Shui-Nee Chow - US grants

Two problems are considered. The first concerns the stability of periodic solutions to delay equations with negative feedback. These equations arise in such fields as population biology, physiology and optics. The second problem concerns the study of heteroclinic and homoclinic orbits representing traveling wave solutions of reaction-diffusion equations. These equations arise in the study of nerve impulses, flame propagation and phase transitions

This research is involved with parallel algorithms for solving for the roots of large-scale algebraic polynomial and certain general nonlinear systems. These algorithms will be developed based on the so-called probability-one homotopy method and they world guarantee finding all the roots. Particularly, the algorithm techniques are to be applied to obtain all steady state solutions of the load flow problem for practical large-scale power systems. A special probability-one homotopy method will be tailored for the load flow to reduce the computational complexity while still guaranteeing the finding of all solutions computationally. A simpler load flow model will be identified and employed first. Then its solutions will be used in order to efficiently obtain the solutions of the full load flow model. More importantly and practically, numerical implementation of the solution techniques will exploit inherent parallelism in the techniques to be efficiently executed on massively parallel distributed-memory multiprocessors. This constitutes the first phase of the proposed research. The second phase of this research project is concerned with local stability and the possible bifurcation of the solutions to the load flow as well as their role and impact on the dynamic behavior of power networks.

This award will support a seminar on "Finite and Infinite Dimensional Dynamical Systems," organized by Prof. Shui-Nee Chow of the Georgia Institute of Technology and Prof. Kyuya Masuda of the University of Tokyo. Participants will meet in Kyoto, Japan, on July 17-21, 1989 to exchange recent research results and promote future joint research projects. The seminar will deal primarily with partial and ordinary differential equations as dynamical systems on finite- and infinite- dimensional spaces. This fairly recent approach has been applied to certain important systems, but the theoretical basis for such application is as yet relatively primitive. This multidisciplinary seminar will bring together researchers in such areas as classical ordinary and partial differential equations, abstract ergodic theory and dynamical systems, and map theory; as well as experimentalists. Discussion will center on general approaches to dynamical systems, treatment of traveling waves, turbulence, chaos, and problems in gas and fluid dynamics. These discussions may have important implications for the development of new and efficient simulation methods for fluidmechanical systems.

The research in this project is a continuation of the efforts of the principal investigator to discuss the qualitative theory of flows and the dependence on parameters of these flows of the attractor for functional differential equations (FDE) and partial differential equations (PDE). The parameters in the PDE are primarily the diffusion coefficients, boundary conditions and the domain. For FDE, the parameters are the delays and rapidly oscillating forces.

The principal investigator will develop and improve methods for understanding bifurcation from invariant tori, semi-linear elliptic problems in unbounded domains, and singularly perturbed differentiable systems. The underlying technique is the theory of invariant manifolds. Dynamical systems encompass both ordinary and partial differential equations, expressing a geometric approach to the study of such equations. An understanding of invariant manifolds is extremely critical as it gives a great deal of information about behavior of solutions in a neighborhood of special solutions.

9306265 Hale The thrust of this proposal is to continue to develop the theory of infinite dimensional dissipative systems by the consideration of specific types of problems. For example, in Partial Differential Equations, we study the dependence of the flows upon parameters, concentrating on the boundary conditions, diffusion coefficients, dissipation coefficients and the shape of the domain. Comparison of flows defined by such equations will be restricted mostly to the global attractors. For Functional Differential Equations, we will be primarily interested in singular perturbation problems which relate the solutions to the properties of maps defined by limit equations. Many physical systems are modeled by partial differential equations defined on a bounded domain. The study of the effects of the domain (its shape, etc.) on the dynamics is of paramount importance. This is especially true if the geometric properties ofl the domain act as a control variable, as, for example, those included under lthe name lof optimal shape design, as well as others. Also, many domains are small in some directions and it is advantageous to make a reduction to a lower dimensional domain for design as well as computational purposes. The present proposal addresses questions of this type. In lazer optics and models in ecology and biology, one encounters singularly perturbed delay differential equations. In such cases, the limit equation for the singularly perturbed parameter equal to zero is mapping. We address in this proposal the relationships between the dynamics of the delay equation and the dynamics of the map. ***

9404199 Afraimovich/Chow Depending on parameters, real systems with dissipation and energy pumping can manifest slow nonessential changes as well as abrupt jumps in their dynamical behavior. In order to explain, describe, and predict phenomena of such a kind in a specific applied system, people need to know a mathematical classification of transitions from simple to complex behavior and a theory of evolution of stationary (established) motions during the changes of parameters. The theory of nonlocal bifurcations of strange attractors is an adequate mathematical tool to study changes in conduct of dissipative systems which allows one to control and, in principal, to govern their behavior. Stationary motions (regimes) of real systems correspond to attractors in their mathematical models. Simple regimes correspond to simple attractors while the complex ones correspond to so called strange attractors. Mathematical image of the onset to complex behavior is a bifurcation of a rising of a strange attractor; mathematical image of evolution of complex behavior is a scenarium (or a chain of bifurcations) of evolution of a strange attractor. In the proposed work we are going to study bifurcations leading to appearance of strange attractors and to describe scenaria of their evolution. We propose to apply the developed mathematical technique to such systems as coupled oscillators, laser systems, circuit systems of electrical engineering and others. In particular, we want to study the problem of stochastic synchronization, i.e., similar behavior of dissipatively coupled dissipative individual subsystems. The phenomenon of stochastic synchronization is interesting, for example, for the problem of secure communications. It also plays a fundamental role in the explanation of deterministic behavior of nonequilibrium media. We are going to describe mechanisms of the occurrence of stochastic synchronization in the language of nonlocal bifurcation theory and strange attractors. The problem of appearance and evolution of strange attractors during the changes of parameters in one-parametrical families of smooth vector fields is very important from the mathematical point of view. It also plays a fundamental role in studying of specific dissipative systems from applications. In the proposed work we are going to study the nonlocal codimension one bifurcations on the boundary of the Morse-Smale systems which may lead to the birth of strange attractors and, also scenaria (i.e., chains of bifurcations) which can be responsible for characteristics of strange attractors. In the first problem, we propose to classify behavior of homoclinic and heteroclinic trajectories at the bifurcation moment, single out situations related to appearance of strange attractors and describe the arising attractors in terms of symbolic dynamics. In the second problem, we want to study scenaria of appearance of new positive Lyapunov exponents in strange attractors and investigate their crises which are related to nontransversal intersections of stable and unstable manifolds. We are going to apply expected results to investigate some specific systems (coupled oscillators, laser systems and others) in the form of dissipatively coupled dissipative individual subsystems. We propose to describe mechanisms of occurrence of stochastic synchronization phenomenon in such systems. For identical subsystems, stability of spatially-homogeneous solutions implies stochastic synchronization. For different individual subsystems, a theory of stochastical synchronization based on bifurcations of strange attractors needs to be developed.