Geodesic
Fast Sampling of Evolving Systems with Periodic Trajectories
I.Yu. Tyukin1, 2 ; A.N. Gorban1 ; T.A. Tyukina1 ; J.M. Al-Ameri1 ; Yu.A. Korablev2
1 University of Leicester, Department of Mathematics, United Kingdom
2 Saint-Petersburg State Electrotechnical University, Department of Automation and Control Processes, Russia
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 4, pp. 73-88.

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We propose a novel method for fast and scalable evaluation of periodic solutions of systems of ordinary differential equations for a given set of parameter values and initial conditions. The equations governing the system dynamics are supposed to be of a special class, albeit admitting nonlinear parametrization and nonlinearities. The method enables to represent a given periodic solution as sums of computable integrals and functions that are explicitly dependent on parameters of interest and initial conditions. This allows invoking parallel computational streams in order to increase speed of calculations. Performance and practical implications of the method are illustrated with examples including classical predator-prey system and models of neuronal cells.
DOI : 10.1051/mmnp/201611406

I.Yu. Tyukin 1, 2 ; A.N. Gorban 1 ; T.A. Tyukina 1 ; J.M. Al-Ameri 1 ; Yu.A. Korablev 2

1 University of Leicester, Department of Mathematics, United Kingdom
2 Saint-Petersburg State Electrotechnical University, Department of Automation and Control Processes, Russia 
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I.Yu. Tyukin; A.N. Gorban; T.A. Tyukina; J.M. Al-Ameri; Yu.A. Korablev. Fast Sampling of Evolving Systems with Periodic Trajectories. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 4, pp. 73-88. doi : 10.1051/mmnp/201611406. https://geodesic-test.mathdoc.fr/articles/10.1051/mmnp/201611406/

[1] H.D.I. Abarbanel, D. Creveling, R. Farisian, M. Kostuk Dynamical state and parameter estimation SIAM J. Appl. Dyn. Sys. 2009 1341 1381

[2] M.W. Adamson, A. Yu. Morozov When can we trust our model predictions? Unearthing structural sensitivity in biological systems Proc. R. Soc. 2013

[3] D. Alonso, F. Bartumeus, J. Catalan Mutual interfeence between predators can give rise to turing spatial patterns Ecology 2002 28 34

[4] L. Chen, B. Liua, Z. Tengb Analysis of a predator-prey model with Holling II functional response concerning impulsive control strategy J. Comp. Appl. Math. 2006 347 362

[5] G. Besancon Remarks on nonlinear adaptive observer design Syst. Contr. Lett. 2000 271 280

[6] D. Brewer, M. Barenco, R. Callard, M. Hubank, J. Stark Fitting ordinary differential equations to short time course data Phil. Trans. R. Soc. A 2008 519 544

[7] J. Distefano, C. Cobelli On parameter and structural identifiabiliy: Nonunique observability/reconstructibility for identifiable systems, other ambiguities, and new definitions IEEE Trans. Automat. Cont. 1980 830 833

[8] D. Fairhurst, I.Yu. Tyukin, H. Nijmeijer, C. Van Leeuwen Observers for canonic models of neural oscillators Math. Model. Nat. Phen. 2010 146 184

[9] M. Farza, M. M'Saad, T. Maatoung, M. Kamoun Adaptive observers for nonlinearly parameterized class of nonlinear systems Automatica 2009 2292 2299

[10] A.N. Gorban. Slow relaxations and bifurcations of omega-limit sets of dynamical systems. PhD thesis, Kuibyshev, Russia, 1980.

[11] A.N. Gorban Singularities of transition processes in dynamical systems: Qualitative theory of critical delays electron Electr. J. Diff. Eqns., Monograph 2004

[12] H.F. Grip, T.A. Johansen, L. Imsland, G.O. Kaasa Parameter estimation and compensation in systems with nonlinearly parameterized perturbations Automatica 2010 19 28

[13] A.L. Hodgkin, A.F. Huxley A quantitative description of membrane current and its application to conduction and excitation in nerve J. Physiol. 1952 500 544

[14] E. Izhikevich. Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, 2007.

[15] T. Johnson, W. Tucker Rigorous parameter reconstruction for differential equations with noisy data Automatica 2008 2422 2426

[16] P. Kuhl, M. Deihl, T. Kraus, J. P. Schloder, H. G. Bock A real-time algorithm for moving horizon state and parameter estimation Comp. Chem. Eng. 2011 71 83

[17] A. Loria, E. Panteley Uniform exponential stability of linear time-varying systems: revisited Syst. Contr. Lett. 2003 13 24

[18] Z.J. Jing, L.S. Chen The existence and uniqueness of limit cycles in general predator-prey differential equations Chineese Sci. Bull. 1984 521 523

[19] R. Marino, P. Tomei Global adaptive observers for nonlinear systems via filtered transformations IEEE Trans. Automat. Contr. 1992 1239 1245

[20] H. Miao, X. Xia, A. Perelson, H. Wu On identifiability of nonlinear ode models and applications in viral dynamics SIAM Rev. 2011 3 39

[21] C. Morris, H. Lecar Voltage oscillations in the barnacle giant muscle fiber Biophys. J. 1981 193 213

[22] J.A. Nelder, R. Mead A simplex method for function minimization Comp. J. 1965 308 313

[23] A. Pavlov, B.G.B. Hunnekens, N.V.D. Wouw, H. Nijmeijer Steady-state performance optimization for nonlinear control systems Automatica 2013 2087 2097

[24] I. Tyukin. Adaptation in Dynamical Systems. Cambridge Univ. Press, 2011.

[25] I. Tyukin, E. Steur, H. Nijmeijer, C. Van Leeuwen Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters Automatica 2013 2409 2423

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