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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 ...
For example, the function / (defined as 0 in x=0) has all derivatives equal to zero. At the same time, this function has a local minimum in x=0, so it is a stable equilibrium. If this function is multiplied by the Sign function, all derivatives will still be zero but it will become an unstable equilibrium.
A ball located at this point, ball 3, is in equilibrium but unstable; the slightest disturbance will cause it to move to one of the stable points. Light switch, a bistable mechanism. In a dynamical system, bistability means the system has two stable equilibrium states. [1] A bistable structure can be resting in either of two states.
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 +.
For example, if the bowl containing a rolling marble was inverted and the marble was balanced on top of the bowl, the center bottom (now top) of the bowl is a fixed state, but not an attractor. This is equivalent to the difference between stable and unstable equilibria. In the case of a marble on top of an inverted bowl (a hill), that point at ...
a) stable, b) turbulent. In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these instabilities will cause the development of turbulence. [1]
In simple terms, if the solutions that start out near an equilibrium point stay near forever, then is Lyapunov stable. More strongly, if x e {\displaystyle x_{e}} is Lyapunov stable and all solutions that start out near x e {\displaystyle x_{e}} converge to x e {\displaystyle x_{e}} , then x e {\displaystyle x_{e}} is said to be asymptotically ...
[6] [non-primary source needed] The summary of the results is that the three-dimensional homogeneous convex (mono-monostatic) body, which has one stable and one unstable equilibrium point, does exist and is not unique. Their form is dissimilar to any typical representative of any other equilibrium geometrical class.