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When is positive, there are two equilibrium points: that is, one saddle point and one node (either an attractor or a repellor). Other examples are in modelling biological switches. [ 4 ] Recently, it was shown that under certain conditions, the Einstein field equations of General Relativity have the same form as a fold bifurcation. [ 5 ]
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.
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 ...
In mechanics and physics, simple harmonic motion (sometimes abbreviated as SHM) is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position.
Partial equilibrium, the equilibrium price and quantity which come from the cross of supply and demand in a competitive market; Radner equilibrium, an economic concept defined by economist Roy Radner in the context of general equilibrium; Recursive competitive equilibrium, an economic equilibrium concept associated with a dynamic program
Vibration (from Latin vibrāre 'to shake') is a mechanical phenomenon whereby oscillations occur about an equilibrium point.Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the oscillations can only be analysed statistically (e.g. the movement of a tire on a gravel road).
Some sink, source or node are equilibrium points. 2-dimensional case refers to Phase plane. In mathematics, an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable. When the variable is time, they are also called time-invariant systems.
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,