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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 ...
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.
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).
The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov . In simple terms, if the solutions that start out near an equilibrium point x e {\displaystyle x_{e}} stay near x e {\displaystyle x_{e}} forever, then x e {\displaystyle x_{e}} is ...
In a dynamical system, multistability is the property of having multiple stable equilibrium points in the vector space spanned by the states in the system. By mathematical necessity, there must also be unstable equilibrium points between the stable points.
Stability generally increases to the left of the diagram. [1] Some sink, source or node are equilibrium points . 2-dimensional case refers to Phase plane . In mathematics , an autonomous system or autonomous differential equation is a system of ordinary differential equations which does not explicitly depend on the independent variable .
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.
Linear approximation of a nonlinear system: classification of 2D fixed point according to the trace and the determinant of the Jacobian matrix (the linearization of the system near an equilibrium point). The roots of the characteristic polynomial det(A - λI) are the eigenvalues of A.