Last update: Feb 17, 2022


Till D. Frank, Department of Psychology and Department of Physics, 406 Babbidge Road, Unit 1020, email:

Interests: Humans, Physics, Synergetics, and Stochastics

Special topic: COVID-19 epidemiology and virology



T. Frank, Determinism and self-organization of human perception and performance (Springer, Berlin, 2019)
ISBN 978-3-030-28820-4


T. D. Frank, Nonlinear Fokker-Planck equations: fundamentals and applications (Springer, Berlin, 2005)
ISBN 3-540-21264-7



Edited Book

T.D. Frank, New Research on Collective Behavior (Nova Publ., New York, 2016)
ISBN 978-1-63483-946-4




  1. Frank, T.D. (in press). SARS-coronavirus-2 infections: biological instabilities characterized by order parameters. Physical Biology, xxx, article xxx
  2. ******************** 2021 **************

  3. Frank, T.D., Stowik, P. (2021). On the search for brain bifurcation parameters: lessons from fMRI studies on visual illusions. Biophysical Reviews and Letters, 16, 77-93
  4. Frank, T.D. (2021). COVID-19 outbreaks follow narrow paths: a computational phase portrait approach based on nonlinear physics and synergetics. International Journal of Modern Physics C, 32, article 2150110
  5. Frank, T.D. (2021). SARS-coronavirus-2 nonlinear dynamics in patients: three-dimensional state and amplitude space descriptions. Journal of the Physical Society of Japan, 90, article 073802
  6. Frank, T.D. (2021). Rise and decay of the COVID-19 epidemics in the USA and the state of New York in the first half of 2020: a nonlinear physics perspective yielding novel insights. BioMed Research International, 2021, article 6645688
  7. Frank, T.D., Chiangga, S. (2021). SEIR order parameters and eigenvectors of the three stages of a completed COVID-19 epidemics: with an illustration for Thailand January to May 2020. Physical Biology, 18, article 046002
  8. Frank, T.D., Galenko, P.I. (2021). Neural oscillations under binocular rivalry: a brain activity pattern formation perspective. International Journal of Physical Research, 9, 68-72
  9. Frank, T.D., Pereira, A. (2021). Swift-Hohenberg model of liquid artificial humans mimicking human graded reactions. Nonlinear Phenomena in Complex Systems, 24, 56-70
  10. Frank, T.D. (2021). Polyrhythmic multifrequency synchronization in coupled oscillators with exactly solvable attractors. International Journal of Modern Physics B, 35, article 2150047
  11. Chiangga, S., Sunpatanon, T., Frank, T.D. (2021). Photon entanglement on a chip, optical instability, and Haken-Zwanzig model. Physica D, 415: article 132760
  12. ******************** 2020 **************

  13. Frank, T.D. (2020). Simplicity from complexity: on the simple amplitude dynamics underlying COVID-19 outbreaks in China. Advances in Complex Systems, 23: article 2050022
  14. Frank, T.D. (2020). Emergence and subsiding of the first-wave COVID-19 pandemic in Pakistan (2020): an eigenvalue analysis based on synergetics. Proceedings of the Pakistan Academy of Science B, 57: 1-7
  15. Frank, T.D. (2020). COVID-19 interventions in some European countries induced bifurcations stabilizing low death states against high death states: an eigenvalue analysis based on the order parameter concept of synergetics. Chaos, Solitons, and Fractals, 140: article 110194
  16. Frank, T.D. (2020). COVID-19 order parameters and order parameter time constants of Italy and China: a modeling approach based on synergetics Journal of Biological Systems, 28: 589-608
  17. Frank, T.D., Mongkolsakulvong, S. (2020). Amplitude equations and bifurcation diagrams for multifrequency synchronization of canonical-dissipative oscillators International Journal of Bifurcation and Chaos, 30: article 2050101
  18. Brooks, T.R., Frank, T.D., Dixon, J. (2020). Grasp affordances in bistable perception of the Necker cube. Nonlinear Dynamics, Psychology and Life Sciences, 24: 143-157
  19. ******************** 2019 **************

  20. Dotov, D.G., Turvey, M.T., Frank, T.D. (2019). Embodied gestalts: unstable visual phenomena become stable when they are stimuli for competitive action selection. Attention, Perception & Psychophysics, 81: 2330-2342
  21. Cox, R.F.A, Den Hartigh, R.J.R, Richardson M.J., Yu C., Frank, T.D. (2019). Editorial: complex dynamical systems in human development. Complexity, 2019: article 5010413 (3 pages)
  22. Chiangga, S., Pitakwongsaporn, S., Frank, T.D. (2019). Simplified P representation operator correspondence applied to quantum systems with generalized Kerr nonlinearity. Modern Physics Letters B, 28: article 1950349 (22 pages)
  23. Varapongpisan, T., Ingsrisawang, L., Frank, T.D. (2019). Taking drift-diffusion analysis from the study of turbulent flows to the study of particulate matter smog and air-pollution dynamics. Condensed Matter Physics, 22: article 24001 (7 pages)
  24. ******************** 2018 **************

