Crisis (dynamical systems)
Encyclopedia
In applied mathematics, in the theory of dynamical systems, a crisis is the sudden appearance or disappearance of a strange attractor as the parameters of a dynamical system
are varied. This global bifurcation occurs when a chaotic
attractor comes into contact with an unstable periodic orbit or its stable manifold
. 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:
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 − pc|γ) near the critical parameter value pc. 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.
Dynamical system
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
are varied. This global bifurcation occurs when a chaotic
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
attractor comes into contact with an unstable periodic orbit or its stable manifold
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 repellor...
. 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
Intermittency
In dynamical systems, intermittency is the irregular alternation of phases of apparently periodic and chaotic dynamics , or different forms of chaotic dynamics ....
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 pointFixed point (mathematics)In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...
, periodic orbit, quasiperiodic orbitQuasiperiodic motionIn mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number of incommensurable frequencies....
, 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 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 − pc|γ) near the critical parameter value pc. 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.