"In order to know the curvature of a curve, the determination of the radius of the osculating circle furnishes us the best measure, where for each point of the curve we find a circle whose curvature is precisely the same. However, when one looks for the curvature of a surface, the question is very equivocal and not at all susceptible to an absolute response, as in the case above. There are only spherical surfaces where one would be able to measure the curvature, assuming the curvature of the sphere is the curvature of its great circles, and whose radius could be considered the appropriate measure. But for other surfaces one doesn’t know even how to compare a surface with a sphere, as when one can always compare the curvature of a curve with that of a circle. The reason is evident, since at each point of a surface there are an infinite number of different curvatures. One has to only consider a cylinder, where along the directions parallel to the axis, there is no curvature, whereas in the directions perpendicular to the axis, which are circles, the curvatures are all the same, and all other oblique sections to the axis give a particular curvature. It’s the same for all other surfaces, where it can happen that in one direction the curvature is convex, and in another it is concave, as in those resembling a saddle." (Leonhard Euler, "Recherches sur la courbure des surfaces", 1767)
"While the minimum and maximum problems of calculus of variations correspond to the problem in the ordinary calculus of finding peaks and pits of a surface, the minimax problems correspond to the problem of finding the saddle points of the surface (the passes of a mountain)."(Karl Menger, "What Is Calculus of Variations and What Are Its Applications?" [James R Newman, "The World of Mathematics" Vol. II], 1956)
"A proven theorem of game theory states that every game with complete information possesses a saddle point and therefore a solution." (Richard A Epstein, "The Theory of Gambling and Statistical Logic" [Revised Edition], 1977)
"So the strategy of mixing the choices with equal likelihood is an equilibrium point for the game, in the same sense that the minimax point is an equilibrium for a game having a saddle point. Thus, using a strategy that randomizes their choices, Max and Min can each announce his or her strategy to the other without the opponent being able to exploit this information to get a larger average payoff for himself or herself." (John L Casti, "Five Golden Rules", 1995)
"What's important about a saddle point is that it represents a decision by the two players that neither can improve upon by unilaterally departing from it. In short, either player can announce such a choice in advance to the other player and suffer no penalty by doing so. Consequently, the best choice for each player is at the saddle point, which is called a 'solution' to the game in pure strategies. This is because regardless of the number of times the game is played, the optimal choice for each player is to always take his or her saddle-point decision. […] the saddle point is at the same time the highest point on the payoff surface in one direction and the lowest in the other direction. Put in algebraic terms using the payoff matrix, the saddle point is where the largest of the row minima coincides with the smallest of the column maxima." (John L Casti, "Five Golden Rules", 1995)
"If sinks, sources, saddles, and limit cycles are coins landing heads or tails, then the exceptions are a coin landing on edge. Yes, it might happen, in theory; but no, it doesn't, in practice." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)
"An attractor is regarded as a stable structure because all the points within it follow the same rules of motion. There are four principal types of attractors: the fixed point, the limit cycle, toroidal attractors, and chaotic attractors. Each type reflects a distinctly different type of movement that occurs within it. Repellors and saddles are closely related structures that are not structurally stable." (Stephen J Guastello & Larry S Liebovitch, "Introduction to Nonlinear Dynamics and Complexity" [in "Chaos and Complexity in Psychology"], 2009)
"In parametrized dynamical systems a bifurcation occurs when a qualitative change is invoked by a change of parameters. In models such a qualitative change corresponds to transition between dynamical regimes. In the generic theory a finite list of cases is obtained, containing elements like ‘saddle-node’, ‘period doubling’, ‘Hopf bifurcation’ and many others." (Henk W Broer & Heinz Hanssmann, "Hamiltonian Perturbation Theory (and Transition to Chaos)", 2009)
"Saddle points have properties of both attractors and repellors. On one hand, they attract traveling points. On the other, the points do not stay on the saddle point very long before they are deflected in another direction. The saddle phenomenon is similar to the flow of people to and from an information booth in an airport or other major facility. The information booth attracts people, but the booth is not their destination. The visitors remain there only long enough to find out what they need to know to get to where they are going. Again, because the points are following different rules of flow, the system is not stable." (Stephen J Guastello & Larry S Liebovitch, "Introduction to Nonlinear Dynamics and Complexity" [in "Chaos and Complexity in Psychology"], 2009)
"[…] what exactly do we mean by a bifurcation? The usual definition involves the concept of 'topological equivalence': if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred. Examples include changes in the number or stability of fixed points, closed orbits, or saddle connections as a parameter is varied."
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