  25. Chiangga, S., Temnuch, W., Frank, T.D. (2018) Entanglement near the optical instability point in damped four wave mixing systems. Physica Scripta, 93: article 065102 (15 pages)
  26. Lopez-Felip, M.A., Davis, T.J., Frank, T.D., Dixon, J.A. (2018). A cluster phase analysis for collective behaviour in team sports. Human Movement Science, 59: 96-111
  27. Kim, S., & Frank, T.D. (2018). Correlation between hysteretic categorical and continuous judgements of perceptual stimuli supporting a unified dynamical systems approach to perception. Perception, 47: 44-66
  28. ******************** 2017 **************

  29. Mongkolsakulvong, S., Frank, T.D (2017). Synchronization and anchoring of two non-harmonic canonical-dissipative oscillators via Smorodinsky-Winternitz potentials. Condensed Matter Physics, 20: article 44001 (7 pages)
  30. Frank, T.D., Kiyatkin, A., Cheong, A., Kholodenko, B.N. (2017). Three factor models versus time series models: quantifying time-dependencies of interactions between stimuli in cell biology and psychobiology for short longitudinal data. Mathematical Medicine and Biology, 34: 177-191
  31. Frank, T.D. (2017). Active and purely dissipative Nambu systems in general thermostatistical setting described by nonlinear partial differential equations involving generalized entropy measures. Entropy, 19: article e19010008 (21 pages)
  32. ******************** 2016 **************

  33. Frank, T.D. (2016). Unstable modes and order parameters of bistable signalling pathways at saddle-node bifurcations: a theoretical study based on synergetics. Advances in Mathematical Physics, 2016: article 8938970 (7 pages)
  34. Chiangga, S., Pornkaveerat, W., Frank, T.D. (2016). Reaction kinetics of the jasmonate-isoleucine complex formation during wound-induced plant defense responses: a model-based re-analysis of published data. Journal of Plant Physiology, 206: 103-113
  35. Frank, T.D. (2016) Formal derivation of Lotka-Volterra-Haken amplitude equations of task-related brain activity in multiple, consecutively performed tasks. International Journal of Bifurcation and Chaos, 26: article 1650164 (17 pages)
  36. Chaikhan, P., Frank, T.D., Mongkolsakulvong, S. (2016). In-phase and anti-phase synchronization in an active Nambu mechanics system. Acta Mechanica, 227: 2703-2717
  37. Gordon, J.M., Kim, S., Frank, T.D. (2016). Linear non-equilibrium thermodznamics of human voluntary behavior: a canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic oscillator case. Condensed Matter Physics, 19: article 34001 (6 pages)
  38. Kim, S., Frank, T.D. (2016). Body-scaled perception is subjected to adaptation when repetitively judging opportunities for grasping. Experimental Brain Research, 234: 2731-2743
  39. Frank, T.D. (2016). A synergetic gait transition model for hysteretic gait transitions from walking to running. Journal of Biological Systems, 14: 51-61
  40. Harrison, H.S., Turvey, M.T, Frank, T.D. (2016). Affordance-based perception-action dynamics: a model of visually guided braking. Review, 123: 305-323
  41. Frank, T.D. (2016). Stochastic systems with delay: perturbation theory for second order statistics. Physics Letters A, 380: 1341-1351
  42. Frank, T.D. (2016). Perception adapts via top-down regulation to task repetition: a Lotka-Volterra-Haken modelling analysis of experimental data. Journal of Integrative Neuroscience, 15: 67-79
  43. Chiangga, S., Pornkaveerat, W., Frank, T.D. (2016). On a Fitzhugh-Nagumo type model for the pulse-like jasmonate defense response in plants. Mathematical Biosciences, 273: 80-90
  44. Frank, T.D. (2016). Front waves in the early RNA world: the Schloegl model and the logistic growth model. Journal of Theoretical Biology, 392: 62-68
  45. ******************** 2015 **************

  46. Nguyen, L.K., Cavadas, M.A.S., Kholodenko, B.N., Frank, T.D., Cheong, A. (2015). Species differential regulation of COX2 can be described by an NFkB-dependent logic AND gate. Cellular and Molecular Life Sciences, 72: 2431-2443
  47. Frank, T.D., Profeta, V.L.S., Harrison, H.S. (2015). Interplay between order-parameter and system parameter dynamics: considerations on perceptual-cognitive-behavioral mode-mode transitions exhibiting positive and negative hysteresis and on response times. Journal of Biological Physics, 41: 257-292
  48. Kim, S., Gordon, J.M., Frank, T.D. (2015). Non-equilibrium thermodynamic state variables of human self-paced rhythmic motions: canonical-dissipative approach, augmented Langevin equation, and entropy maximization. Open Systems & Information Dynamics, 22: article 1550007 (22 pages)
  49. Frank, T.D., Mongkolsakulvong, S. (2015). Order-disorder transitions in time-discrete mean field systems with memory: a novel approach via nonlinear autoregressive models. Physica Scripta, 90: article 055202 (16 pages)
  50. Frank, T.D. (2015). On the interplay between order parameter and system parameter dynamics in human perceptual-cognitive-behavioral systems. Nonlinear Dynamics, Psychology, and Life Sciences, 19: 111-146
  51. Frank, T.D. (2015). Domains of attraction of walking and running attractors are context dependent: illustration for locomotion on tilted floors. International Journal of the Scientific World, 3: 81-90
  52. Abdolvahab, M., Carello, C. Pinto, C., Turvey, M. T., Frank, T.D. (2015) Symmetry and order parameter dynamics of the human odometer. Biological Cybernetics, 109: 63-73
  53. Dotov, D. G., Kim, S., Frank, T.D. (2015). Non-equilibrium thermodynamical description of rhythmic motion patterns of active systems: a canonical-dissipative approach. BioSystems, 128: 26-36
  54. ******************** 2014 **************

