
In
applied mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
and
astrodynamics
Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the Newton's law of univ ...
, 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 crisis is the sudden appearance or disappearance of a
strange attractor
In the mathematics, 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 va ...
as the parameters of a
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 ...
are varied. This
global bifurcation occurs when a
chaotic attractor comes into contact with an
unstable
In dynamical systems instability means that some of the outputs or internal state (controls), states increase with time, without bounds. Not all systems that are not Stability theory, stable are unstable; systems can also be marginal stability ...
periodic orbit
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
Iterated functions
Given ...
or its
stable manifold
In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repell ...
. As the orbit approaches the unstable orbit it will diverge away from the previous attractor, leading to a qualitatively different behaviour. Crises can produce
intermittent behaviour.
Grebogi, Ott, Romeiras, and Yorke distinguished between three types of crises:
* The first type, a boundary or an exterior crisis, the attractor is suddenly destroyed as the parameters are varied. In the postbifurcation state the motion is transiently chaotic, moving chaotically along the former attractor before being attracted to a
fixed point, periodic orbit,
quasiperiodic orbit, another strange attractor, or diverging to infinity.
* In the second type of crisis, an interior crisis, the size of the chaotic attractor suddenly increases. The attractor encounters an unstable fixed point or periodic solution that is inside the
basin of attraction
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 ...
.
* In the third type, an attractor merging crisis, two or more chaotic attractors merge to form a single attractor as the critical parameter value is passed.
Note that the reverse case (sudden appearance, shrinking or splitting of attractors) can also occur. The latter two crises are sometimes called explosive bifurcations.
While crises are "sudden" as a parameter is varied, the dynamics of the system over time can show long transients before orbits leave the neighbourhood of the old attractor. Typically there is a time constant τ for the length of the transient that diverges as a power law (τ ≈ , ''p'' − ''p''
c,
''γ'') near the critical parameter value ''p''
c. The exponent ''γ'' is called the critical crisis exponent. There also exist systems where the divergence is stronger than a power law, so-called super-persistent chaotic transients.
See also
*
Intermittency
In dynamical systems, intermittency is the irregular alternation of phases of apparently periodic and chaotic dynamics ( Pomeau–Manneville dynamics), or different forms of chaotic dynamics (crisis-induced intermittency).
Experimentally ...
*
Bifurcation diagram
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically ( fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the ...
*
Phase portrait
References
{{Reflist
External links
Scholarpedia: Crises
Dynamical systems
Nonlinear systems
Bifurcation theory