Bifurcation theory, computation and applications

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Many problems arising in science and engineering can be modelled by mathematical systems that dynamically evolve in time. Among them, nonlinear differential equations are widely used to describe complex phenomena, such as vibration of machinery, weather patter, animal locomotion, population growth, financial markets, spread of diseases, recurrent infection, etc. Such dynamical processes often exhibit qualitative changes, called bifurcations, like earthquakes, population explosions, stock market crashes, disease outbreaks and so on. Bifurcation theory is a powerful tool to study bifurcation phenomena and explain how small changes in the system parameters can lead to these qualitative changes. The methods of bifurcation theory enable us to identify the parameter values where bifurcations occur, and to predict the changes in behaviour of the system near the bifurcation points. To facilitate the application of bifurcation theory in solving real-world complex problems, which are usually high dimensional and nonlinear, the developments of efficient algorithms and computer programs to reduce system complexity are required. Center manifold theory and normal form theory will be used to greatly simplify system equations by extracting the "key" information (or terms) to form a simple system, while keeping the qualitative behaivour of the system unchanged near bifurcation points. This proposed research program has three objectives. One of them is to develop the theory and efficient computational methods for the parametric normal forms of general delay differential equations, which can be easily used to find, for example, periodic solutions (limit cycles) of a complex system and determine their stability. The other two objectives are focused on applications. We intend to apply the methods of dynamical systems and bifurcation theory with the efficient computational tools to analyze models of specific biological processes: HIV-1 therapy, and recurrent infection. In particular, we shall further study our early developed HIV-1 therapy model to find new mechanisms which can be used to identify other types of viral blips. We also want to apply the bifurcation theory and normal forms for delay differential systems to investigate the blips phenomenon in HIV-1 models with delays. These applications are not only significant for increasing scientific understanding, but progress may yield practical, clinical benefits. In addition, the theory and methodology we develop for these specific applications will be useful for many applications from other disciplines, as mentioned above.