  55. Frank, T.D. (2014). Exact solutions for chemical concentration waves of self-propelling camphor particles racing on a ring: a novel potential dynamics perspective. Condensed Matter Physics, 17: article 43002 (12 pages)
  56. Frank, T.D. (2014). Secondary bifurcations in a Lotka-Volterra model for N competitors with applications to action selection and compulsive behaviors. International Journal of Bifurcation and Chaos, 24: article 1450156 (20 pages)
  57. Frank, T.D. (2014). Interpersonal distances are the consequence of the self-organization of human spatial behavior: a theoretical study based on synergetics. Universal Journal of Psychology, 2: 285-289
  58. Frank, T.D., Yupapin, P.P. (2014). Quantum theoretical approach to the integrate-and-fire model of human decision making. International Journal of Psychological Studies, 6: 95-105
  59. Frank, T.D. (2014). A nonlinear physics model based on extended synergetics for the flow of infant actions during infant-mother face-to-face communication. International Journal of the Scientific World, 2: 62-74
  60. Frank, T.D. (2014) Action flow in obsessive-compulsive disorder rituals: a model based on extended synergetics and a comment on the 4th law. Journal of Advances in Physics, 5: 845-853
  61. Frank, T.D. (2014). Decision-making in physical intelligent systems regulated by growth rate factors. Computer and Information Science, 7: 55-64
  62. Frank, T.D. (2014) Multistable perception in schizophrenia a model-based analysis via coarse-grained order parameter dynamics and a comment on the 4th law. Universal Journal of Psychology, 2: 231-240
  63. Frank, T.D., Gifford, T.D., Chiangga, S. (2014). Minimalistic model for navigation of mobile robots around obstacles based on complex-number calculus and inspired by human navigation behavior. Mathematics and Computers in Simulation, 97: 108-122
  64. ******************** 2013 **************

  65. Frank, T.D., Mongkolsakulvong, S. (2013). On strongly nonlinear autoregressive models: implications for the theory of transient and stationary responses of many-body systems. Fluctuation and Noise Letters, 12: article 1350022 (27 pages)
  66. Dotov, D.G., Frank, T.D., Turvey, M.T. (2013) Balance affects prism adaptation: evidence from the latent aftereffect. Experimental Brain Research, 231: 425-432
  67. Sarapat, N., Frank, T.D., Yupapin, P.P (2013). Conjugate mirror design and simulation using a nonlinear coupling microring circuit. Journal of Nonlinear Optical Physics and Materials, 22: article 1350024 (11 pages)
  68. Frank, T.D., Kim, S., Dotov, D.G. (2013). Canonical-dissipative non-equilibrium energy distributions: parameter estimation via implicit moment method, implementation, and application. International Journal of Modern Physics B, 27: article 1350156 (19 pages)
  69. Frank, T.D. (2013). RNA polymerase on DNA as a Fermi system: internal energy, entropy, heat capacity, and negative temperature. Journal of Biological Systems, 21: article 1350021 (14 pages)
  70. Frank, T.D., Collins, A. J. F., Cheong, A. (2013). Effective single-step post-transcriptional dynamics allowing for a direct maximum likelihood estimation of transcriptional activity and the quantification of sources of gene expression variability with an illustration for the hypoxia and TNFalpha regulated inflammatory pathway. ISRN Computational Biology, 2013: article 719138 (11 pages)
  71. Lopresti-Goodman, S., Turvey, M.T., Frank, T.D. (2013). Negative hysteresis in the behavioral dynamics of the affordance “graspable”. Attention, Perception & Psychophysics, 75: 1075-1091
  72. Frank, T.D. (2013). Strongly nonlinear stochastic processes in physics and the life sciences. ISRN Mathematical Physics, 2013: article 149169 (28 pages)
  73. Chiangga, S., Pitakwongsaporn, S., Frank, T.D., Yupapin, P.P. (2013). Optical bistability investigation in a nonlinear microring resonantors.Journal of Lightwave Technology, 31: 1101-1105
  74. Frank, T.D. (2013). A limit cycle model for cycling mood variations of bipolar disorder patients derived from cellular biochemical reaction equations Communications in Nonlinear Science and Numerical Simulation, 18: 2107-2119
  75. ******************** 2012 **************

