Bifurcation Analysis In Heterodimensional Cycles And Cancer Models

This thesis aims at deep theoretical analysis of bifurcations of heterodimensionalcycles,which are more general but less discussed in nonlinear continuous dynamicalsystems with higher dimension,and the explanation of bifurcation phenomena in somecancer models by the current techniques and results on bifurcation theory of differentialequations.The thesis consists of five chapters.Chapter 1 presents the background and significance of the problems and summarizesthe main results.Dynamical systems with heterodimensional cycles are very common([66]) and the existence of heterodimensional cycle always leads to complex dynamicalbehavior.Therefore,research on bifurcation problem of heterodimensional cycles ex-hibits prominent potential for both practical and theoretical values,which is presented inChapter 2 and Chapter 3.Concretely speaking,in Chapter 2 the degenerate bifurcations ofnontwisted heterodimensional cycles in the 3-dimensional space are investigated,wheretwo cases are contained,one with nonresonance and the other with resonance.Followingthe method developed by Zhu [114],we construct a moving frame formed by the tangentvector tangles of invariant manifolds and the principle normal of the heterodimensionalcycle F in some regular tubular neighborhood ofΓ.Since such a coordinate frame itselfinherits the geomeric invariance of the corresponding manifolds and transmits the dy-namical properties including the intrinsic contraction and expansion of the systems,onepartial Poincar(?) mapping (global mapping) induced by solutions of systems in the regu-lar neighborhood ofΓunder this frame becomes extraordinarily simple.Furthermore,theother partial Poincar(?) mapping (local mapping) induced by solutions near the equilibria isestablished by introducing Shilnikov coordinates and the local normal form ([22]).Thenby composing these two mappings,we obtain the Poincar(?) mapping and the bifurcationequations nearΓ.Due to the solutions of bifurcation equations for degenerated nontwisted heterodi-mensional cycles,we achieve quite rich bifurcation results involving a family of bifurca-tion surfaces ofheterodimensional cycles,which indicates that there are great differencesin bifurcation behaviors between the heterodimensional cycles and the equidimensionalcycles.Complete bifurcation diagrams in the two kinds of nontwisted cases are also dis- played when no resonance occurs in heterodimensional cycles.After the improvement in the literature [51],the above method is more applicable andmuch easier to carry out by solving the bifurcation equations.In Chapter 3,we apply thisgeneralized technique to analyze bifurcations near heterodimensional cycles with stronginclination flip in 4-dimensional nonlinear systems.The highly degenerate bifurcationequations elicit the coexistence of a heterodimensional cycle and a homoclinic loop,thecoexistence of a heterodimensional cycle and a periodic orbit,the possible coexistenceof loops homoclinic to different saddles and so on.It is obvious to see that the bifur-cation results from heterodimensional cycles in 4-dimensional spaces are far richer thanthat of both 3-dimensional cases and higher dimensional equidimensional cases,whichwell illuminate the extreme complexity of dynamical behaviors on the bifurcations ofheterodimensional cycles in higher dimensional systems.The bifurcation analysis in cancer models can be found in the last chapters.In Chap-ter 4 we analyze qualitatively the bifurcation phenomena near the degenerate positiveequilibrium in 2-dimensional cancer models of interactions between lymphocyte cellsand solid tumor cells.By using planar bifurcation theory,we observe that the cancermodels undergo codimension-2 Bogdanov-Takens bifurcation near the degenerate equi-librium,whether before vascularization or after vascularization.The analytic bifurcationresults correct some wrong conclusions near the degenerate equilibrium in the literature([1,68]).In Chapter 5,the degeneracy of the Hopfbifurcation point B is firstly discussedin a 3-dimensional cancer model.Then we utilize the Hopfbifurcation theorem for higherdimensional systems to conclude the occurrance of Hopf bifurcation.Moreover,an ap-plicable Hopf bifurcation formula([49]) is imposed to obtain sufficient conditions for theexistence of stable periodic solutions bifurcated from point B.Finally detailed numeri-cal simulations of trajectories are given to support the bifurcation analysis of the cancermodels.