Connecting period-doubling cascades to chaos

The appearance of infinitely-many period-doubling cascades is one of the most prominent features observed in the study of maps depending on a parameter. They are associated with chaotic behavior, since bifurcation diagrams of a map with a parameter often reveal a complicated intermingling of period-doubling cascades and chaos. Period doubling can be studied at three levels of complexity. The first is an individual period-doubling bifurcation. The second is an infinite collection of period doublings that are connected together by periodic orbits in a pattern called a cascade. It was first described by Myrberg and later in more detail by Feigenbaum. The third involves infinitely many cascades and a parameter value $\mu_2$ of the map at which there is chaos. We show that often virtually all (i.e., all but finitely many) ``regular'' periodic orbits at $\mu_2$ are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired -- connected to exactly one other cascade, or solitary -- connected to exactly one regular periodic orbit at $\mu_2$. The solitary cascades are robust to large perturbations. Hence the investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of $F(\mu_2, \cdot)$. Examples discussed include the forced-damped pendulum and the double-well Duffing equation.Comment: 29 pages, 13 figure

Ambassadors of the game: do famous athletes have special obligations to act virtuously?

Do famous athletes have special obligations to act virtuously? A number of philosophers have investigated this question by examining whether famous athletes are subject to special role model obligations (Wellman 2003; Feezel 2005; Spurgin 2012). In this paper we will take a different approach and give a positive response to this question by arguing for the position that sport and gaming celebrities are ‘ambassadors of the game’: moral agents whose vocations as rule-followers have unique implications for their non-lusory lives. According to this idea, the actions of a game’s players and other stakeholders—especially the actions of its stars—directly affect the value of the game itself, a fact which generates additional moral reasons to behave in a virtuous manner. We will begin by explaining the three main positions one may take with respect to the question: moral exceptionalism, moral generalism, and moral exemplarism. We will argue that no convincing case for moral exemplarism has thus far been made, which gives us reason to look for new ways to defend this position. We then provide our own ‘ambassadors of the game’ account and argue that it gives us good reason to think that sport and game celebrities are subject to special obligations to act virtuously