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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 ...
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, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically, for functions, a fixed point is an element that is mapped to itself by the function. Any set of fixed points of a transformation is also an invariant set.
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 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.
If < there are two equilibrium points, a stable equilibrium point at and an unstable one at +. At = (the bifurcation point) there is exactly one equilibrium point. At this point the fixed point is no longer hyperbolic. In this case the fixed point is called a saddle-node fixed point.
On the Lémeray diagram, a stable fixed point corresponds to the segment of the staircase with progressively decreasing stair lengths or to an inward spiral, while an unstable fixed point is the segment of the staircase with growing stairs or an outward spiral.
In several typical examples, the system has only one stable fixed point at low values of the parameter. A saddle-node bifurcation gives rise to a pair of new fixed points emerging, one stable and the other unstable, at a critical value of the parameter. The unstable solution can then form another saddle-node bifurcation with the initial stable ...