# Mark Paul

## Professor

### Mechanical Engineering

Ph.D., Mechanical Engineering, University of California, Los Angeles, 2000

M.S., Mechanical Engineering, University of California, Los Angeles, 1994

B.S., Aerospace Engineering, University of California, Los Angeles, 1993

**Welcome to the Paul Research Group at Virginia Tech**

We are exploring the engineering and scientific implications of nonlinear dynamics, nonequilibrium physics, and pattern formation on the physical and biological worlds. My goal is to improve our physical understanding of the world around us by employing analytical and numerical methods, from simple models to large-scale parallel numerical simulations. Most interesting are the questions posed today by spatiotemporal chaos, the rapid advance of experimental science to the nanoscale, and the complex behavior of biological systems.

### Lab News

- Congratulations to Mu Xu for successful PhD defense
- Persistant homology paper in Physica D
- 2016 Fall Fluid Mechanics Symposium
- Fall 2016 ME Distinguished Speakers
- Mu Xu’s paper in Physical Review E
- ME 6744 Chaos and Nonlinear Dynamics

### TOPICS

- Computational Systems Biology
- Fluid Dynamics
- General News
- Micro and Nanoscale Physics
- Nonequilibrium Physics
- Nonlinear Dynamics
- Pattern Formation
- Statistical Physics

