| Boundary value problems for second order nonlinear ordinary differential equations (in short, BVP), implicit second order differential systems (in short, IBVP), and functional differential equations (in short, FDE) have been studied using different methods and techniques for many decades. Using equivariant degree based methods, this dissertation discusses three differential systems.;The first example is boundary value problems for symmetric second order IBVP of the form.; &cubl4;y&d3; =ht,y,y,y&d3;&d2; , a,e,t∈0,1 ,y 0=y1= 0, .;where V:=Rn is an orthogonal G-representation and h : [0,1] x V x V x V → V is a G-equivartiant map satisfying the so-called Caratheodory condition. Under the Hartman-Nagumo conditions, one can analyse the solvability and symmetric classification of solutions to the above system by equivariant degree methodology, whose routine formula is applied in computer programs by us. Three concrete examples of large scale in R24,R60 and R105 , generated by programs, are also presented after the theory discussion.;The second example is about local and global bifurcation of IBVP problems of the form.; &cubl4;y&d3;=h a,t,y,y,y&d3;&d2; , t∈0,1, y∈V,y0 =0=y1. We are interested in finding bifurcating branches of non-zero solutions for the above parametrized family of IBVPs, bifurcating from the zero solution (alphao,0) for some ao∈R . The dissertation describes the theory and routines to find local and global bifurcations for this system and provides a concrete system as an example.;In the third example, this dissertation studies symmetric properties of multiple branches of non-constant periodic solutions to IBVP of the form.; ddt&sqbl0;xt -ba,xt&sqbr0; =fa,xt,ddt xt-b a,xt . in the presence of group symmetries using the twisted equivariant degree based method and analyses the occurrence and symmetric classification of branches of non-constant periodic solutions as a result of the Hopf bifurcation. A concrete example in R24 and its bifurcation analysis, again generated by our programs, is presented as well.;As these results show, the formulas in this dissertation have been programmed directly in computer languages for dihedral groups, and one can expect that in the near future these algorithms, with minor modifications, can also be applied to other symmetric groups and in large scale systems as well. |
