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In mathematics, a phase portrait is a geometric representation of the orbits of a dynamical system in the phase plane. ... Phase Portrait Behavior [1] Eigenvalue ...
The signs of the eigenvalues indicate the phase plane's behaviour: If the signs are opposite, the intersection of the eigenvectors is a saddle point . If the signs are both positive, the eigenvectors represent stable situations that the system diverges away from, and the intersection is an unstable node .
Phase portrait showing saddle-node bifurcation. Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.
But the topological conjugacy in this context does provide the full geometric picture. In effect, the nonlinear phase portrait near the equilibrium is a thumbnail of the phase portrait of the linearized system. This is the meaning of the following regularity results, and it is illustrated by the saddle equilibrium in the example below.
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]
The Federal Reserve cut its key interest rate Wednesday by a quarter-point — its third cut this year — but also signaled that it expects to reduce rates more slowly next year than it ...
Israel occupied the Golan Heights during the 1967 Mideast War and later annexed the roughly 460-square-mile area in 1981 in a unilateral decision that was not recognized by the international ...
MOSCOW (Reuters) -Russia said on Wednesday that relations with Washington were so confrontational that Russian citizens should not visit the United States, Canada and some EU countries in coming ...