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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 ...
This means that the rank at the critical point is lower than the rank at some neighbour point. In other words, let k be the maximal dimension of the open balls contained in the image of f; then a point is critical if all minors of rank k of f are zero. In the case where m = n = k, a point is critical if the Jacobian determinant is zero.
Stability generally increases to the left of the diagram. [1] Some sink, source or node are equilibrium points. In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation.
Instead of considering stability only near an equilibrium point (a constant solution () =), one can formulate similar definitions of stability near an arbitrary solution () = (). However, one can reduce the more general case to that of an equilibrium by a change of variables called a "system of deviations".
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. If > there are no equilibrium points. [2] Saddle node bifurcation. In fact, this is a normal form of a saddle-node bifurcation.
Randomly selected points of the 2D phase space converge exponentially to a 1D center manifold on which dynamics are slow (non exponential). Studying dynamics of the center manifold determines the stability of the non-hyperbolic fixed point at the origin. The center manifold of a dynamical system is based upon an equilibrium point of that
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
Thus the criterion provides a way to determine if the equations of motion of a linear system have only stable solutions, without solving the system directly. For discrete systems, the corresponding stability test can be handled by the Schur–Cohn criterion, the Jury test and the Bistritz test. With the advent of computers, the criterion has ...