(secondary Hopf bifurcation) near the bifurcation parameter value as using delay as parameter.Consider the sunflower equationa b .a + -a + -sin aft — r) = 0, (1)r rwhere a, b > 0 are constants, r > 0 is delay. Define x = a, y = a,then (1) can be rewritten as the following delay differential equationb . \ ay(t) = —sin re (t — r)----r rIntroduce the notations0 1(2)A(r) =o -a-r JB(r) =0 0and let X(t) = (x(t),y(t)), we obtain the characteristic equation of system (2) isdetd(A,r) = 0,where d(X, r) = e~Xr{\I - A(r)) - B{r), that is A a b_r rLet cuq be the unique solution ofa(3)(4)a tana;= —7i\in (0, —). Wei Junjie et.al. proved that(5)b sin ljq is the Hopf bifurcation parameter value of system (2).For stepsize \/h > 0, we can represent delay r as r = (m + S)h, where m > 0 is an integer, 0 < 5 < 1. So, the forward Euler method for system (2) is= Xn +in-.(1----h)yn----hsm[Sxn^m+i + (1 - 6)xn-m(6)The characteristic equation of discretization (6) isdet[zm({z - 1)1 - hA(r)) - z6hB(r) - (1 - 5)hB{r)] = 0. (7)In this dissertation, we through the comparison and analysis of the characteristic equations between system (6) and (2) directly, provedTheoreml There is a Neimark-Sacker bifurcation at parameter value rh = ro + O(h) for the forward Euler scheme (6) of the sunflower equation as the stepsize h sufficiently small.Theoreml is the main conclusion of this dissertation, which shows that, for the forward Euler scheme for sunflower equation, a Neimark-Sacker bifurcation (known as secondary Hopf bifurcation) will occur near the sunflower equation's Hopf bifurcation parameter value r0 (in fact at the parameter value r^ = ro + O{h)) when using delay as a parameter. There are two innovations in this dissertation:1. The methods used in our paper can be applied to delay differential equations with arbitrary dimension conveniently;2. The methods used in our paper can be applied to deal with the Neimark-Sacker bifurcation problems in the numerical discretization for multi-delay systems using delay as parameter.
|
PDF Full Text Request