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CLINT SPROTT earned his bachelor's degree from MIT in 1964 and his PhD in physics from the University of Wisconsin - Madison in 1969. His professional interests are in experimental plasma physics and nonlinear dynamics.

Most of his professional career has been devoted to experimental plasma physics with an application to the development of controlled nuclear fusion; since fusion promises an inexhaustible supply of energy, and its attainment would revolutionize society.

This interest began in graduate school where he studied electron cyclotron resonance heating of plasmas confined in a toroidal octupole magnetic field; and continued in subsequent employment at the Oak Ridge National Laboratory, where he worked on an electron cyclotron heated mirror device (ELMO) and its toroidal successor (Elmo Bumpy Torus). Upon return to the University of Wisconsin, he continued and expanded these studies to include ion cyclotron resonance heating in octupole and tokamak devices.

Since 1989 his work has been mostly in nonlinear dynamics and chaos. He developed several computer programs to demonstrate chaos and to perform time-series analysis of experimental data with the aim of clarifying the underlying dynamics. These studies may have application to plasma turbulence and anomalous transport, but they are of much more general interest. He has done statistical analyses of large collections of numerically simulated chaotic systems.

SEE: Images of a Complex World By Robin Chapman & J.C. Sprott.

Sprott's Fractals

Chaos and Time-Series Analysis
By J.C. "Clint" Sprott

Like so many terms in nonlinear dynamics, there is no universally accepted definition of a strange attractor, but most such objects of practical interest have the following properties, many of which are shared by other attractors:

• It is a limit set as time goes to infinity. It is called an omega set in contrast to an alpha set, which is the limit set as time goes to minus infinity. (Alpha and omega are the first and last letters of the Greek alphabet.) Note that it takes an infinite time for an arbitrary initial condition to reach the attractor, but it is usually approached very closely in a few times the inverse of the least negative Lyapunov exponent.
  
• It is an invariant set. Any orbit or trajectory that starts on it stays on it for all time.
  
• It is bounded. It does not stretch to infinity but can be enclosed within a region of finite (hyper)volume, but there are some exceptions. It is contained within a basin of attraction that may stretch to infinity but often has a finite and sometimes fractal boundary.
  
• It is a set of measure zero in space. If the space is two-dimensional, the attractor will have a dimension less than two and hence zero area. If you were to throw darts at it, you would never hit it. Neither could you see it, just as you cannot see a line in a plane unless you draw it with a finite width, in which case it is not really a line.
  
• It is a fractal. Fractals are self-similar objects with structure on all size scales and usually a noninteger dimension.
  
• It is dense in periodic orbits. Every point on the attractor is arbitrarily close to one of these orbits, but they comprise a set of measure zero, and they are all unstable. Most of these orbits have extremely long periods.}
  
• It is transitive. This means that if you start almost anywhere on the attractor (other than one of the periodic orbits), the dynamics will take you arbitrarily close to every other point on the attractor. The nonperiodic orbits are dense, which means that every point on the attractor is arbitrarily close to one.
  
• It is measure invariant. This means that every transitive orbit spends the same fraction of its time in a given region of the attractor after infinite time, but the measure may not be uniform (not the same everywhere on the attractor). No matter where you put the drop of cream in your coffee, stirring chaotically will eventually produce a uniform density of cream. Almost no orbit is special; almost all are typical. The exceptions are the periodic orbits, but they are a set of measure zero on the attractor, albeit infinitely numerous.
  
• It is indecomposable (or ergodic). This means that it is not made of smaller attractors that come close or even touch one another. Some authors (e.g., Ruelle) distinguish attractors from the more general attracting sets, which are produced by a cloud of initial conditions and can be decomposed.
  
• It is structurally stable. Although most attractors violate the strict requirement that there be a nonvanishing neighborhood in parameter space that gives topologically equivalent attractors, as a practical matter, they usually remain intact for most perturbations of the coefficients or the addition of other small terms. As a result, their structure is not sensitive to small numerical errors, although the sequence of points on the attractor visited by an orbit usually is.
  
• It is usually chaotic. This means that most nearby initial conditions separate exponentially on average. Not all chaotic systems, such as the logistic map, have strange attractors.
  
• It is aesthetically appealing. While this is not a mathematical property, it is one of their most evident characteristics and the reason many people are interested in them.