Intermittency
Encyclopedia
In dynamical system
s, 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).
Pomeau and Manneville described three routes to intermittency where a nearly periodic system show irregularly spaced bursts of chaos. These (type I, II and III) correspond to the approach to a saddle-node bifurcation
, a subcritical Hopf bifurcation
, or an inverse period-doubling bifurcation
. In the apparently periodic phases the behaviour is only nearly periodic, slowly drifting away from a unstable periodic orbit. Eventually the system gets far enough away from the periodic orbit to be affected by chaotic dynamics in the rest of the state space
, until it gets close to the orbit again and returns to the nearly periodic behaviour. Since the time spent near the periodic orbit depends sensitively on how closely the system entered its vicinity (in turn determined by what happened during the chaotic period) the length of each phase is unpredictable.
Another kind, on-off intermittency, occurs when a previously transversally stable chaotic attractor with dimension less than the embedding space begins to lose stability. Near unstable orbits within the attractor orbits can escape into the surrounding space, producing a temporary burst before returning to the attractor.
In crisis-induced intermittency a chaotic attractor suffers a crisis
, where two or more attractors cross the boundaries of each other's basin of attraction. As an orbit moves through the first attractor it can cross over the boundary and become attracted to the second attractor, where it will stay until its dynamics moves it across the boundary again.
Intermittent behaviour is commonly observed in turbulent or close to turbulent fluid flows. It has also been experimentally demonstrated in circuit oscillators and chemical reactions.
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...
s, intermittency is the irregular alternation of phases of apparently periodic and 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...
dynamics (Pomeau–Manneville dynamics), or different forms of chaotic dynamics (crisis-induced intermittency).
Pomeau and Manneville described three routes to intermittency where a nearly periodic system show irregularly spaced bursts of chaos. These (type I, II and III) correspond to the approach to a saddle-node bifurcation
Saddle-node bifurcation
In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to...
, a subcritical Hopf bifurcation
Hopf bifurcation
In the mathematical theory of bifurcations, a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of...
, or an inverse period-doubling bifurcation
Period-doubling bifurcation
In mathematics, a period doubling bifurcation in a discrete dynamical system is a bifurcation in which the system switches to a new behavior with twice the period of the original system...
. In the apparently periodic phases the behaviour is only nearly periodic, slowly drifting away from a unstable periodic orbit. Eventually the system gets far enough away from the periodic orbit to be affected by chaotic dynamics in the rest of the state space
State space
In the theory of discrete dynamical systems, a state space is a directed graph where each possible state of a dynamical system is represented by a vertex, and there is a directed edge from a to b if and only if ƒ = b where the function f defines the dynamical system.State spaces are...
, until it gets close to the orbit again and returns to the nearly periodic behaviour. Since the time spent near the periodic orbit depends sensitively on how closely the system entered its vicinity (in turn determined by what happened during the chaotic period) the length of each phase is unpredictable.
Another kind, on-off intermittency, occurs when a previously transversally stable chaotic attractor with dimension less than the embedding space begins to lose stability. Near unstable orbits within the attractor orbits can escape into the surrounding space, producing a temporary burst before returning to the attractor.
In crisis-induced intermittency a chaotic attractor suffers a crisis
Crisis (dynamical systems)
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...
, where two or more attractors cross the boundaries of each other's basin of attraction. As an orbit moves through the first attractor it can cross over the boundary and become attracted to the second attractor, where it will stay until its dynamics moves it across the boundary again.
Intermittent behaviour is commonly observed in turbulent or close to turbulent fluid flows. It has also been experimentally demonstrated in circuit oscillators and chemical reactions.
See also
- Crisis (dynamical systems)Crisis (dynamical systems)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...
- Turbulent flowTurbulenceIn fluid dynamics, turbulence or turbulent flow is a flow regime characterized by chaotic and stochastic property changes. This includes low momentum diffusion, high momentum convection, and rapid variation of pressure and velocity in space and time...
- Fluorescence intermittencyFluorescence intermittencyFluorescence intermittency, or blinking, is the phenomenon of random switching between ON and OFF states of the emitter under its continuous excitation...
(blinking) of organic molecules and colloidal quantum dots (nanocrystals)