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  2. Stability theory - Wikipedia

    en.wikipedia.org/wiki/Stability_theory

    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 ...

  3. Mechanical equilibrium - Wikipedia

    en.wikipedia.org/wiki/Mechanical_equilibrium

    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

  4. Ecological stability - Wikipedia

    en.wikipedia.org/wiki/Ecological_stability

    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.

  5. Saddle-node bifurcation - Wikipedia

    en.wikipedia.org/wiki/Saddle-node_bifurcation

    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 +.

  6. Equilibrium point (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Equilibrium_point...

    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.

  7. Stability (algebraic geometry) - Wikipedia

    en.wikipedia.org/wiki/Stability_(algebraic_geometry)

    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.

  8. Lyapunov stability - Wikipedia

    en.wikipedia.org/wiki/Lyapunov_stability

    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 ...

  9. Marginal stability - Wikipedia

    en.wikipedia.org/wiki/Marginal_stability

    In the theory of dynamical systems and control theory, a linear time-invariant system is marginally stable if it is neither asymptotically stable nor unstable.Roughly speaking, a system is stable if it always returns to and stays near a particular state (called the steady state), and is unstable if it goes further and further away from any state, without being bounded.