Dynamical systemIn mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured.
Dynamical systems theoryDynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle.
Limit cycleIn mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912).
Limit setIn mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can be used to understand the long term behavior of a dynamical system. A system that has reached its limiting set is said to be at equilibrium.
Measure-preserving dynamical systemIn mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special case of conservative systems. They provide the formal, mathematical basis for a broad range of physical systems, and, in particular, many systems from classical mechanics (in particular, most non-dissipative systems) as well as systems in thermodynamic equilibrium.
AutonomyIn developmental psychology and moral, political, and bioethical philosophy, autonomy is the capacity to make an informed, uncoerced decision. Autonomous organizations or institutions are independent or self-governing. Autonomy can also be defined from a human resources perspective, where it denotes a (relatively high) level of discretion granted to an employee in his or her work. In such cases, autonomy is known to generally increase job satisfaction. Self-actualized individuals are thought to operate autonomously of external expectations.
Integrable systemIn mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.
Collision avoidance systemA collision avoidance system (CAS), also known as a pre-crash system, forward collision warning system (FCW), or collision mitigation system, is an advanced driver-assistance system designed to prevent or reduce the severity of a collision. In its basic form, a forward collision warning system monitors a vehicle's speed, the speed of the vehicle in front of it, and the distance between the vehicles, so that it can provide a warning to the driver if the vehicles get too close, potentially helping to avoid a crash.
Autonomous administrative divisionAn autonomous administrative division (also referred to as an autonomous area, entity, unit, region, subdivision, or territory) is a subnational administrative division or internal territory of a sovereign state that has a degree of autonomy—self-governance—under the national government. Autonomous areas are distinct from the constituent units of a federation (e.g. a state, or province) in that they possess unique powers for their given circumstances. Typically, it is either geographically distinct from the rest of the state or populated by a national minority.
