Skip to main content
SpringerLink
Log in

Detection of coarse-grained unstable states of microscopic/stochastic systems: a timestepper-based iterative protocol

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We address an iterative procedure that can be used to detect coarse-grained hyperbolic unstable equilibria (saddle points) of microscopic simulators when no equations at the macroscopic level are available. The scheme is based on the concept of coarse timestepping (Kevrekidis et al. in Commun. Math. Sci. 1(4):715–762, 2003) incorporating an adaptive mechanism based on the chord method allowing the location of coarse-grained saddle points directly. Ultimately, it can be used in a consecutive manner to trace the coarse-grained open-loop saddle-node bifurcation diagrams of complex dynamical systems and large-scale systems of ordinary and/or partial differential equations. We illustrate the procedure through two indicative examples including (i) a kinetic Monte Carlo simulation (kMC) of simple surface catalytic reactions and (ii) a simple agent-based model, a financial caricature which is used to simulate the dynamics of buying and selling of a large population of interacting individuals in the presence of mimesis. Both models exhibit coarse-grained regular turning points which give rise to branches of saddle points.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adalsteinsson, D., Sethian, J.: A level set approach to a unified model for etching, deposition, and lithography III: Redeposition, reemission, surface diffusion, and complex simulations. J. Comput. Phys. 138, 193–223 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alder, B.J., Wainwright, T.E.: Studies in molecular dynamics. I. General method. J. Chem. Phys. 31(2), 459 (1959)

    Article  MathSciNet  Google Scholar 

  3. Andersen, H.C.: Molecular dynamics simulations at constant pressure and/or temperature. J. Chem. Phys. 72, 2384–2393 (1980)

    Article  Google Scholar 

  4. Frenkel, D., Smit, B.: Understanding Molecular Simulation: From Algorithms to Applications. Academic Press, San Diego (2002)

    Google Scholar 

  5. Gungor, M.R., Maroudas, D., Zhou, S.J.: Molecular-dynamics study of the mechanism and kinetics of void growth in ductile metallic thin films. Appl. Phys. Lett. 77, 343–345 (2000)

    Article  Google Scholar 

  6. Mendelev, M.I., Han, S., Srolovitz, D.J., Ackland, G.J., Sun, D., Asta, M.: Development of new interatomic potentials appropriate for crystalline and liquid iron. Philos. Mag. 83(35), 3977–3994 (2003)

    Article  Google Scholar 

  7. Pantelides, D.M.S.T., Laks, D.B.: Defects in heterogeneous solids: from microphysics to macrophysics. Mater. Sci. Forum 143, 1–8 (1994)

    Article  Google Scholar 

  8. Parinello, M., Rahman, A.: Crystal structure and pair potential: a molecular dynamics study. Phys. Rev. Lett. 45, 1196–1199 (1980)

    Article  Google Scholar 

  9. Rapaport, D.C.: The Art of Molecular Dynamics Simulation. Cambridge University Press, Cambridge (1995)

    Google Scholar 

  10. Allen, M.P., Tildesley, D.J.: Computer Simulation of Liquids. Oxford University Press, London (1989)

    Google Scholar 

  11. Doi, M.: Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci., Part B, Polym. Phys. 19, 229–243 (1981)

    Google Scholar 

  12. Doi, M., Edwards, S.F.: The Theory of Polymer Dynamics. Clarendon Press, Oxford (1988)

    Google Scholar 

  13. Frisch, U., Hasslacher, B., Pomeau, Y.: Lattice-gas automata for the Navier-Stokes equation. Phys. Rev. Lett. 56, 1505–1508 (1986)

    Article  Google Scholar 

  14. Larson, R.G.: The Structure and Rheology of Complex Fluids. Oxford University Press, New York (1999)

    Google Scholar 

  15. Padding, J., Boek, E., Briels, W.: Rheology of wormlike micellar fluids from Brownian and molecular dynamics simulations. J. Phys., Condens. Matter 17, S3347 (2005)