  76. Richardson, M.J., Garcia, R.L., Frank, T.D., Gregor, M., Marsh, K.L. (2012). Measuring group synchrony: a cluster-phase method for analyzing multivariate movement time series. Frontiers in Physiology, 3: article 405 (10 pages)
  77. Isenhower, R.W., Frank, T.D., Kay, B.A., Carello, C. (2012). Capturing and quantifying the dynamics of valenced emotions and valenced events of the organism-environment system. Nonlinear Dynamics, Psychology, and Life Sciences, 16: 397-427
  78. Frank, T.D. (2012). Multistable pattern-formation systems: candidates for physical intelligence? Ecological Psychology, 24: 220-240
  79. Turvey, M. T., Harrison, S.J., Frank, T. D., Carello, C. (2012). Human odometry verifies the symmetry perspective of biped gaits. Journal of Experimental Psychology: Human Perception and Performance, 38: 1014-1025
  80. Frank, T.D., Cheong, A., Okada-Hatakeyama, M., Kholodenko, B.N. (2012). Catching transcriptional regulation by thermostatistical modeling. Physical Biology, 9: 045007
  81. Frank, T.D. Silva, P.L., Turvey, M. T. (2012). Symmetry axiom of Haken-Kelso-Bunz coordination dynamics revisited in the context of cognitive activity. Journal of Mathematical Psychology, 56: 149-165
  82. Frank, T. D., Blau, J. J. C., Turvey, M. T. (2012). Symmetry breaking analysis of prism adaptation’s latent aftereffect. Cognitive Science, 36: 674-697
  83. Frank, T.D. Carmody, A.M., Kholodenko, B.N. (2012). Versatility of cooperative transcriptional activation: a thermodynamical modeling analysis for greater-than-additive and less-than-additive effect. PLoS One, 7: e34439 (15 pages)
  84. Bruning, U., Fitzpatrick, S. F., Birtwistle, M., Frank, T., Taylor, C. T., Cheong, A. (2012). NFkappaB and HIF display synergistic behavior during hypoxic stimulation. Cellular and Molecular Life Science, 69: 1319-1329
  85. Frank, T. D. (2012). Nambu bracket formulation of nonlinear biochemical reactions beyond mass action kinetics. Journal of Nonlinear Mathematical Physics, 19: 1250007 (17 pages)
  86. Mongkolsakulvong, S., Chaikan P., Frank, T.D. (2012). Oscillatory nonequilibrium Nambu systems: the canonical-dissipative Yamaleev oscillator. European Physical
    Journal B, 85:
    90 (10 pages)
  87. Isenhower, R.W., Kant, V., Frank, T. D., Pinto, C. M. A., Carello, C., Turvey, M. T. (2012). Equivalence of human odometry by walk and run is indifferent to self-selected speed. Journal of Motor Behavior, 44: 47-52
  88. ******************** 2011 **************

  89. Dotov, D. G., Frank, T. D. (2011). From the W-method to the canonical-dissipative method for studying uni-manual rhythmic behavior. Motor Control, 15: 550-567
  90. Frank T. D. (2011). Nonlinear physics approach to DNA cross-replication: marginal stability, generalized logistic growth, and impacts of degradation.Physics Letters A, 375: 3851-3857
  91. Frank, T.D., (2011). Unifying mass-action kinetics and Newtonian mechanics by means of Nambu brackets.Journal of Biological Physics, 37: 375-385
  92. Lopresti-Goodman, S.M., Turvey M.T., Frank, T. D. (2011). Behavioral dynamics of the affordance ‘graspable’.Attention, Perception, &Psychophysics, 73: 1948-1965
  93. Frank, T. D. (2011). Stochastic processes and mean field systems defined by nonlinear Markov chains: an illustration for a model of evolutionary population dynamic.Brazilian Journal of Physics, 41: 129-134
  94. Frank T. D. (2011). Fractional Brownian motion analysis does not provide evidence for neurophysiological feedback mechanisms: a comment on ‘White matter hyperintensities and dynamics of postural control’. Magnetic Resonance Imaging, 29: 887-888
  95. Frank T. D. (2011). Rate of entropy production as a physical selection principle for mode-mode transitions in non-equilibrium systems: with an application to a non-algorithmic dynamic message buffer. European Journal of Scientific Research, 54: 59-74
  96. Frank T. D. (2011). Virial theorem and non-equilibrium canonical-dissipative distributions characterizing Parkinson tremor. International Journal of Modern Physics B, 25: 243-253
  97. Frank T. D. (2011). Collective behavior of biophysical systems with thermodynamic feedback loops: a case study for a nonlinear Markov model — the Takatsuji system.Modern Physics Letters B, 25: 551-568
  98. Frank T. D. (2011). Multistable selection equations of pattern formation type in the case of inhomogeneous growth rates: with applications to two-dimensional assignment problems. Physics Letters A, 375: 1465-1469
  99. Frank, T. D., Rhodes, T. (2011). Micro-dynamic associated with two-state nonlinear Markov processes: with an application to free recall. Fluctuation and Noise Letters, 10: 41-58
  100. Patanarapeelert, K., Frank T. D., Tang, I.M. (2011). From a cellular automata model of tumor-immune interactions to its macroscopic dynamical equation: a drift-diffusion analysis approach.Mathematical and Computer Modelling, 53: 122-130
  101. ******************** 2010 **************

