The Construction of Arbitrary Stable Dynamics in Non-Linear Neural Networks

In this paper, two methods for constructing systems of ordinary differential equations realizing any fixed finite set of equilibria in any fixed finite dimension are introduced; no spurious equilibria are possible for either method. By using the first method, one can construct a system with the fewest number of equilibria, given a fixed set of attractors. Using a strict Lyapunov function for each of these differential equations, a large class of systems with the same set of equilibria is constructed. A method of fitting these nonlinear systems to trajectories is proposed. In addition, a general method which will produce an arbitrary number of periodic orbits of shapes of arbitrary complexity is also discussed. A more general second method is given to construct a differential equation which converges to a fixed given finite set of equilibria. This technique is much more general in that it allows this set of equilibria to have any of a large class of indices which are consistent with the Morse Inequalities. It is clear that this class is not universal, because there is a large class of additional vector fields with convergent dynamics which cannot be constructed by the above method. The easiest way to see this is to enumerate the set of Morse indices which can be obtained by the above method and compare this class with the class of Morse indices of arbitrary differential equations with convergent dynamics. The former set of indices are a proper subclass of the latter, therefore, the above construction cannot be universal. In general, it is a difficult open problem to construct a specific example of a differential equation with a given fixed set of equilibria, permissible Morse indices, and permissible connections between stable and unstable manifolds. A strict Lyapunov function is given for this second case as well. This strict Lyapunov function as above enables construction of a large class of examples consistent with these more complicated dynamics and indices. The determination of all the basins of attraction in the general case for these systems is also difficult and open.

Ubuntu, Jesus, and Earth: Integrating African Religion and Christianity in Ecological Ethics

Africa faces problems of ecological devastation caused by economic exploitation, rapid population growth, and poverty. Capitalism, residual colonialism, and corruption undermine Africa's efforts to forge a better future. The dissertation describes how in Africa the mounting ecological crisis has religious, political, and economic roots that enable and promote social and environmental harm. It presents the thesis that religious traditions, including their ethical expressions, can effectively address the crisis, ameliorate its impacts, and advocate for social and environmental betterment, now and in the future. First, it examines African traditional religion and Christian teaching, which together provide the foundation for African Christianity. Critical examination of both religious worldviews uncovers their complementary emphases on human responsibility toward planet Earth and future generations. Second, an analysis of the Gwembe Tonga of Chief Simamba explores the interconnectedness of all elements of the universe in African cosmologies. In Africa, an interdependent, participatory relationship exists between the world of animals, the world of humans, and the Creator. In discussing the annual lwiindi (rain calling) ceremony of Simamba, the study explores ecological overtones of African religions. Such rituals illustrate the involvement of ancestors and high gods in maintaining ecological integrity. Third, the foundation of the African morality of abundant life is explored. Across Sub-Saharan Africa, ancestors' teachings are the foundation of morality; ancestors are guardians of the land. A complementary teaching that Christ is the ecological ancestor of all life can direct ethical responses to the ecological crisis. Fourth, the eco-social implications of ubuntu (what it means to be fully human) are examined. Some aspects of ubuntu are criticized in light of economic inequalities and corruption in Africa. However, ubuntu can be transformed to advocate for eco-social liberation. Fifth, the study recognizes that in some cases conflicts exist between ecological values and religious teachings. This conflict is examined in terms of the contrast between awareness of socioeconomic problems caused by population growth, on the one hand, and advocacy of a traditional African morality of abundant children, on the other hand. A change in the latter religious view is needed since overpopulation threatens sustainable living and the future of Earth. The dissertation concludes that the identification of Jesus with African ancestors and theological recognition of Jesus as the ecological ancestor, woven together with ubuntu, an ethic of interconnectedness, should characterize African consciousness and promote resolution of the socio-ecological crisis.

The Synthesis of Arbitrary Stable Dynamics in Non-linear Neural Networks II: Feedback and Universality

We wish to construct a realization theory of stable neural networks and use this theory to model the variety of stable dynamics apparent in natural data. Such a theory should have numerous applications to constructing specific artificial neural networks with desired dynamical behavior. The networks used in this theory should have well understood dynamics yet be as diverse as possible to capture natural diversity. In this article, I describe a parameterized family of higher order, gradient-like neural networks which have known arbitrary equilibria with unstable manifolds of known specified dimension. Moreover, any system with hyperbolic dynamics is conjugate to one of these systems in a neighborhood of the equilibrium points. Prior work on how to synthesize attractors using dynamical systems theory, optimization, or direct parametric. fits to known stable systems, is either non-constructive, lacks generality, or has unspecified attracting equilibria. More specifically, We construct a parameterized family of gradient-like neural networks with a simple feedback rule which will generate equilibrium points with a set of unstable manifolds of specified dimension. Strict Lyapunov functions and nested periodic orbits are obtained for these systems and used as a method of synthesis to generate a large family of systems with the same local dynamics. This work is applied to show how one can interpolate finite sets of data, on nested periodic orbits.