Stability Loss Delay In Slow-fast Systems And Dynamics Of Population Models

This thesis studies stability loss delay phenomenon of abstract planar slow-fast systems with higher degenerate critical manifolds,and analyses dynamics of planar slow-fast prey-predator models via geometric singular perturbation theory.The thesis consists of four chapters.The main contents are the following:Chapter 1 outlines the history of geometric singular perturbation theory and de-scribes the problems to be solved.Chapter 2 provides definitions and preliminaries,including basic concepts of slow-fast systems,Fenichel's theory,canard phenomenon,the blowup method,fold point theory,and slow divergence integral theory.Chapter 3 discusses dynamics of a abstract planar slow-fast system near a turn-ing point,especially stability loss delay phenomenon.Pioneering work on using the blowup method to study dynamics of slow-fast systems is Dumortier and Roussarie[Mem.Amer.Math.Soc.,121(577):x+100,1996],where the behavior of slow-fast Van der Pol equation near the turning points was analyzed.In[J.Differential Equa-tions,260(8):6697-6715,2016],De Maesschalck and Schecter investigate stability loss delay phenomenon of a well-known slow-fast system near a turning point using the blowup method.Here,we consider dynamics of a more degenerate version of the sys-tem.Relying mainly on Fenichel's theory,the quasihomogeneous polar and Euclidean blowup method,and slow-fast normal form,we investigate the entry-exit function,the expression of Poincare map,and the asymptotic expansions of the solutions in detail.This result is an essential extension of De Maesschalck and Schecter's one via some new techniques.We end this chapter by establishing a bioeconomic model and proving the existence of relaxation oscillation applying the theory obtained above.Our result explains well the bioeconomic phenomena with periodic motionChapter 4 focuses on the dynamics of a classical Holling-Leslie prey-predator mod-el.It was previously investigated in[SIAM J.Appl.Math.,55(3):763-783,1995]by Hsu and Huang for global stability of an positive equilibrium,and in[J.Differential Equations,257(6):1721-1752,2014]by Huang,Ruan and Song for subcritical Hopf and Bogdanov-Takens bifurcations.At the moment these are the only known results on this system.Employing geometric singular perturbation theory,we achieve much rich-er new dynamics on this classical Holling-Leslie model.First,we discuss the singular Hopf bifurcation,the existence of canard cycles,of canard explosion,of relaxation os-cillations,and of heteroclinic and homoclinic orbits,and then study the cyclicity of all possible slow-fast cycles.These phenomena are also simulated via numerical calcu-lations.We conclude that at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the slow-fast cycles under small perturbations.Furthermore,we discuss the maximal number of the limit cycles of this system.