Gloval Bifurcation & Asymptotics Behavior in Nonlinear Differential Equations

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501497 Ward Systems of ordinary or parabolic partial differential equations serve in modeling many physical systems. The differential equations typically contain parameters whose values correspond to physical properties of the system. Such a system generates a dynamical system in an appropriate phase space. To understand a dynamical system we should determine its compact invariant sets, which contain the non-transient solutions such as equilibria, periodic solutions, or ones of greater complexity. It is important also to determine the behavior of the transient solutions with respect to the non-transients. Finally, one should know how all of this depends upon the parameters. For example, a parameter change may induce a stable or hyperbolic equilibrium state to bifurcate into one or more time-periodic solutions, or ones with a more complex time structure. The Conley homotopy index theory contains concepts and methods that can be used to prove existence, and determine the properties, of compact isolated invariant sets in a dynamical system. Like topological degree the index is locally constant, and if the index changes as a parameter crosses a value, a bifurcation is implied. A major part of this project is the application of Conley homotopy index methods to study bifurcation in systems of reaction-diffusion equations. The investigators will: (I) Continue investigations of continua of full bounded solutions bifurcating from trivial solutions in infinite dimensional dynamical systems, especially in nonlinear reaction- diffusion systems. Using Conley homotopy index methods Ward recently demonstrated conditions for the existence of such bifurcating continua. Two of the main goals of this project are to fully describe the global aspects of these continua, and to apply the results to the study of global bifurcation of equilibria in gradient systems. (II) Study bifurcation and persistence in reaction-diffusion models arising in population biology an d chemistry. Physically meaningful solutions must have non-negative components. In this connection the investigators propose the development of a homotopy index bifurcation theory for nonlinear ordinary and parabolic partial differential systems which have an invariant non-negative cone. (III) Build on their earlier work on nonautonomous differential equations. In particular, they will continue the development of a homotopy index based theory for nonlinear time-dependent ordinary and parabolic partial differential systems. In this case a dynamical system is not generated in the space of initial values, but one can associate a skew-product flow and use the Conley theory to study its invariant sets. From this one can derive information regarding the original system. The investigators will focus on existence, dynamics, and bifurcation phenomena. (IV) Study related problems for semilinear parabolic partial differential equations with nonlinear boundary conditions, such the existence and asymptotic stability of periodic solutions, and the effect of interactions between the spectrum of the linear part of such equations with the nonlinearities. %%% Differential equations model physical phenomena such as mechanical systems, chemical reactions, and the spread of disease. An initial state (e.g., initial number of infected individuals) then evolves over time according to the laws of the system expressed by its equations. This constitutes a dynamical system. A goal of dynamical systems theory is to describe how the states of the system evolve. Thus, the number of fish of a given species in a lake might tend toward a steady, fixed population, while the number of individuals infected at any given time with a transmittable disease, such as chicken pox, may exhibit a periodic behavior, peaking every six or seven years. To understand a complex dynamical system one should determine the non-transient or long-term phenomena such as equilibria, time-periodic regimes, and others perhaps of a more complex nature, the relation between the transient solutions and the non-transients, and how this all depends on system parameters. Mathematical models depend upon certain parameters fom the physical system, such as the volume of a lake in a study of its fish population or diffusion rates in a chemical reaction. The nature of the long-term behavior of a system may depend upon its parameter values. If the parameters change, the number or structure of the non-transient solutions can change. Such a parameter dependent change is a bifurcation. Another kind of bifurcation occurs when the behavior of the transients changes, as when an equilibrium point (a non-transient) loses stablity with a parameter change. Non-transient solutions correspond to invariant sets in the dynamical system. The homotopy index theory was developed by Charles Conley and others to analyze the existence and properties of isolated invariant sets in dynamical systems. It also can be used to show that bifurcations must take place at certain parameter values. A main part of this project involves the application of Conley homotopy index methods to bifurcation phenomena in systems of reaction-diffusion equations. Reaction-diffusion systems of partial differential equations relate diffusion phenomena, such as diffusion of a chemical in a medium, with reaction rates, as in a chemical reaction. They serve as the mathematical models of a great many phenomena, including chemical reactions, population densities, and climate. The investigators will: (I) Continue investigations of continua (families of solutions joined together) of bounded solutions bifurcating from equilibrium solutions in nonlinear reaction-diffusion systems. By using Conley homotopy index methods Ward recently found conditions for the existence of such bifurcating continua. Two of the main goals of this project are to fully describe the global aspects of these families of bifurcat ing solutions, and to apply these results to study global bifurcation of equilibria in gradient systems (systems which, in the long run, will reach an equilibrium state). (II) Study bifurcation and persistence in reaction-diffusion models arising in population biology and chemistry. (III) Build on their earlier work on time-varying systems of differential equations. Such systems exhibit phenomena different from the time-independent systems discussed above. The investigators will continue to develop a homotopy index based theory for nonlinear time-dependent ordinary and parabolic partial differential systems. ***