    Article  Google Scholar 

  16. Tao, Y., den Otter, W.K., Briels, W.J.: Kayaking and wagging of rods in shear flow. Phys. Rev. Lett. 95, 2378021-4 (2005)

    Google Scholar 

  17. Iori, G.: A microsimulation of traders activity in the stock market: the role of heterogeneity, agents? Interactions and trade frictions. J. Econ. Behav. Organ. 49, 269–285 (2002)

    Article  Google Scholar 

  18. Liu, X., Gregor, S., Yang, J.: The effects of behavioral and structural assumptions in artificial stock market. Physica A 387, 2535 (2008)

    Google Scholar 

  19. Raberto, M., Cincotti, S., Focardi, S., Marchesi, M.: Agent-based simulation of a financial market. Physica A 299, 319–327 (2001)

    Article  MATH  Google Scholar 

  20. Samanidou, E., Zschischang, E., Stauffer, D., Lux, T.: Agent-based models of financial markets. Rep. Prog. Phys. 70, 409–450 (2007)

    Article  Google Scholar 

  21. Wang, S., Zhang, C.: Microscopic model of financial markets based on belief propagation. Physica A 354, 496–504 (2005)

    Article  Google Scholar 

  22. Bressloff, P.C., Coombes, S.: Travelling waves in chains of pulse-coupled integrate-and-fire oscillators with distributed delays. Physica D 130, 232–254 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Casti, A.R.R., Omurtag, A., Sornborger, A., Kaplan, E., Knight, B., Sirovich, L., Victor, J.: A population study of integrate-and-fire-or-burst neurons. Neural Comput. 14, 957–986 (2002)

    Article  MATH  Google Scholar 

  24. Coombes, S., Osbaldestin, A.H.: Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator. Phys. Rev. E 62, 4057–4066 (2000)

    Article  Google Scholar 

  25. Ermentrout, B.G., Chow, C.C.: Modeling neural oscillations. Physiol. Behav. 77, 629–633 (2002)

    Article  Google Scholar 

  26. Kozma, R., Puljic, M., Balister, P., Bollobas, B., Freeman, W.J.: Phase transitions in the neuropercolation model of neural populations with mixed local and non-local interactions. Biol. Cybern. 92, 367–379 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Laing, C.R., Chow, C.: Stationary bumps in networks of spiking neurons. Neural Comput. 13, 1473–1494 (2001)

    Article  MATH  Google Scholar 

  28. Laing, C.R., Chow, C.: A spiking neuron model for binocular rivalry. J. Comput. Neurosci. 12, 39–53 (2002)

    Article  Google Scholar 

  29. Omurtag, A., Knight, B.W., Sirovich, L.: On the simulation of large populations of neurons. J. Comput. Neurosci. 8, 51–63 (2000)

    Article  MATH  Google Scholar 

  30. Burke, D.S., Epstein, J.S., Cummings, D.A.T., Parker, J.I., Kline, K.C., Singa, R.M., Chakravarty, S.: Individual-based computational modeling of smallpox epidemic control strategies. Acad. Emerg. Med. 13, 1142–1149 (2006)

    Article  Google Scholar 

  31. Eubank, S.H., Guclu, V.S.A., Kumar, M., Marathe, M.V., Srinivasan, A., Toroczkai, Z., Wang, N.: Modelling disease outbreaks in realistic urban social networks. Nature 429, 180–184 (2004)

    Article  Google Scholar 

  32. Ferguson, N.M., Cummings, D.A.T., Cauchemez, S., Fraser, C., Riley, S., Meeyai, A., Iamsirithaworn, S., Burke, D.S.: Strategies for containing an emerging influenza pandemic in southeast Asia. Nature 437, 209–214 (2005)

