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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 ...
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.
However, such a fixed point interchanges its stability with another fixed point as the parameter is varied. [1] In other words, both before and after the bifurcation, there is one unstable and one stable fixed point. However, their stability is exchanged when they collide. So the unstable fixed point becomes stable and vice versa.
Symmetry breaking in pitchfork bifurcation as the parameter ε is varied. ε = 0 is the case of symmetric pitchfork bifurcation.. In a dynamical system such as ¨ + (;) + =, which is structurally stable when , if a bifurcation diagram is plotted, treating as the bifurcation parameter, but for different values of , the case = is the symmetric pitchfork bifurcation.
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.
Another name is blue sky bifurcation in reference to the sudden creation of two fixed points. [1] If the phase space is one-dimensional, one of the equilibrium points is unstable (the saddle), while the other is stable (the node). Saddle-node bifurcations may be associated with hysteresis loops and catastrophes.
The unstable solution can then form another saddle-node bifurcation with the initial stable solution at a higher value of the parameter, leaving only the higher fixed solution. Thus, at values of the parameter between the two critical values, the system has two stable solutions. An example of a dynamical system that demonstrates similar features is
For r < 1, exists outside [0, 1] as an unstable fixed point, but for r = 1, the two fixed points collide, and for r > 1, appears between [0, 1] as a stable fixed point. When the parameter r = 1, the trajectory of the logistic map converges to 0 as before, but the convergence speed is slower at r = 1.