  102. Frank T. D. (2010). Comment on “Dynamic analysis of postural profiles in quiet stance on carpets through fractional Brownian Motion”.Textile Research Journal, 80: 2115-2116
  103. Frank T. D. (2010). Pumping and entropy production in non-equilibrium drift-diffusion systems: a canonical-dissipative approach.European Journal of Scientific Research, 46: 136-146
  104. Frank T. D., Richardson, M.J. (2010). On a test statistic for the Kuramoto order parameter of synchronization: with an illustration for group synchronization during rocking chairs. Physica D, 239:2084-2092
  105. Chiangga, S., Frank T. D. (2010). Stochastic properties of single-transverse-mode vertical-cavity surface-emitting lasers. Nonlinear Phenomena in Complex Systems, 13:32-37
  106. Frank T. D. (2010). A Fokker-Planck approach to canonical-dissipative Nambu systems: with an application to human motor control during dynamic haptic perception.Physics Letters A, 374: 3136-3142
  107. Bödeker, H.U., Beta, C., Frank, T.D., Bodenschatz, E. (2010). Quantitative analysis of random ameboid motion. Europhysics Letters, 90: 28005 (5 pages)
  108. Frank, T.D., van der Kamp, J., Savelsbergh, G.J.P. (2010). On a multistable dynamic model of behavioral and perceptual infant development. Developmental Psychobiology, 52: 352-371
  109. Frank, T.D. (2010). Active systems with Nambu dynamics: with a applications to rod wielding for haptic length perception and self-propagating systems on two-spheres. European Physical Journal B, 74: 195-203
  110. Mongkolsakulvong, S., Frank, T.D. (2010). Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space.Condensed Matter Physics, 13: 13001 (18 pages)
  111. Frank, T.D. (2010). On a moment-based data analysis method for canonical-dissipative oscillatory systems.Fluctuation and Noise Letters, 9: 69-87
  112. Bonnet, C.T., Kinsella-Shaw, J.M., Frank, T.D., Bubela, D., & Turvey, M.T. (2010). Deterministic and stochastic postural processes: effects of age, task, environment.Journal of Motor Behavior, 42: 85-97
  113. ******************** 2009 **************

  114. Frank, T.D. (2009). On a multistable competitive network model in the case of an inhomogeneous growth rate spectrum: with an application to priming.Physics Letters A, 373: 4127-4133
  115. Frank, T.D. (2009). Numeric and exact solutions of the nonlinear Chapman-Kolmogorov equation: a case study for a nonlinear semi-group Markov model.International Journal of Modern Physics B, 23: 3829-3843
  116. Frank, T.D. (2009). Deterministic and stochastic components of nonlinear Markov models with an application to decision making during the bailout votes 2008 (USA).European Physical Journal B, 70: 249-255
  117. Frank, T.D. (2009). Chaos from nonlinear Markov processes: why the whole is different from the sum of its parts.Physica A, 388: 4241-4247
  118. Frank, T.D., Mongkolsakulvong, S. (2009). Parametric solution methods for self-consistency equations and order parameter equations derived from nonlinear Fokker-Planck equations.Physica D, 238: 1186-1196
  119. Frank, T.D., Richardson, M.J., Lopresti-Goodman, S.M., Turvey M.T. (2009). Order parameter dynamics of body-scaled hysteresis and mode transitions in grasping behavior,Journal of Biological Physics, 35: 127-147.
  120. Frank, T.D. (2009). Nonextensive cutoff distributions of postural sway for the old and the youngPhysica A, 388: 2503-2510
  121. Frank, T.D. (2009). On the linear discrepancy model and risky shifts in group behaviour: a nonlinear Fokker-Planck perspective. Journal of Physica A, 42: 155001
  122. Frank, T.D., Blau. J.J.C., Turvey, M.T. (2009). Nonlinear attractor dynamics in the fundamental and extended prism adaptation paradigm,Physics Letters A, 373: 1022-1030
  123. Stepp, N., Frank, T.D. (2009). A data analysis method for decomposing synchronization variability of anticipatory systems into stochastic and deterministic components,European Physical Journal B, 67: 251-257
  124. ******************** 2008 **************

