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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 ...
The potential energy is at a local maximum, which means that the system is in an unstable equilibrium state. If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away. Diagram of a ball placed in a stable equilibrium. Second derivative > 0
It is possible for an ecosystem or a community to be stable in some of their properties and unstable in others. For example, a vegetation community in response to a drought might conserve biomass but lose biodiversity. [3] Stable ecological systems abound in nature, and the scientific literature has documented them to a great extent.
An equilibrium point is hyperbolic if none of the eigenvalues have zero real part. If all eigenvalues have negative real parts, the point is stable. If at least one has a positive real part, the point is unstable.
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form / =, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part.
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 +.
When the curve goes down to zero the point is unstable, and will flow down to zero along the action of . When the flow stays between zero and infinity, the point is in an unstable equilibrium (semi-stable). This analogy with mechanical equilibrium motivates the terminology of stability and instability.
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 ...