He lived 1903 to 1987, invented Kolmogorov probability, and developed measure theory [1965]. System-complexity measures {algorithmic complexity, Kolmogorov} {Kolmogorov complexity, Kolmogorov} {algorithmic information content, Kolmogorov} can be number of bits for smallest program that can run on universal Turing machines and produce same output. In turbulence, low frequencies transfer energy to higher frequencies throughout fluid.
She lived 1900 to 1998. For non-linear radio amplifiers, equations {Van der Pol equation} can calculate output for sine-wave input. At higher amplifier gains, output period doubles input period and then becomes non-periodic. Van-der-Pol-equation solutions were early chaos-theory ideas.
He lived 1908 to 1968. He proposed neutron stars [1932], and J. Robert Oppenheimer and G. M. Volkov found mass limit {Landau-Oppenheimer-Volkov limit, Landau} for making black holes instead of neutron stars, 2.5 times Sun mass.
Turbulence begins when new frequencies appear in fluid at overlapping velocities and masses. Turbulent motions include oscillatory, skewed varicose, cross-roll, knot, and zigzag. Turbulence is like white noise, with all frequencies.
He lived 1917 to ? and studied complex systems. He invented non-periodic weather-system computer models that were sensitive to initial conditions {butterfly effect, Lorenz}.
He studied fluid convections with circular motions {Rayleigh-Bénard convection, Lorenz}. Equations are dx / dt = 10 * (y - x), dy / dt = x * z + 28 * x - y, and dz / dt = x * y - (8/3) * z.
Paths through phase space never cross. Attractor can move to another surface when it moves to another phase-space region, so surfaces do not intersect.
Complex non-linear systems can have different final states that are not interchangeable {intransitive system}. Systems can be almost intransitive and can flip spontaneously from one state to another.
One-dimensional objects with cycle of period three have all periods.
He lived 1930 to ? and studied non-linear oscillators that had stable, non-repeating, periodic patterns. He studied topology in five or higher dimensions and Poincaré conjecture. He invented topological phase-space transformations {Smale's horseshoe}, in which space stretches, shrinks, and folds multiple times in any dimension. Transformations are sensitive to initial conditions.
He lived 1923 to 2002 and studied catastrophe theory.
He used three independent motions to describe turbulence. However, this was wrong. Phase-space centers can be not equilibria or periodic loops but infinitely long lines in confined space {strange attractor, Ruelle}. Strange attractors are stable, can have few dimensions, and are periodic but not exactly periodic.
He studied feedback systems and devised how to calculate order in one-dimensional-system chaos [1973], using quantum-field-theory renormalization group, stochastic processes, and fractals to remove infinities. Using y = r * (x - x^2) and x(t) = r * sin(pi * x(t - 1)), doubling oscillation period converges geometrically and so scales with constant ratio = 4.6692016090, to predict all doubling values. Functions are recursive {self-referential} and so introduce higher frequencies that indicate turbulence.
He studied conductivity [1973], with Jerry Gollub. He studied phase transitions. Rotating one cylinder inside another causes intervening liquid to flow {Couette-Taylor flow, Swinney}. First, flow streamlines. At faster speed, fluid cylinder separates into layers along cylinder axis, so fluid goes up and down cylinder. At higher frequency, flow is chaotic, with no defined frequencies. Vapor at critical point gives off white glow {opalescence, vapor}.
He analyzed work of Robert May. In one-dimensional systems, regular cycle of period three implies regular cycles of other periods, as well as chaotic behavior.
Assign initial number to logistic difference equation. Low rate values make number go to zero. Medium values make number go higher steady-state numbers. After high initial value, system oscillates between two values. After even higher initial value, system oscillates among four values. After even higher initial values, system oscillates among 8, 16, 32, and so on, values, with smaller differences between rates, until chaos starts {point of accumulation} {accumulation point, complexity}. After that point, oscillations are among all values. However, at higher points, oscillations are among 3 or 7 values, then oscillations are among 6, 9, 12, 14, 21, 28, and so on, values, then chaos returns again.
He lived 1909 to 1984 and studied chaos in vibrating strings {Fermi-Pasta-Ulam theorem}.
He studied stretching, compressing, and folding phase space to get self-similarity {Hénon attractor} [1976]: x(t) = y(t - 1) + 1 - 1.4 * (x(t - 1))^2 and y(t) = 0.3 * x(t - 1). He predicted that globular clusters have center that experienced gravitational collapse {gravothermal collapse} [2002].
They studied catastrophe theory.
He studied random graphs and Boolean networks to try to find complex-system, chaos, and self-organization laws. Most algorithms are their shortest descriptions {incompressibility}. Element physical interactions can order systems {self-organization, Kauffman} [Kauffman, 1995].
Self-organizing systems follow physical laws and describe living-system energy flows.
Brain has hierarchical structure and new properties can arise at highest levels.
He used a liquid-helium box to study turbulence onset and found that it had period doubling, as in other complex non-linear systems [1996]. First, system reaches steady state as cylinders roll, then convection rolls become toruses, then those bifurcate, making 1, 2, 4, 8, 16, and so on, rolls as convection coil goes faster, and turbulence increases.
Phase transitions and critical points can be hierarchies of phase regions that affect neighbors {phase scaling} [1999].
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Date Modified: 2022.0225