    Article  Google Scholar 

  33. Keeling, M.J., Eames, K.T.D.: Networks and epidemic models. J. R. Soc. Interface 2, 295–307 (2005)

    Article  Google Scholar 

  34. Longini, I.M., Fine, P.E., Thacker, S.B.: Predicting the global spread of new infectious agents. Am. J. Epidemiol. 123, 383–391 (1986)

    Google Scholar 

  35. Bonabeau, E., Dorigo, M., Theraulaz, G.: Inspiration for optimization from social insect behaviour. Nature 406, 39–42 (2000)

    Article  Google Scholar 

  36. DeAngelis, D., Rose, K., Huston, M.: Frontiers in Mathematical Biology. Springer, Berlin (1994)

    Google Scholar 

  37. Grimm, V.: Ten years of individual-based modelling in ecology: what we have learned, and what could we learn in the future? Ecol. Model. 115, 129–148 (1999)

    Article  Google Scholar 

  38. Levine, H., Rappel, W.J.: Self-organization in systems of self-propelled particles. Phys. Rev. E 63, 0171011-4 (2001)

    Article  Google Scholar 

  39. Liu, Y., Passino, K.M.: Stable social foraging swarms in a noisy environment. IEEE Trans. Autom. Control 49, 30–44 (2004)

    Article  MathSciNet  Google Scholar 

  40. Balmer, M., Nagel, K., Raney, B.: Large scale multi-agent simulations for transportation applications. J. Intell. Transport. Syst. 8, 205–221 (2004)

    Article  Google Scholar 

  41. Dijkstra, J., Jessurun, A., Timmermans, H.: Pedestrian and Evacuation Dynamics. Springer, Berlin (2001)

    Google Scholar 

  42. Helbing, D.: Traffic and related self-driven many-particle systems. Rev. Mod. Phys. 73, 1067–1141 (2001)

    Article  Google Scholar 

  43. Low, D.J., Addison, P.S.: A nonlinear temporal headway model of traffic dynamics. Nonlinear Dyn. 16, 127–151 (1998)

    Article  MATH  Google Scholar 

  44. Raney, B., Cetin, N., Vollmy, A., Vrtic, M., Axhausen, K., Nagel, K.: An agentbased microsimulation model of swiss travel: first results. Netw. Spat. Econ. 3, 23–41 (2003)

    Article  Google Scholar 

  45. Dhooge, A., Govaerts, W., Kuznetsov, Y.A.: MATCONT: A MATLAB package form numerical bifurcation analysis of ODEs. ACM Trans. Math. Softw. 29, 141–164 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Doedel, E.J., Govaerts, W., Kuznetsov, Y.A., Dhooge, A.: Numerical continuation of branch points of equilibria and periodic orbits. Int. J. Bifurc. Chaos 15, 841–860 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  47. Doedel, E., Tuckerman, L.S. (eds.): Numerical methods for bifurcation problems and large-scale dynamical systems. IMA Volumes in Mathematics and its Applications, vol. 119. Springer, Berlin (2000)

    MATH  Google Scholar 

  48. Govaerts, J.: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia (2000)

    Book  MATH  Google Scholar 

  49. Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, New York (1989)

    MATH  Google Scholar 

  50. Seydel, R.: Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos, 2nd edn. Springer, Berlin (1994)

    MATH  Google Scholar 

  51. Gear, C.W., Kevrekidis, I.G., Theodoropoulos, C.: Coarse integration/bifurcation analysis via microscopic simulators: micro-Galerkin methods. Comput. Chem. Eng. 26, 941–963 (2002)

    Article  Google Scholar 

  52. Haataja, M., Srolovitz, D.J., Kevrekidis, I.G.: Apparent hysteresis in a driven system with self-organized drag. Phys. Rev. Lett. 92, 1606031-4 (2004)

    Article  Google Scholar 

  53. Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., Theodoropoulos, C.: Equation-free coarse-grained multiscale computation: enabling microscopic simulators to perform system-level tasks. Commun. Math. Sci. 1(4), 715–762 (2003)

