Dynamical systems theory in the context of "Equations of motion"

A sequence of events is isochronous if the events occur regularly, or at equal time intervals. The term isochronous is used in several technical contexts, but usually refers to the primary subject maintaining a constant period or interval (the reciprocal of frequency), despite variations in other measurable factors in the same system. Isochronous timing is a characteristic of a repeating event, whereas synchronous timing refers to the relationship between two or more events.

• In dynamical systems theory, an oscillator is called isochronous if its frequency is independent of its amplitude.
• In horology, a mechanical clock or watch is isochronous if it runs at the same rate regardless of changes in its drive force, so that it keeps correct time as its mainspring unwinds or chain length varies. Isochrony is important in timekeeping devices. Simply put, if a power-providing device (e.g. a spring or weight) provides constant torque to the wheel train, it will be isochronous, since the escapement will experience the same force regardless of how far the weight has dropped or the spring has unwound.
• In electrical power generation, isochronous means that the frequency of the electricity generated is constant under varying load; there is zero generator droop. (See Synchronization (alternating current).)
• In telecommunications, an isochronous signal is one where the time interval separating any two corresponding transitions is equal to the unit interval or to a multiple of the unit interval; but phase is arbitrary and potentially varying.
  • The term is also used in data transmission to describe cases in which corresponding significant instants of two or more sequential signals have a constant phase relationship.
  • Isochronous burst transmission is used when the information-bearer channel rate is higher than the input data signaling rate.
• In the Universal Serial Bus used in computers, isochronous is one of the four data flow types for USB devices (the others being Control, Interrupt and Bulk). It is commonly used for streaming data types such as video or audio sources. Similarly, the IEEE 1394 interface standard, commonly called Firewire, includes support for isochronous streams of audio and video at known constant rates.
• In particle accelerators an isochronous cyclotron is a cyclotron where the field strength increases with radius to compensate for the relativistic increase in mass with speed.
• An isochrone is a contour line of equal time, for instance, in geological layers, tree rings or wave fronts. An isochrone map or diagram shows such contours.
• In linguistics, isochrony is the postulated rhythmic division of time into equal portions by a language.
• In neurology, isochronic tones are regular beats of a single tone used for brainwave entrainment.

Large-scale brain networks (also known as intrinsic brain networks) are collections of widespread brain regions showing functional connectivity by statistical analysis of the fMRI BOLD signal or other recording methods such as EEG, PET and MEG. An emerging paradigm in neuroscience is that cognitive tasks are performed not by individual brain regions working in isolation but by networks consisting of several discrete brain regions that are said to be "functionally connected". Functional connectivity networks may be found using algorithms such as cluster analysis, spatial independent component analysis (ICA), seed based, and others. Synchronized brain regions may also be identified using long-range synchronization of the EEG, MEG, or other dynamic brain signals.

The set of identified brain areas that are linked together in a large-scale network varies with cognitive function. When the cognitive state is not explicit (i.e., the subject is at "rest"), the large-scale brain network is a resting state network (RSN). As a physical system with graph-like properties, a large-scale brain network has both nodes and edges and cannot be identified simply by the co-activation of brain areas. In recent decades, the analysis of brain networks was made feasible by advances in imaging techniques as well as new tools from graph theory and dynamical systems.

In atmospheric science, an atmospheric model is a mathematical model constructed around the full set of primitive, dynamical equations which govern atmospheric motions. It can supplement these equations with parameterizations for turbulent diffusion, radiation, moist processes (clouds and precipitation), heat exchange, soil, vegetation, surface water, the kinematic effects of terrain, and convection. Most atmospheric models are numerical, i.e. they discretize equations of motion. They can predict microscale phenomena such as tornadoes and boundary layer eddies, sub-microscale turbulent flow over buildings, as well as synoptic and global flows. The horizontal domain of a model is either global, covering the entire Earth (or other planetary body), or regional (limited-area), covering only part of the Earth. Atmospheric models also differ in how they compute vertical fluid motions; some types of models are thermotropic, barotropic, hydrostatic, and non-hydrostatic. These model types are differentiated by their assumptions about the atmosphere, which must balance computational speed with the model's fidelity to the atmosphere it is simulating.

Forecasts are computed using mathematical equations for the physics and dynamics of the atmosphere. These equations are nonlinear and are impossible to solve exactly. Therefore, numerical methods obtain approximate solutions. Different models use different solution methods. Global models often use spectral methods for the horizontal dimensions and finite-difference methods for the vertical dimension, while regional models usually use finite-difference methods in all three dimensions. For specific locations, model output statistics use climate information, output from numerical weather prediction, and current surface weather observations to develop statistical relationships which account for model bias and resolution issues.