  125. Frank, T.D. (2008). Fokker-Planck equations are more than just partial differential equations: a comment on a study by Dehghan and Tatari (Phys. Scr. 74:2006: 310), Physica Scripta , 78: 067001
  126. Frank, T.D. (2008). Nonlinear Markov processes: deterministic case,Physics Letters A, 372: 6235-6239
  127. Frank, T.D. (2008). Markov chains of nonlinear Markov processes and an application to a winner-takes-all model for social conformity,Journal of Physics A, 41: 282001
  128. Frank, T.D., Mongkolsakulvong, S. (2008). A nonextensive thermostatistical approach to the Haissinski theory of accelerator beams, Physica A , 387: 4828-4838 (Erratum available).
  129. Frank, T.D. (2008). Nonlinear Markov processes, Physics Letters A, 372: 4553-4555
  130. Frank, T.D., Patanarapeelert, K., Beek, P.J. (2008) Portfolio theory of optimal isometric force production: variability predictions and nonequilibrium fluctuation-dissipation theorem, Physics Letters A, 372: 3562-3568
  131. Frank, T.D., Michelbrink, M., Beckmann, H., Schöllhorn, W.I. (2008). On a quantitative dynamical systems approach to differential learning: self-organization principle and order parameter equations,Biological Cybernetics, 98: 19-31.
  132. Frank, T.D. (2008). Green functions and Langevin equations for nonlinear diffusion equations: a comment on ‘Markov processes, Hurst exponents, and nonlinear diffusion equations’ by Bassler et al.Physica A, 387: 773-778
  133. ******************** 2007 **************

  134. Wilmer, A., Frank, T.D., Beek, P.J., Friedrich, R. (2007). A data-analysis method for identifying differential effects of time-delayed feedback forces and periodic driving forces in stochastic systems, European Physical Journal B, 60: 203-215
  135. Frank, T.D. (2007). Towards a time-continuous MA(1) model: application to stock index returns, Nonlinear phenomena in complex systems, 10(3): 256-263.
  136. Frank, T.D. (2007). Kramers-Moyal expansion for stochastic differential equations with single and multiple delays: applications to financial physics and neurophysics.Physics Letters A, 360: 552-562.
  137. Müller, H., Frank, T.D., Sternad, D. (2007). Variability, covariation and invariance with respect to coordinate systems in motor control: reply to Smeets and Louw (2007). Journal of
    Experimental Psychology: Human Perception and Performance
    :33: 250-255.
  138. Frank, T.D. (2007). Exact solutions and Monte Carlo simulations of self-consistent Langevin equations: a case study for the collective dynamics
    of stock prices. International Journal of Modern Physics B, 21: 1099-1112.
  139. Frank, T.D. (2007). A mini-tutorial on measure-valued Markov processes and nonlinear martingale problems, Physica A, 382: 453-464, 2007
  140. Mongkolsakulvong, S., Frank, T.D., Tang, I.M. (2007). Phenomenological generalization of the Maier-Saupe theory for nematic liquid crystals in the framework of the dynamical mean field approach, Phase Transitions, 80: 967-980.
  141. ******************** 2006 **************

  142. Frank, T.D., Friedrich, R., Beek, P.J. (2006). Stochastic order parameter equation of isometric force production revealed by drift-diffusion estimates. Physical Review E, 74: 051905.
  143. Patanarapeelert, K., Frank, T.D., Friedrich, R., Beek, P.J.,Tang I.M. (2006). A data analysis method for identifying deterministic components of stable and unstable time-delayed
    systems with colored noise. Physics Letters A, 360: 190-198.
  144. Frank, T.D. (2006). Smoluchowski approach to nonlinear Vlasov-Fokker-Planck equations: stability analysis of beam dynamics and Haissinski theory. Physical Review ST-AB, 9: 084401.
  145. Patanarapeelert, K., Frank, T.D., Friedrich, R., Beek, P.J.,Tang, I.M. (2006). Theoretical analysis of destabilization resonances in time-delayed stochastic second order dynamical systems and some implications for human motor control. Physical Review E, 73: 021901.
  146. Frank, T.D. (2006). Time-dependent solutions for stochastic systems with delays: perturbation theory and applications to financial physics. Physics Letters A, 357: 275-283.
  147. ******************** 2005 **************

