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Swinney graduated from Rhodes College in 1961 with a bachelor's degree and obtained his Ph.D. from Johns Hopkins University in 1968. After positions at New York University and City College of New York, he came to the University of Texas at Austin in 1978 and founded the Center of Nonlinear Dynamics, of which he is the director.
He is a Sid Richardson Foundation Regents Chair, University of Texas at Austin (1990–present). He is a member of the National Academy of Sciences (1992) and a Fellow of the American Academy of Arts and Sciences (1991). He was awarded the American Physical Society Fluid Dynamics Prize (1995), the Society for Industrial and Applied Mathematics Jürgen Moser Prize (2007), the European Geosciences Union Richardson Medal (2012), and the Boltzmann Medal (2013). He is Fellow of the American Physical Society (1977), the American Association for the Advancement of Science (1999), and the Society for Industrial and Applied Mathematics (2009). He was a Guggenheim Fellow (1983–84) and he was inducted into The Johns Hopkins University Society of Scholars (1984). He was awarded honorary doctoral degrees by Rhodes College (2002), The Hebrew University of Jerusalem (2008), and the University of Buenos Aires (2010).
Harry Swinney is one of the pioneers in the study of chaos theory, and he has played a leading role in the study of nonlinear dynamics during the last 40 years. He and Jerry Gollub first showed that the temporal behavior of fluid held between rotating cylinders, (Taylor–Couette flow) becomes chaotic after a small number of bifurcations, and that this transition in temporal behavior leads directly to turbulence. This early turbulence work provided the first experimental evidence of deterministic chaos in a dynamical system and triggered studies of chaos in many other fields in the early 1980s. Swinney then played a pioneering role in the development of new ideas and tools using the methods of dynamical systems theory for the analysis of time series: his paper about the determination of Lyapunov exponents  has been quoted more than 3000 times by scientists working in many different fields.
Swinney also made outstanding scientific contributions on pattern-forming instabilities, granular flows and dynamics of fracture. He was the first to achieve the observation of Turing instabilities in chemical dynamics. He made the first observation of coherent structures in granular flows and pioneering contributions to dynamic fracture and acoustic emission from rapidly moving cracks. During his career, Swinney has designed many other elegant experiments to model nonlinear phenomena in geophysics: the first laboratory study of the dynamics of Jovian-type vortices, a study of anomalous turbulent diffusion in two-dimensional rotating flows, a laboratory model of the atmospheric blocking phenomenon. Most recently, he has conducted experiments modeling the dynamics of internal waves over topography in the ocean. Swinney has been an international leader in the study of nonlinear processes since the mid-70s. His contributions have not only been scientific, but also he has organized and promoted the study of nonlinear dynamics throughout the world.
His research has focused on instabilities, chaos, pattern formation, and turbulence in systems driven away from equilibrium by the imposition of gradients in temperature, velocity, concentration, etc. and past research projects include:
- chaos and pattern formation in flow between concentric rotating cylinders (the Taylor–Couette system)
- chaos and strange attractors in oscillating chemical reactions
- a laboratory model of Jupiter's Great Red Spot
- a laboratory model of the atmospheric "blocking" phenomenon
- turbulence in buoyancy driven convection; pattern formation in surface-tension-driven (Marangoni) convection
- growth of metallic fractal clusters in electrodeposition
- chemical patterns of the type predicted by Alan Turing in his 1952 paper "The Chemical Basis for Morphogenesis"
- other patterns in chemical reaction-diffusion systems, including reactions that are periodically forced in time, where Arnold tongue type phase diagrams have been found
- vertically oscillated containers of grains (sand, metallic particles, etc.), which exhibit square, stripe, hexagon, spiral, and oscillon (localized) patterns.
- shock waves in supersonic sand
- the determination of statistical properties of rapid granular flows, where the observations are compared to the predictions of kinetic theory and continuum theory
- instabilities in fluidized beds, where a fluid flows upward through a granular bed, such as in a gasoline refinery catalytic cracker
- viscous fingering patterns at the interface between immiscible fluids
- buckling of thin sheets (plastic, leaves of plants)
- scaling and transport in rapidly rotating turbulent flows, such as those in oceans and atmospheres.
- generation of internal wave beams by three-dimensional topography
- resonant generation of intense boundary flows in tidal flow over model continental slopes
- harmonic generation by reflecting internal waves
- turning depths in the ocean and their impact on oceanic processes
- reflection of internal wave beams from regions where they become evanescent
- propagation of internal wave beams in stratified fluids with nonlinear density gradients
- A 2012 review upon receiving the Lewis Fry Richardson medal from the European Geosciences Union serves as a general reference to this article.