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In a conservative force field, bistability stems from the fact that the potential energy has two local minima, which are the stable equilibrium points. [2] These rest states need not have equal potential energy. By mathematical arguments, a local maximum, an unstable equilibrium point, must lie between the two minima. At rest, a particle will ...
English: Graph of the potential energy of a conservative bistable system. There are two local minima of the potential energy at x 1 and x 2 which are stable equilibrium points at which the red object can rest. Between them is a local maximum of energy x 3 which is an unstable equilibrium point. If the red object is placed there it is in ...
Diagram of a ball placed in an unstable equilibrium. Second derivative < 0 The potential energy is at a local maximum, which means that the system is in an unstable equilibrium state. If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away.
An attractor is a stable point which is also called a "sink". The repeller is considered as an unstable point, which is also known as a "source". A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and
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 activated complex is an arrangement of atoms in an arbitrary region near the saddle point of a potential energy surface. [1] The region represents not one defined state, but a range of unstable configurations that a collection of atoms pass through between the reactants and products of a reaction. Activated complexes have partial reactant ...
The second method, which is now referred to as the Lyapunov stability criterion or the Direct Method, makes use of a Lyapunov function V(x) which has an analogy to the potential function of classical dynamics. It is introduced as follows for a system ˙ = having a point of equilibrium at =.
The simplest non-trivial examples are the exponential growth model/decay (one unstable/stable equilibrium) and the logistic growth model (two equilibria, one stable, one unstable). The phase space of a two-dimensional system is called a phase plane , which occurs in classical mechanics for a single particle moving in one dimension, and where ...