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Asymptotic stability of fixed points of a non-linear system can often be established using the Hartman–Grobman theorem. Suppose that v is a C 1 - vector field in R n which vanishes at a point p , v ( p ) = 0 .
Complex eigenvalues of an arbitrary map (dots). In case of the Hopf bifurcation, two complex conjugate eigenvalues cross the imaginary axis. In the mathematical theory of bifurcations, a Hopf bifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic solution arises. [1]
If is a hyperbolic periodic point, the stable manifold theorem assures that for some neighborhood of , the local stable and unstable sets are embedded disks, whose tangent spaces at are and (the stable and unstable spaces of ()), respectively; moreover, they vary continuously (in a certain sense) in a neighborhood of in the topology of () (the ...
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.
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 ...
In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. In discrete systems (described by maps), this corresponds to a fixed point having a Floquet multiplier with modulus equal to one. In both cases, the equilibrium is non-hyperbolic at the bifurcation point. The topological changes in ...
Let : be a smooth map with hyperbolic fixed point at .We denote by () the stable set and by () the unstable set of .. The theorem [2] [3] [4] states that is a smooth manifold and its tangent space has the same dimension as the stable space of the linearization of at .
For example, when one point is plotted for r, it is a fixed point, and when m points are plotted for r, it corresponds to an m-periodic orbit. When an orbital diagram is drawn for the logistic map, it is possible to see how the branch representing the stable periodic orbit splits, which represents a cascade of period-doubling bifurcations.