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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.
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 ...
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 ...
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 .
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form / =, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part.
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.
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 discontinuity in , and other properties, e.g. internal energy, , and entropy,, of the substance, is called a first order phase transition. [12] [13] In order to specify the unique experimentally observed pressure, (), at which it occurs another thermodynamic condition is required, for from Fig.1 it could clearly occur for any pressure in the range .