    MathSciNet  MATH  Google Scholar 

  54. Kevrekidis, I.G., Gear, C.W., Hummer, G.: Equation-free: the computer-assisted analysis of complex multiscale systems. AIChE J. 50(7), 1346–1354 (2004)

    Article  Google Scholar 

  55. Makeev, A., Maroudas, D., Kevrekidis, I.G.: Coarse stability and bifurcation analysis using stochastic simulators: kinetic Monte Carlo examples. J. Chem. Phys. 116, 10083–10091 (2002)

    Article  Google Scholar 

  56. Mooller, J., Runborg, O., Kevrekidis, P.G., Lust, K., Kevrekidis, I.G.: Equationfree, effective computation for discrete systems: a time stepper based approach. Int. J. Bifurc. Chaos Appl. Sci. Eng. 15(3), 975–996 (2005)

    Article  Google Scholar 

  57. Moon, S.J., Ghanem, R., Kevrekidis, I.G.: Coarse graining the dynamics of coupled oscillators. Phys. Rev. Lett. 96(14), 144101-4 (2006)

    Article  Google Scholar 

  58. Runborg, O., Theodoropoulos, C., Kevrekidis, I.G.: Effective bifurcation analysis: a timestepper-based approach. Nonlinearity 15, 491–511 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  59. Russo, L., Siettos, C.I., Kevrekidis, I.G.: Reduced computations for nematic-liquid crystals: a timestepper approach for systems with continuous symmetries. J. Non-Newton. Fluid Mech. 146, 51–58 (2007)

    Article  MATH  Google Scholar 

  60. Siettos, C.I., Graham, M., Kevrekidis, I.G.: Coarse Brownian dynamics for nematic liquid crystals: bifurcation diagrams via stochastic simulation. J. Chem. Phys. 118, 10149–10156 (2003)

    Article  Google Scholar 

  61. Makeev, A.G., Maroudas, D., Panagiotopoulos, A., Kevrekidis, I.G.: Coarse bifurcation analysis of kinetic Monte Carlo simulations: a lattice-gas model with lateral interactions. J. Chem. Phys. 117, 8229–8240 (2002)

    Article  Google Scholar 

  62. Siettos, C.I., Armaou, A., Makeev, A.G., Kevrekidis, I.G.: Microscopic/stochastic timesteppers and coarse control: a kinetic Monte Carlo example. AIChE J. 49, 1922–1926 (2003)

    Article  Google Scholar 

  63. Armaou, A., Siettos, C.I., Kevrekidis, I.G.: Time-steppers and coarse control of microscopic distributed processes. Int. J. Robust Nonlinear Control 14, 89–111 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  64. Samaey, G., Vanroose, W., Roose, D., Kevrekidis, I.G.: Newton-Krylov solvers for the equation-free computation of coarse traveling waves. Comput. Methods Appl. Mech. Eng. 197(43–44), 3480–3491 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  65. Kavousanakis, M., Russo, L., Siettos, C., Boudouvis, A.G., Georgiou, G.C.: A timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics. J. Non-Newton. Fluid Mech. 151, 59–68 (2008)

    Article  MATH  Google Scholar 

  66. Siettos, C.I., Maroudas, D., Kevrekidis, I.G.: Coarse bifurcation diagrams via microscopic simulators: a state-feedback control-based approach. Int. J. Bifurc. Chaos Appl. 14, 207–220 (2004)

    Article  MATH  Google Scholar 

  67. Siettos, C., Rico-Martinez, R., Kevrekidis, I.G.: A systems-based approach to multiscale computation: equation-free detection of coarse-grained bifurcations. Comput. Chem. Eng. 30, 1632–1642 (2006)

    Article  Google Scholar 

  68. Christofides, P.D., Armaou, A.: Control and optimization of multiscale process systems. Comput. Chem. Eng. 30, 1670–1686 (2006)