### Archives

Archives

## Publications

- H.N. Patel, I. Carroll, R. Lopez, S. Sankararaman, C. Etienne, S. Kodigala, M.R. Paul, and H. W.Ch. Postma, DNA-graphene interactions during translocation through nanogaps, PLOS One, (2017).
- M. Xu and M.R. Paul, Covariant Lyapunov Vectors of Chaotic Rayleigh-Benard Convection,
*Physical Review E*,**93**, 062208, (2016). - M. Radiom, M.R. Paul, and W.A. Ducker, Dynamics of Single-Stranded DNA Tethered to a Solid,
*Nanotechnology,***27**, 255701, (2016). - M. Kramar, R. Levanger, J. Tithof, B. Suri, M. Xu, M.R. Paul, M.F. Schatz, and K. Mischaikow, Analysis of Kolmogorov Flow and Rayleigh-Benard Convection using Persistent Homology,
*Physica D,*, (2016).**334** - K. Subramanian, M.R. Paul, and J.J. Tyson, Dynamical localization of DivL and PleC in the asymmetric division cycle of
*Caulobacter crescentus*: A theoretical investigation of alternate models,*PLOS Computational Biology*,**11**, e1004348, (2015). - M. Radiom, B.A. Robbins, M.R. Paul, and W.A. Ducker, Hydrodynamic interactions of two nearly touching Brownian spheres in a stiff potential: effect of fluid inertia,
*Physics of Fluids*,**27**, 022002, (2015). - B.A. Robbins, M. Radiom, W.A. Ducker, J.Y. Walz, and M.R. Paul, The Stochastic Dynamics of Tethered Microcantilevers in a Viscous Fluid,
*Journal of Applied Physics*,**116**, 164905, (2014). - C. Lissandrello, F. Inci, M. Francom, M. R. Paul, U. Demirci, and K. L. Ekinci, Nanomechanical Motion of
*Escherichia coli*Adhered to a Surface,*Applied Physics Letters*,**105**, 113701, (2014). - C.O. Mehrvarzi and M.R. Paul, Front propagation in a chaotic flow field,
*Physical Review E*,**90**, 012905, (2014). - M.R. Paul, M.T. Clark and M.C. Cross, Coupled motion of microscale and nanoscale elastic objects in a viscous fluid,
*Physical Review E*,**88**, 043012, (2013). - S.E. Epstein and M.R. Paul, The stochastic dynamics of a nanobeam near an optomechanical resonator in a viscous fluid,
*Journal of Applied Physics*,**114**, 144901, (2013). - K. Subramanian, M.R. Paul, and J.J. Tyson, Bistable histidine kinase switches in
*Caulobacter crescentus*,*PLOS Computational Biology*, (2013). - A. Karimi and M.R. Paul, Bioconvection in Spatially Extended Domains,
*Physical Review E*,**87**, (2013). - A. Karimi and M.R. Paul, Length Scale of a Chaotic Element in Rayleigh-Benard Convection,
*Physical Review E*,**86**, (2012). - M. Radiom, C. Honig, J.Y. Walz, M.R. Paul, and W.A. Ducker, A Correlation Force Spectrometer for Single Molecule Measurements under Tensile Load,
*Journal of Applied Physics*,**113**, (2012). - D. Seo, M.R. Paul, and W.A. Ducker, A Pressure Gauge based on Gas Density Measurement from Analysis of the Thermal Noise of an AFM Cantilever,
*Review of Scientific Instruments*,**83**, (2012). - A. Karimi and M.R. Paul, Quantifying Spatiotemporal Chaos in Rayleigh-Benard Convection,
*Physical Review E*,**85**, (2012). - M. Radiom, B. Robbins, C. Honig, J.Y. Walz, M.R. Paul, and W.A. Ducker, Rheology of Fluids Measured by Correlation Force Spectroscopy,
*Review of Scientific Instruments*,**83**, (2012). - C. Honig, M. Radiom, B. Robbins, J.Y. Walz, M.R. Paul, and W.A. Ducker, Correlation Force Spectroscopy,
*Applied Physics Letters*,**100**, (2012). - A. Karimi, Z-F. Huang, M.R. Paul, Exploring Spiral Defect Chaos in Generalized Swift-Hohenberg Models with Mean Flow,
*Physical Review E*,**84**, (2011). - D. Barik, W.T. Baumann, M.R. Paul, B. Novak, and J.J. Tyson, A Model of Yeast Cell Cycle Regulation Based on Multisite Phosphorylation,
*Molecular Systems Biology*,**8**, (2010). - A. Karimi, and M.R. Paul, Extensive Chaos in the Lorenz-96 Model,
*Chaos*,**20**, (2010). - A. Duggleby and M.R. Paul, Computing the Karhunen-Loeve dimension of an Extensively Chaotic Flow Field Given a Finite Amount of Data,
*Computers and Fluids*,**39**, 9, (2010). - M.T. Clark, J.E. Sader, J.P. Cleveland, and M.R. Paul, The Spectral Properties of Microcantilevers in Viscous Fluid,
**81**, 046306,*Physical Review E*(2010). - S. Misra, H. Dankowicz, and M.R. Paul, Degenerate Discontinuity-Induced Bifurcations in Tapping-Mode Atomic-Force Microscopy,
**239**,*Physica D*, (2010). - M.M. Villa and M.R. Paul, The Stochastic Dynamics of Micron Scale Doubly-Clamped Beams in a Viscous Fluid,
*Physical Review E*,**79**, 056314, (2009). - H. Dankowicz and M.R. Paul, Discontinuity-Induced Bifurcations in Systems with Hysteretic Force Interactions,
*Journal of Computational and Nonlinear Dynamics*,**4**(2009). - S. Kar, W.T. Baumann, M.R. Paul, and J.J. Tyson, Exploring the Roles of Noise in the Eukaryotic Cell Cycle,
*Proceedings of the National Academy of Sciences*, (2009). - N. Hashemi, M.R. Paul, H. Dankowicz, M. Lee, W. Jhe, The Dissipated Power in Atomic Force Microscopy due to Interactions with a Capillary Fluid Layer,
*Journal of Applied Physics*,**104**, 063518, (2008). - D. Barik, M.R. Paul, W.T. Baumann, Y. Cao, and J.J. Tyson, Stochastic Simulation of Enzyme-Catalyzed Reactions with Disparate Time Scales,
*Biophysical Journal*,**95**, (2008). - M.T. Clark and M.R. Paul, The Stochastic Dynamics of Rectangular and V-shaped Atomic Force Microscope Cantilevers in a Viscous Fluid and Near a Solid Boundary,
*Journal of Applied Physics*,**103**, 094910, (2008). - N. Hashemi, H. Dankowicz and M.R. Paul, The Nonlinear Dynamics of Tapping Mode Atomic Force Microscopy with Capillary Force Interactions,
*Journal of Applied Physics*,**103**, 093512, (2008). - S. Misra, H. Dankowicz, and M.R. Paul, Event-Driven Feedback Tracking and Control of Tapping-Mode Atomic Force Microscopy,
*Proceedings of the Royal Society A*,**2095**, (2008). - M.R. Paul, M.I. Einarsson, M.C. Cross and P.F. Fischer, Extensive Chaos in Rayleigh-Benard Convection,
*Physical Review E*,**75**, 045203, (2007). - A. Duggleby, K.S. Ball, M.R. Paul, and P.F. Fischer, Dynamical Eigenfunction Decomposition of Turbulent Pipe Flow,
*Journal of Turbulence*,**8**, 43, (2007). - M.T. Clark and M.R. Paul, The Stochastic Dynamics of an Array of Atomic Force Microscopes in a Viscous Fluid,
*International Journal of Nonlinear Dynamics*,**42**, (2006). - A. Duggleby, K.S. Ball, and M.R. Paul, The Effect of Spanwise Wall Oscillation on Turbulent Pipe Flow Structures Resulting in Drag Reduction,
*Physics of Fluids*,**19**, (2007). - J.L. Arlett, M.R. Paul, J. Solomon, M.C. Cross, S.E. Fraser, and M.L. Roukes, BioNEMS: Nanomechanical Devices for Single Molecule Biophysics, Lecture Notes in Physics,
**711**, (2007). - M.R. Paul, M.T. Clark, and M.C. Cross, The stochastic dynamics of micron and nanoscale elastic cantilevers in fluid: fluctuations from dissipation,
*Nanotechnology*,**17**, (2006). - J.E. Solomon and M.R. Paul, The Kinetics of Analyte Capture on Nanoscale Sensors,
*Biophysical Journal*,**90**, (2006). - M.R. Paul and J.E. Solomon, The Physics and Modeling of BioNEMS, in Nanodevices for Life Sciences, Wiley-VCH, (2006).
- M.R. Paul, K.-H. Chiam, M.C. Cross, and P.F. Fischer, Rayleigh-Benard Convection in Large-Aspect-Ratio Domains,
*Physical Review Letters*,**93**, (2004). - M.R. Paul and M.C. Cross, Stochastic Dynamics of Nanoscale Mechanical Oscillators Immersed in a Viscous Fluid,
*Physical Review Letters*,**92**, 235501, (2004). - M.R. Paul and I. Catton, The Relaxation of Two-Dimensional Rolls in Rayleigh-Benard Convection,
*Physics of Fluids*,**16**, 1262, (2004). - J.D. Scheel, M.R. Paul, M.C. Cross, and P.F. Fischer, Traveling Waves in Rotating Rayleigh-Benard Convection: Analysis of Modes and Mean Flow,
*Physical Review E*,**68**, 066216, (2003). - M.R. Paul, K.-H. Chiam, M.C. Cross, P.F. Fischer, and H.S. Greenside, Pattern Formation and Dynamics in Rayleigh-Benard Convection: Numerical Simulations of Experimentally Realistic Geometries,
*Physica D*,**184**, 114-126, (2003). - K.-H. Chiam, M. R. Paul, M. C. Cross, and H. S. Greenside, Mean flow and spiral defect chaos in Rayleigh-Benard convection,
*Physical Review E*,**67**, 056206, (2003). - M.R. Paul, M. C. Cross, and P. F. Fischer, Rayleigh-Benard Convection with a Radial Ramp in Plate Separation,
*Physical Review E*,**66**, 046210 (2002). - M. R. Paul, M. C. Cross, P. F. Fischer and H. S. Greenside, Power Law Behavior of Power Spectra in Low Prandtl Number Rayleigh-Benard Convection,
*Physical Review Letters*,**87**, (2001). - M.R. Paul, F. Issacci, I. Catton, and G.E. Apostolakis, Characterization of Smoke Particles Generated in Terrestrial and Microgravity Environments,
*Fire Safety Journal*,**28**, 233-252 (1997). - G.E. Apostolakis, I. Catton, F. Issacci, S. Jones, M.R. Paul, T. Paulos, and K. Paxton, Risk Based Fire Safety Experiments,
*Reliability Engineering and System Safety*,**49**, 275-291 (1995).