  148. Frank, T.D., Friedrich, R. (2005). Estimating the nonextensivity of systems from experimental data: a nonlinear diffusion equation approach.Physica A, 347: 65-76.
  149. Frank, T.D., Patanarapeelert, K., Tang, I.M.(2005). Delay- and noise-induced transitions: a case study for a Hongler model with time delay.Physics Letters A, 339: 246-251.
  150. Frank, T.D., Friedrich, R., Beek, P.J. (2005). Identifying and comparing states of time-delayed systems: phase diagrams and applications to human movements.Physics Letters A, 338: 74-80.
  151. Frank, T.D. (2005). Short-time correlations of many-body systems described by nonlinear Fokker-Planck equations and Vlasov-Fokker-Planck equations.Physics Letters A 337:224-234.
  152. Frank, T.D., Friedrich, R., Beek, P.J. (2005). Time series analysis for multivariate time-delayed systems with noise: applications to laser physics and human movements.Stochastics and dynamics, 5(2): 297-306.
  153. Frank, T.D., Sondermann, M., Ackemann, T., Friedrich, R. (2005). Parametric data analysis of bistable stochastic systems. Nonlinear phenomena in complex systems, 8(2):193-199.
  154. Frank, T.D. (2005). On the characterization of nanoporous materials by means of empirical and intrinsic transport coefficients.Science and Technology of Advanced Materials, 6: 221-223.
  155. Frank, T. D. (2005). On the Maier-Saupe model of liquid crystals: isotropic-nematic phase transitions and second order statistics studied by Shiino’s perturbation theory and strongly nonlinear Smoluchowski equations.Physical Review E, 72: 041703 (Erratum available).
  156. Frank, T.D. (2005). Delay Fokker-Planck equations, Novikov’s theorem, and Boltzmann distributions as small delay approximations.Physical Review E, 72: 011112.
  157. Frank T.D. (2005). Delay Fokker-Planck equations, perturbation theory, and data analysis for nonlinear stochastic systems with time delays.Physical Review E, 71:031106.
  158. Frank, T.D., Beek, P.J. (2005). Fokker-Planck equations for globally coupled many-body systems with time delays. Journal of Statistical Mechanics: Theory and Experiment, 10: P10010
  159. Patanarapeelert, K., Frank, T.D., Friedrich, R., Tang, I.M. (2005). On reducible nonlinear time-delayed stochastic systems: fluctuation-dissipation relations, transitions to bistability, and secondary bifurcations to non-stationarityJournal of Physics A, 38: 10069-10083.
  160. Frank, T.D. (2005). Stationary distributions of stochastic processes described by a linear neutral delay differential equation.Journal of Physics A, 38: L485-L490.
  161. Frank, T.D.(2005). Modelling the stochastic single particle dynamics of relativistic fermions and bosons using nonlinear drift-diffusion equations. Mathematical and Computer Modelling,42: 1057-1062.
  162. Hutt, A., Frank, T.D. (2005). Stability, critical fluctuations and $1/f^\alpha$-activity of neural fields involving transmission delays.Acta Physica Polonica B, 108: 1021-1040.
  163. ******************** 2004 **************

  164. Frank T.D. (2004). Classical Langevin equations for the free electron gas and blackbody radiation.Journal of Physics A, 37: 3561-3567.
  165. Frank T.D. (2004). Autocorrelation functions of nonlinear Fokker-Planck equations.European Physics Journal B, 37: 139-142.
  166. Frank T.D. (2004). Asymptotic properties of nonlinear diffusion, nonlinear drift-diffusion, and nonlinear reaction-diffusion equations. Annalen der Physik, 13: 461-469.
  167. Bödeker H.U., Liehr A.W., Röttger M.C., Frank T.D., Friedrich R., Purwins H.G. (2004). Measuring the interaction law of dissipative solitons.New Journal of Physics, 6: 62.1-62.18.
  168. Frank T.D. (2004). Analytical results for fundamental time-delayed feedback systems subjected to multiplicative noise.Physical Review E, 69: 061104.
  169. Frank T.D. (2004). On a nonlinear master equation and the Haken-Kelso-Bunz model.Journal of Biological Physics, 30: 139-159.
  170. Frank T.D. (2004). Complete description of a generalized Ornstein-Uhlenbeck process related to the nonextensive Gaussian entropy.Physica A, 340:251-256.
  171. Frank T.D. (2004). Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker-Planck equations.Physica A, 331: 391-408.
  172. Frank T.D. (2004). Dynamic mean field models: H-theorem for stochastic processes and basins of attraction of stationary processes.Physica D, 195: 229-243.
  173. Frank T.D. (2004). Stability analysis of stationary states of mean field models described by Fokker-Planck equations.Physica D, 189: 199-218.
  174. Frank T.D. (2004). Fluctuation-dissipation theorems for nonlinear Fokker-Planck equations of the Desai-Zwanzig type and Vlasov-Fokker-Planck equations.Physics Letters A, 329: 475-485.
  175. Frank T.D., Beek P.J., Friedrich R. (2004). Identifying noise sources of time-delayed feedback systems.Physics Letters A, 328: 219-224.
  176. Frank T.D. (2004). Stability analysis of nonequilibrium mean field models by means of self-consistency equations.Physics Letters A, 327: 146-151.
  177. ******************** 2003 **************

  178. Frank T.D. (2003). A note on the Markov property of stochastic processes described by nonlinear Fokker-Planck equations.Physica A, 320:204-210.
  179. Frank T.D., Beek P.J., Friedrich R. (2003). Fokker-Planck perspective on stochastic delay systems: exact solutions and data analysis of biological systems.Physical Review E, 68: 021912.
  180. Bödeker H.U., Röttger M.C., Liehr A.W., Frank T.D., Friedrich R., Purwins H.G. (2003). Noise-covered drift bifurcation of dissipative solitons in a planar gas-discharge system.Physical Review E, 67: 056220.
  181. Liehr A.W, Bödeker H.U., Röttger M.C., Frank T.D., Friedrich R., Purwins H.G. (2003). Drift-bifurcation detection for dissipative solitons.New Journal of Physics, 5: 89.1-89.9.
  182. Frank T.D. (2003). On the second variation of free energies and nonlinear Fokker-Planck equations involving periodic variables. Progress of Theoretical Physics Supplement, 150: 48-56.
  183. Frank T.D. (2003). A Markov approach to nonlinear multivariate delay systems with noise.Physica Scripta, 68: 333-336.
  184. Frank T.D. (2003). On the boundedness of free energy functionals.Nonlinear phenomena in complex systems, 6(3): 696-704.
  185. Frank T.D. (2003). Single particle dynamics of many-body systems described by Vlasov-Fokker-Planck equations.Physics Letters A, 319:173-180.
  186. ******************** 2002 **************

