Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs-Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain a new analytical formula for the first Lyapunov’s (focal) value at the boundary of stability of the corresponding equilibrium state. On the other hand, the global analysis reveals that the system may exhibit the phenomena of Shilnikov chaos. Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator. Finally, it is found that for some new combinations of parameters, the system demonstrates chaotic behavior and transition from chaos to regular behavior is realized through inverse period-doubling bifurcations.

Values Across the Lifespan

Chapter 1 introduces the chief claim and main argument of the book, which we call the life stage relativity of values. This is the claim that different values matter more at different stages of our lives. During early life, caring, trust, and nurturing ought to figure prominently, due to the vulnerabilities and needs that characterize infancy and childhood. By young adulthood, the capacity to develop greater physical and emotional independence makes autonomy a focal value. During later life, we face heightened risk for chronic disease and disability, which makes maintaining capabilities central, and, in the face of loss, keeping dignity intact. Chapter 1 raises the concern that moral theories reflect life stage bias, in particular, midlife bias. Midlife bias consists of applying the values central during midlife to all life stages. Countering it requires addressing empirical, conceptual, and psychological naïveté and situating values within the context of life stages.

ON THE CENTER PROBLEM FOR DEGENERATE SINGULAR POINTS OF PLANAR VECTOR FIELDS

The center problem for degenerate singular points of planar systems (the degenerate-center problem) is a poorly-understood problem in the qualitative theory of ordinary differential equations. It may be broken down into two problems: the monodromy problem, to decide if the singular point is of focus-center type, and the stability problem, to decide whether it is a focus or a center. We present an outline on the status of the center problem for degenerate singular points, explaining the main techniques and obstructions arising in the study of the problem. We also present some new results. Our new results are the characterization of a family of vector fields having a degenerate monodromic singular point at the origin, and the computation of the generalized first focal value for this family V1. This gives the solution of the stability problem in the monodromic case, except when V1 = 1. Our approach relies on the use of the blow-up technique and the study of the blow-up geometry of singular points. The knowledge of the blow-up geometry is used to generate a bifurcation of a limit cycle.