In the theory of

dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

, a bailout embedding is a system defined as :

&\frac = u. \end

Here the function ''k''(''x'') < 0 on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

of unwanted

orbits In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...

; otherwise ''k''(''x'') > 0. The

trajectories A trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traje ...

of the full system of a bailout embedding ''bail out''—that is, detach—from the

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group (mathematics), group that is a subgroup. When some object X is said to be embedded in another object Y ...

, into a larger space, in which they move around. If, after some time these orbits arrive at a

stable A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use tod ...

neighbourhood of the embedding, ''k''(''x'') > 0, they collapse once more onto the embedding; that is, onto the original dynamics. The bailout embedding forms in this way an enlarged version of the

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

, one in which particular sets of orbits are cut from the asymptotic or

limit set In 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 c ...

, while maintaining the dynamics of a different set of orbits—the wanted set—as

attractor In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain c ...

s of the larger dynamical system. With a choice of ''k''(''x'') = −(''γ'' + ∇''f''), these dynamics are seen to detach from unstable regions such as

saddle point In mathematics, a saddle point or minimax point is a Point (geometry), point on the surface (mathematics), surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a Critical point (mathematics), ...

s in

conservative system In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink o ...

s. One important application of the bailout embedding concept is to

divergence-free In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf ...

flows; the most important class of these are

Hamiltonian system A Hamiltonian system is a dynamical system governed by Hamilton's equations. In physics, this dynamical system describes the evolution of a physical system such as a planetary system or an electron in an electromagnetic field. These systems can ...

References

Dynamical systems Nonlinear systems {{chaos-stub