Some Bifurcation Problems And Their Numerical Analysis In Delay Differential Systems

In this dissertation, we lay our efforts on the research of T-B singularity and accompanying bifurcation properties in delay differential systems, and the retentivity of Hopf bifurcation, T-B singularity and homoclinic orbit in numerical discretization of delay differential systems.T-B point is a class of equilibria with multiple 2 singularity in differential systems. In the first instance, F.Takens and R.I.Bogdanov revealed the mechanism of the T-B singularity in planar system induce homoclinic bifurcation and Hopf bifurcation by the method of versal unfolding. This is an important discovery in the bifurcation theory of ordinary differential systems, particularly, this is a hard-to-get result with universality on the the existence of homoclinic orbit.The dimension of ordinary differential systems with T-B singularity is necessarily to be greater than 1, but it is not true for delay differential systems, T.Faria et.al. has investigated the definition and description of T-B singularity for scalar delay differential equations. However, to describe and determine the T-B singularity in delay differential systems certainly is much more harder than in the scalar case, because of the complexity of the generalized eigenspace of the linearized delay differential equations. Besides, there is no definite results by now on the bifurcation behavior near T-B point in delay differentialequations.In chapter 2 of this paper, we consider the following parameterized delay differential systemsx(t) = A(a)x(t) + B(a)x{t-l) + F(x(t),x(t-l),a), xeW1 (1)where a € M2 are bifurcation parameters, A(a),B(a) are CT(r > 2) smooth matrix value functions from R2 to Rnxn, and bounded uniformly with respect to a. F(x, y, a):rxRBxl2^ Rn is Cr(r > 2) smooth in all variables, and satisfiesor 0771F(0,0, a) = 0, —(0,0, a) = 0, —(0,0, a) = 0, Va € M2.In §2.2, we characterize the T-B singularity of systems (1) by the semi-group theory of retarded functional differential equations, and obtain a feasible criterion for the determination of T-B singularity. In §2.3, we reduce the systems (1) with T-B singularity on the center manifold into the following simple 2-dimensional ordinary differential systemswhere k^ , k2 are parameters linearly relying on a = (ai, a^), a, b are parameters relying on the original delay differential systems (1).In §2.4, through a detailed analysis of bifurcation behavior of the reduced ordinary differential equations, we obtain and prove the bifurcation properties of the delay differential systems (1) near its T-B point on ai-a2 plane (there exist a Hopf bifurcation curve and a homoclinic bifurcation curve emanating from T-B point).In §2.5, we studied a concrete example, its bifurcation diagram on parameter plane is obtained by the methods developed in our paper.In this chapter, we give a description and setup a feasible criterion for T-B singularity of delay differential systems (1), in addition, we obtain definite results on the bifurcation properties of the delay differential systems (1) near T-B point. These results are universal and novel on the existence of homoclinic orbit with respect to delay differential systems. In the analysis, a series of difficulties such as to determine the generalized eigenspace and computation of normal forms have been conquered. Above theoretical studies are part of the innovative works of our paper.In recent 20 years, the numerical methods and numerical analysis of dynamical systems is an important research field concerned by the scholars in applied and computational mathematics, which contains two aspects: (i) to construct the numerical discretization of continuous systems and to study their convergence and stability properties (orbit computation oriented); (ii)studies on retentivity (succession) of the dynamical behavior in numerical discretization with respect to the original differential systems. Concerning delay differential equations, the former aspect of numerical methods has been investigated widely, and has been summarized by monographs, while the latter aspect is a fresh subject of numerical analysis in dynamical systems appeared recently. In the mean time, there is few research work on retentivity of dynamical behavior in numerical discretization of delay differential systems, and correlative works existed mostly concentrate on Hopf bifurcation, specially on individual delay differential equations. Due to the complexity of the characteristic equation of linearized delay differential equations, to discuss the retentivity of Hopf bifurcation in discretization of delay differential systems is more harder than the case of scalar delay equations. Besides, to our knowledge, at present there is no published results on the retentivity of T-B singularity and connecting orbits in numerical discretization of delay differential systems.In chapter 3, we discuss the discretization and numerical methods of delay differential equations, and a detailed discussion on the convergence analysis and error estimate of the forward Euler method is given, as a foundation ofa further study on the retentivity of bifurcation behavior in discretization of delay differential