Search results
Results From The WOW.Com Content Network
In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. Formal definition The ...
Strogatz notes that "hyperbolic is an unfortunate name—it sounds like it should mean 'saddle point'—but it has become standard." [1] Several properties hold about a neighborhood of a hyperbolic point, notably [2] Orbits near a two-dimensional saddle point, an example of a hyperbolic equilibrium. A stable manifold and an unstable manifold exist,
A typical example of a differential equation with a saddle-node bifurcation is: = +. Here is the state variable and is the bifurcation parameter.. If < there are two equilibrium points, a stable equilibrium point at and an unstable one at +.
Hyperbolic paraboloid A model of an elliptic hyperboloid of one sheet A monkey saddle. A saddle surface is a smooth surface containing one or more saddle points.. Classical examples of two-dimensional saddle surfaces in the Euclidean space are second order surfaces, the hyperbolic paraboloid = (which is often referred to as "the saddle surface" or "the standard saddle surface") and the ...
The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov . In simple terms, if the solutions that start out near an equilibrium point x e {\displaystyle x_{e}} stay near x e {\displaystyle x_{e}} forever, then x e {\displaystyle x_{e}} is ...
The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand ...
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
Varying the payoffs in the matrix can change the equilibrium point. For example, in the table shown on the right, Even has a chance to win 7 if both he and Odd play Heads. To calculate the equilibrium point in this game, note that a player playing a mixed strategy must be indifferent between his two actions (otherwise he would switch to a pure ...