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A typical example of a differential equation with a saddle-node bifurcation is: = +. Here is the state variable and is the bifurcation parameter.. If < there are two equilibrium points, a stable equilibrium point at and an unstable one at +.
In mathematics, specifically in differential equations, an equilibrium point is a constant solution to a differential equation. Formal definition The ...
In mechanics and physics, simple harmonic motion (sometimes abbreviated as SHM) is a special type of periodic motion an object experiences by means of a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position.
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 ...
Hyperbolic equilibrium point, a mathematical concept in physics; Mechanical equilibrium, the state in which the sum of the forces, and torque, on each particle of the system is zero; Radiative equilibrium, the state where the energy radiated is balanced by the energy absorbed
For example, at an amplitude of θ 0 = 0.4 radians (23°) it is 1% larger than given by (1). The period increases asymptotically (to infinity) as θ 0 approaches π radians (180°), because the value θ 0 = π is an unstable equilibrium point for the pendulum.
An example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system.
The harmonic oscillator model is very important in physics, because any mass subject to a force in stable equilibrium acts as a harmonic oscillator for small vibrations. Harmonic oscillators occur widely in nature and are exploited in many manmade devices, such as clocks and radio circuits. They are the source of virtually all sinusoidal ...