  187. Frank T.D. (2002). On a general link between anomalous diffusion and nonextensivity.Journal of Mathematical Physics, 43:344-350.
  188. Frank T.D. (2002). Generalized Fokker-Planck equations derived from generalized linear nonequilibrium thermodynamics.Physica A, 310: 397-412.
  189. Frank T.D. (2002). Multivariate Markov processes for stochastic systems with delays: Application to the stochastic Gompertz model with delay.Physical Review E, 66: 011914.
  190. Frank T.D., Daffertshofer A., Beek P.J. (2002). Impacts of statistical feedback on the flexibility-accuracy trade-off of biological systems.Journal of Biological Physics, 28: 39-54.
  191. Frank T.D. (2002). Stability analysis of mean field models described by Fokker-Planck equations.Annalen der Physik, 11:707-716.
  192. Frank T. D. (2002). Generalized multivariate Fokker-Planck equations derived from kinetic transport theory and linear nonequilibrium thermodynamicsPhysics Letters A, 305: 150-159. (Erratum available).
  193. Frank T.D. (2002). Interpretation of Lagrange multipliers of generalized maximum-entropy distributions.Physics Letters A, 299: 153-158.
  194. Frank T.D., Plastino, A.R. (2002). Generalized thermostatistics based on the Sharma-Mittal entropy and escort mean values.European Physical Journal B, 30: 543-549.
  195. Frank T.D. (2002). On a mean field Haken-Kelso-Bunz model and a free energy approach to relaxation processes.Nonlinear phenomena in complex systems, 5(4): 332-341.
  196. ******************** 2001 **************

  197. Frank T.D., Beek P.J. (2001). Stationary solutions of linear stochastic delay differential equations: Applications to biological systems.Physical Review E, 64: 021917.
  198. Frank T.D., Daffertshofer A., Beek P.J. (2001). Multivariate Ornstein-Uhlenbeck processes with mean field dependent coefficients: Applications to postural sway.Physical Review E, 63: 011905.
  199. Frank T.D. (2001). Lyapunov and free energy functionals of generalized Fokker-Planck equations.Physics Letters A, 290: 93-100.
  200. Frank T.D. (2001). H-theorem for Fokker-Planck equations with drifts depending on process mean values.Physics Letters A, 280: 91-96.
  201. Frank T.D., Daffertshofer A., Peper C.E., Beek P.J., Haken H. (2001). H-theorem for a mean field model describing coupled oscillator systems under external forces.Physica D, 150: 219-236.
  202. Frank T.D. (2001). A Langevin approach for the microscopic dynamics of nonlinear Fokker-Planck equations.Physica A, 301: 52-62.
  203. Frank T.D., Daffertshofer A. (2001). H-theorem for nonlinear Fokker-Planck equations related to generalized thermostatistics. Physica A, 295: 455-474.
  204. Frank T. D. and Daffertshofer A. (2001). Multivariate nonlinear Fokker-Planck equations and generalized thermostatistics.Physica A, 292: 392-410.
  205. Frank T. D., Daffertshofer A., and Beek P. J. (2001). Interpreting screw displacement apparent motion as a self-organizing process.Behavioral and Brain Sciences, 24: 668-669.
  206. ******************** 2000 **************

  207. Frank, T.D. (2000). On nonlinear and nonextensive diffusion and the second law of thermodyanmics.Physics Letters A, 267: 298-304.
  208. Frank, T.D., Daffertshofer, A., Peper, C.E., Beek, P.J., Haken, H.(2000). Towards a comprehensive theory of brain activity: coupled oscillator systems under external forces.Physica D, 144: 62-86.
  209. Frank, T.D., Daffertshofer, A. (2000). Exact time-dependent solutions of the Renyi Fokker-Planck equation and the Fokker-Planck equations related to the entropies proposed by Sharma and Mittal.Physica A, 285: 129-144.
  210. Daffertshofer, A., Frank, T.D., Peper, C.E., Beek, P.J. (2000). Three pertinent issues in the modeling of brain activity: Nonlinearities, time scales and neural underpinnings.Behavioral and Brain Sciences, 23:400-401.
  211. Daffertshofer A., Peper C.E., Frank T.D., Beek P.J. (2000). Spatio-temporal patterns of encephalographic signals during polyrhythmic tapping.Human Movement Science, 19: 474-498.
  212. ******************** 1999 **************

  213. Frank, T.D., Daffertshofer, A. (1999). Nonlinear Fokker-Planck equations whose stationary solutions make entropy-like functionals stationaryPhysica A, 272: 497-508. (Erratum available).
  214. Frank, T.D., Daffertshofer, A., Beek, P.J., Haken, H. (1999). Impacts of noise on a field theoretical model of the human brain. Physica D, 127: 233-249.