Transverse Contraction Criteria for Existence, Stability, and Robustness of a Limit Cycle

This paper derives a differential contraction condition for the existence of an orbitally-stable limit cycle in an autonomous system. This transverse contraction condition can be represented as a pointwise linear matrix inequality (LMI), thus allowing convex optimization tools such as sum-of-squares programming to be used to search for certificates of the existence of a stable limit cycle. Many desirable properties of contracting dynamics are extended to this context, including preservation of contraction under a broad class of interconnections. In addition, by introducing the concepts of differential dissipativity and transverse differential dissipativity, contraction and transverse contraction can be established for large scale systems via LMI conditions on component subsystems.Comment: 6 pages, 1 figure. Conference submissio

Nonlinear and Complex Dynamics in Economics

This paper is an up-to-date survey of the state-of-the-art in dynamical systems theory relevant to high levels of dynamical complexity, characterizing chaos and near chaos, as commonly found in the physical sciences. The paper also surveys applications in economics and �finance. This survey does not include bifurcation analyses at lower levels of dynamical complexity, such as Hopf and transcritical bifurcations, which arise closer to the stable region of the parameter space. We discuss the geometric approach (based on the theory of differential/difference equations) to dynamical systems and make the basic notions of complexity, chaos, and other related concepts precise, having in mind their (actual or potential) applications to economically motivated questions. We also introduce specifi�c applications in microeconomics, macroeconomics, and �finance, and discuss the policy relevancy of chaos

Hydrodynamics of Core-Collapse Supernovae at the Transition to Explosion. I. Spherical Symmetry

We study the transition to runaway expansion of an initially stalled core-collapse supernova shock. The neutrino luminosity, mass accretion rate, and neutrinospheric radius are all treated as free parameters. In spherical symmetry, this transition is mediated by a global non-adiabatic instability that develops on the advection time and reaches non-linear amplitude. Here we perform high-resolution, time-dependent hydrodynamic simulations of stalled supernova shocks with realistic microphysics to analyze this transition. We find that radial instability is a sufficient condition for runaway expansion if the neutrinospheric parameters do not vary with time and if heating by the accretion luminosity is neglected. For a given unstable mode, transition to runaway occurs when fluid in the gain region reaches positive specific energy. We find approximate instability criteria that accurately describe the behavior of the system over a wide region of parameter space. The threshold neutrino luminosities are in general different than the limiting value for a steady-state solution. We hypothesize that multidimensional explosions arise from the excitation of unstable large-scale modes of the turbulent background flow, at threshold luminosities that are lower than in the laminar case.Comment: Accepted for publication in ApJ. Added an appendix describing implementation and testing of a grid of variable spacing in FLASH3.2. Minor changes otherwis