    Article  Google Scholar 

  69. Lou, Y.M., Christofides, P.D.: Feedback control of surface roughness in sputtering processes using the stochastic Kuramoto-Sivashinsky equation. Comput. Chem. Eng. 29, 741–759 (2005)

    Article  Google Scholar 

  70. Ni, D., Christofides, P.: Dynamics and control of thin film surface microstructure in a complex deposition process. Chem. Eng. Sci. 60, 1603–1617 (2005)

    Article  Google Scholar 

  71. Ni, D., Christofides, P.: Multivariable predictive control of thin film deposition using a stochastic PDE model. Ind. Eng. Chem. Res. 44, 2416–2427 (2005)

    Article  Google Scholar 

  72. Gallivan, M.A., Murray, R.M.: Reduction and identification methods for Markovian control systems, with application to thin film deposition. Int. J. Robust Nonlinear Control 14, 113–132 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  73. Braatz, R., Alkire, R., Seebauer, E., Rusli, E., Gunawan, R., Drews, X.L.T.O., HE, Y.: Perspectives on the design and control of multiscale systems. J. Process Control 16, 193–204 (2006)

    Article  Google Scholar 

  74. Rusli, E., Drews, T.O., Ma, D.L., Alkire, R.C., Braatz, R.D.: Robust nonlinear feedback-feedforward control of a coupled kinetic Monte Carlo—finite difference simulation. J. Process Control 16, 409–417 (2006)

    Article  Google Scholar 

  75. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31, 53–98 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  76. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 2nd edn. Springer, Berlin (1998)

    MATH  Google Scholar 

  77. Kelley, C.T.: Iterative Methods for Linear and Nonlinear Equations. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  78. Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434 (1976)

    Article  MathSciNet  Google Scholar 

  79. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361 (1977)

    Article  Google Scholar 

  80. Ertl, G.: In: Anderson, J.R., Boudart, M. (eds.) Catalysis, Science and Technology, vol. 4. Springer, Berlin (1983). p. 209

    Google Scholar 

  81. Eiswirth, M., Ertl, G.: Kinetic oscillations in the catalytic CO oxidation on a Pt(110) surface. Surf. Sci. 177, 90–100 (1986)

    Article  Google Scholar 

  82. Eiswirth, R.M., Krischer, K., Ertl, G.: Nonlinear dynamics in the CO-oxidation on Pt single crystal surfaces. Appl. Phys. A, Mater. Sci. Process. 51, 79–90 (1990)

    Article  Google Scholar 

  83. Rosè, H., Hempel, H., Schimansky-Geier, L.: Stochastic dynamics of catalytic CO oxidation on Pt(100). Physica A 206, 42–440 (1994)

    Article  Google Scholar 

  84. Omurtag, A., Sirovich, L.: Modeling a large population of traders: mimesis and stability. J. Econ. Behav. Organ. 61, 562–576 (2006)

    Article  Google Scholar 

  85. Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Nadler, B., Warner, F., Zucker, S.: Geometric diffusions as a tool for harmonic analysis and structure definition of data. Part I: Diffusion maps. Proc. Natl. Acad. Sci. USA 102(21), 7426–7431 (2005)

    Article  Google Scholar 

  86. Goussis, D., Valorani, M.: An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems. J. Comput. Phys. 214, 316–346 (2006)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Applied Mathematics & Physical Sciences, National Technical University of Athens, 157 80, Athens, Greece

    A. C. Tsoumanis & C. I. Siettos

Authors
  1. A. C. Tsoumanis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. I. Siettos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. I. Siettos.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tsoumanis, A.C., Siettos, C.I. Detection of coarse-grained unstable states of microscopic/stochastic systems: a timestepper-based iterative protocol. Nonlinear Dyn 67, 103–117 (2012). https://doi.org/10.1007/s11071-011-9962-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-9962-0

Keywords

Search

Navigation