Abstract
Differential equations describe many interesting phenomena arising from various disciplines. This includes many important models, e.g. predator-prey in population biology or the Van der Pol oscillator in electrical engineering. Complete Lyapunov functions allow for the systematic study of the qualitative behaviour of complicated systems. In this paper, we extend the analysis of the algorithm presented in [1]. We study the efficiency of our algorithm and discuss important sections of the code.
The first author in this paper is supported by the Icelandic Research Fund (Rannís) grant number 163074-052, Complete Lyapunov functions: Efficient numerical computation.
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Argáez, C., Giesl, P., Hafstein, S.: Iterative construction of complete Lyapunov functions. In: Proceedings of 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2018), pp. 211–222 (2018). https://doi.org/10.5220/0006835402110222, ISBN 978-989-758-323-0
Argáez, C., Giesl, P., Hafstein, S.: Analysing dynamical systems towards computing complete Lyapunov functions. In: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, SIMULTECH 2017, Madrid, Spain (2017)
Argáez, C., Giesl, P., Hafstein, S.: Computational approach for complete Lyapunov functions. Accepted in Springer Proceedings in Mathematics and Statistics (2018)
Argáez, C., Hafstein, S., Giesl, P.: Wendland functions a C++ code to compute them. In: Proceedings of the 7th International Conference on Simulation and Modeling Methodologies, Technologies and Applications, SIMULTECH 2017, Madrid, Spain, pp. 323–330 (2017)
Argáez, C., Giesl, P., Hafstein, S.: Computation of complete Lyapunov functions for three-dimensional systems (2018, submitted)
Argáez, C., Giesl, P., Hafstein, S.: Computation of complete Lyapunov functions for three-dimensional systems. In: 57th IEEE Conference on Decision and Control (CDC) (2018, to be published)
Conley, C.: Isolated Invariant Sets and the Morse Index. CBMS Regional Conference Series, vol. 38. American Mathematical Society (1978)
Conley, C.: The gradient structure of a flow I. Ergodic Theory Dyn. Syst. 8, 11–26 (1988)
Dellnitz, M., Junge, O.: Set oriented numerical methods for dynamical systems. In: Handbook of Dynamical Systems, vol. 2, pp. 221–264. North-Holland, Amsterdam (2002)
Giesl, P.: Construction of Global Lyapunov Functions Using Radial Basis Functions. Lecture Notes in Mathematics, vol. 1904. Springer, Berlin (2007)
Giesl, P., Hafstein, S.: Review of computational methods for Lyapunov functions. Discrete Contin. Dyn. Syst. Ser. B 20(8), 2291–2331 (2015)
Giesl, P., Wendland, H.: Meshless collocation: error estimates with application to Dynamical Systems. SIAM J. Numer. Anal. 45(4), 1723–1741 (2007)
Hsu, C.S.: Cell-to-Cell Mapping. Applied Mathematical Sciences, vol. 64. Springer, New York (1987)
Hurley, M.: Chain recurrence, semiflows, and gradients. J. Dyn. Diff. Equ. 7(3), 437–456 (1995)
Hurley, M.: Lyapunov functions and attractors in arbitrary metric spaces. Proc. Am. Math. Soc. 126, 245–256 (1998)
Iske, A.: Perfect centre placement for radial basis function methods. Technical report TUM-M9809, TU Munich, Germany (1998)
Krauskopf, B., Osinga, H., Doedel, E.J., Henderson, M., Guckenheimer, J., Vladimirsky, A., Dellnitz, M., Junge, O.: A survey of methods for computing (un)stable manifolds of vector fields. Int. J. Bifur. Chaos Appl. Sci. Eng. 15(3), 763–791 (2005)
Lyapunov, A.M.: The general problem of the stability of motion. Int. J. Control 55(3), 521–790 (1992). Translated by A. T. Fuller from Édouard Davaux’s French translation (1907) of the 1892 Russian original
Narcowich, F.J., Ward, J.D., Wendland, H.: Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting. Math. Comput. 74, 743–763 (2005)
Wendland, H.: Error estimates for interpolation by compactly supported radial basis functions of minimal degree. J. Approx. Theory 93, 258–272 (1998)
Wendland, H.: Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)
Acknowledgement
First author wants to thank Dr. A. Argáez for nice discussions on normed spaces as well as the Icelandic Research Fund (Rannís) for funding this work under the grant: number 163074-052, Complete Lyapunov functions: Efficient numerical computation.
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Argáez, C., Giesl, P., Hafstein, S. (2020). Iterative Construction of Complete Lyapunov Functions: Analysis of Algorithm Efficiency. In: Obaidat, M., Ören, T., Rango, F. (eds) Simulation and Modeling Methodologies, Technologies and Applications. SIMULTECH 2018. Advances in Intelligent Systems and Computing, vol 947. Springer, Cham. https://doi.org/10.1007/978-3-030-35944-7_5
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