equations.In chapter 4, we concentrate our research on the retentivity of Hopf bifurcation and T-B singularity in numerical discretization of delay differential systems.Consider parameterized delay differential systemsx(t) = f(x{i),x{t-l),a), xeW1 (3)where a e Rq are bifurcation parameters, f(x,y,a) is Cr(r > 2) smooth in all variables. In this chapter, we take the forward Euler method as an example, utilizing the theory of strong continuous semi-group and the tools of infinitesimal generator etc., prove the retentivity of Hopf bifurcation and T-B singularity in discretization of systems (3) through the analysis and comparison between the characteristic equations of the linearized equations of systems (3) and their discretization's. To be more precise, we proved: if the systems (3) possess a Hopf bifurcation when a — a* € R, then the discrete delay systems corresponding to the forward Euler method also possess a Hopf bifurcation as the parameter a = a* + 0{e) when the step-size e is sufficiently small; if the systems (3) have a T-B singularity when a = a* G R2, then the discrete delay systems corresponding to the forward Euler method also have a T-B singularity as the parameter a = a* when the step-size e is sufficiently small.In this chapter, we have extended the numerical studies of Hopf bifurcation from individual delay differential equations to general delay differential systems in any finite dimension, and it is the first piece of work to discuss the preservation property of T-B singularity in numerical discretization of delay differential equations. The results obtained in this chapter are novel and substantial for the numerical analysis of delay differential systems with T-B singularity. In addition, we also discussed the numerical Hopf bifurcation of a delayed predator-prey model and the numerical analysis of a class of delay differential equations with T-B singularity, and verified the theoretical results.In chapter 5, we made efferts to study the retentivity of homoclinic orbits in numerical discretization of delay differential systems. Consider the following delay differential systemsx{t) = f(x{t), x{t - 1), a, u), xeW1 (4)where a G R is the bifurcation parameter, v G 1R9 are additional parameters. Our assumptions on systems (4) are:(HI) f(x, y, a, v) is Cr(r > 2) smooth in all variables, and Cr+1smooth in a, and the (r + l)th order derivative with respect to x and y exist, furthermore, /, together with all its derivatives, is uniformly bounded by some constant C\.(H2) xe is hyperbolic equilibria of systems (4) at (a, v) = (a, u).According to these two assumptions, there exists a Cr smooth function x(ct, u) with x(a, u) = xe as a —> a, u —> v, such that x(a, u) is hyperbolic equilibria of systems (4).(H3) At (a,f) = (a.t'), systems (4) possess a homoclinic orbit x(t), which satisfieslim x(t) = xe.(H4) For all tel, fy(x(t),x(t — l),o;, v) are nonsingular, and there exists a constant a > 0 such that | det fy(x(t), x(t — 1), <5, u)\ > a.(H5) Homoclinic orbit x{t) is nondegenerate with respect to bifurcation parameter a.Under these assumptions, in §5.2, we describe the concept of regular homoclinic orbit pair (x(t), a) through the exponential dichotomy and Fredholm alternative theory, and proof some useful lemmas. Based on the regular properties, we obtain a family of smooth regular homoclinic orbit pairs (x(v,t),a(v)) of systems (4) which satisfy limf^±oo x(i/, t) = x(a(v), u).In §5.3, we prove the sampling of the exact homoclinic orbitsxz{u, e, t*) = {xk{u, e, t*))kex, xk(u, e, t*) = x{u, t* + fee), W ￡ Rare 1-tangential homoclinic orbits of the time-e-flow of systems (4), and they are nondegenerate with respect to bifurcation parameter a. Besides, we proved the homoclinic orbit pairs {xz{v, e, t*), a(u)) defined by above sampling are regular solutions of some operator equation G = 0.In §5.4, take the forward Euler method (step-size e = ^,m G Z+) as an example, we proved the preservation properties of homoclinic orbits in numerical discretization of delay differential systems by means of an implicit function theorem with parameters, and the 1-order approximate property of the numerical homoclinic orbits with respect to x-iiy, e, ￡*). In another words, there exist a family of discrete homoclinic orbits xz(i/, e, ￡*) in the discretization systems near xz{y, e, t*) when step-size e is sufficiently small, and these orbits tend to 5z(i/, e, t*) by the rate of O(e), meanwhile, the corresponding parameters a(u, e, t*) tend to a (is) by the rate of O(s) as e —> 0; And these orbits are unique determined up to a phase shift controlled by t*. We also proved some properties of the numerical homoclinic orbits: (i) (xz(u, e, t*), a(u, ￡, t*)) have propertiesxk{v,e,t*) = xk-i{i',e,t* + e),a(v, e, t*) = a(v, e, t* + e).(ii) xz(i/, e, t*) are either transversal or 1-tangential with respect to corresponding bifurcation parameter a(v, e, t*).In addition, a concrete example and some numerical results of the simulation of homoclinic orbits in delay differential equations by forward Euler method are given.In this chapter, under certain regular assumptions, we have identified the preservation properties of homoclinic orbits in numerical discretization of de-...