**Spring 2018**

Chaos and Nonlinear Dynamics (ME6744)

**Course Announcement:**

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ME 6744: Chaos and Nonlinear Dynamics, Spring 2018

Instructor: Prof. Mark Paul

Time: Tuesday and Thursday, 11-12:15

Location: 241 Goodwin Hall

Course Description:

Many open challenges in science and engineering involve nonlinear dynamical systems that exhibit chaotic dynamics. Important examples include fluid turbulence, the dynamics of the weather and climate, excitable media such as cardiac tissue and nerve fibers, population dynamics, transport in complex flow fields, the dynamics of the biomass in the oceans, and the complex motion of the cantilever of an atomic force microscope. Despite the great importance of these systems the aperiodic nature of their dynamics makes them difficult to control, design, analyze, and predict. This course will discuss analytical and numerical approaches to gain fundamental physical insights into these systems, and others, using both simplified mathematical models and physical examples that can be directly compared with experimental measurements.

Course Content:

Overview of theoretical and numerical approaches for the study of nonlinear and chaotic dynamics in science and engineering. Fractals, bifurcation analysis, predictability, strange attractors, and routes to chaos. Roles of dissipation and noise in deterministic chaos. Use of Lyapunov spectra, fractal dimension, information, entropy, correlation functions, and attractor reconstruction to describe chaos.