# American Institute of Mathematical Sciences

June  2016, 3(2): 191-210. doi: 10.3934/jcd.2016010

## Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation

 1 Department of Mathematics, University of Sussex, Falmer BN1 9QH, United Kingdom

Received  December 2016 Revised  February 2017 Published  April 2017

A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of a periodic orbit without requiring information about its position or stability. Moreover, it is robust to small perturbations of the system.
In two-dimensional systems, a contraction metric can be characterised by a scalar-valued function. In [9], the function was constructed as solution of a first-order linear Partial Differential Equation (PDE), and numerically constructed using meshless collocation. However, information about the periodic orbit was required, which needed to be approximated.
In this paper, we overcome this requirement by studying a second-order PDE, which does not require any information about the periodic orbit. We show that the second-order PDE has a solution, which defines a contraction metric. We use meshless collocation to approximate the solution and prove error estimates. In particular, we show that the approximation itself is a contraction metric, if the collocation points are dense enough. The method is applied to two examples.
Citation: Peter Giesl, James McMichen. Determination of the basin of attraction of a periodic orbit in two dimensions using meshless collocation. Journal of Computational Dynamics, 2016, 3 (2) : 191-210. doi: 10.3934/jcd.2016010
##### References:
 [1] D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Trans. Automat. Contr., 47 (2002), 410-421. doi: 10.1109/9.989067. [2] E. Aylward, P. Parrilo and J.-J. Slotine, Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 44 (2008), 2163-2170. doi: 10.1016/j.automatica.2007.12.012. [3] V. Boichenko, G. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations, volume 141 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 2005. doi: 10.1007/978-3-322-80055-8. [4] G. Borg, A Condition for the Existence of Orbitally Stable Solutions of Dynamical Systems, Kungliga Tekniska Högskolan Handlingar Stockholm 153, 1960. [5] M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543241. [6] F. Forni and R. Sepulchre, A differential Lyapunov framework for Contraction Analysis, IEEE Trans. Automat. Control, 59 (2014), 614-628. doi: 10.1109/TAC.2013.2285771. [7] P. Giesl, Necessary conditions for a limit cycle and its basin of attraction, Nonlinear Anal., 56 (2004), 643-677. doi: 10.1016/j.na.2003.07.020. [8] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, volume 1904 of Lecture Notes in Mathematics. Springer, Berlin, 2007. [9] P. Giesl, On the determination of the basin of attraction of a periodic orbit in two-dimensional systems, Journal of Mathematical Analysis and Applications, 335 (2007), 461-479. doi: 10.1016/j.jmaa.2007.01.069. [10] P. Giesl, On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems, J. Math. Anal. Appl., 354 (2009), 606-618. doi: 10.1016/j.jmaa.2009.01.027. [11] P. Giesl and S. Hafstein, Construction of a CPA contraction metric for periodic orbits using semidefinite optimization, Nonlinear Anal., 86 (2013), 114-134. doi: 10.1016/j.na.2013.03.012. [12] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291. [13] P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741. doi: 10.1137/060658813. [14] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964. [15] P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104 (1962), 154-178. doi: 10.2307/1993939. [16] A. Iske, Perfect Centre Placement for Radial Basis Function Methods, Technical report, Technical report TUM M9809, 1999. [17] G. Leonov, I. Burkin and A. Shepelyavyi, Frequency Methods in Oscillation Theory, Ser. Math. and its Appl., Vol. 357, Kluwer, 1996. doi: 10.1007/978-94-009-0193-3. [18] D. Lewis, Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312. doi: 10.2307/2372245. [19] W. Lohmiller and J.-J. Slotine, On contraction analysis for non-linear systems, Automatica, 34 (1998), 683-696. doi: 10.1016/S0005-1098(98)00019-3. [20] I. Manchester and J.-J. Slotine, Transverse contraction criteria for existence, stability, and robustness of a limit cycle, Systems Control Lett., 63 (2014), 32-38. doi: 10.1016/j.sysconle.2013.10.005. [21] J. McMichen, Determination of Areas and Basins of Attraction in Planar Dynamical Systems using Meshless Collocation, PhD thesis, University of Sussex, 2016. [22] B. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155. doi: 10.7146/math.scand.a-10661. [23] H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Journal of Approximation Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137. [24] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2005.

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##### References:
 [1] D. Angeli, A Lyapunov approach to incremental stability properties, IEEE Trans. Automat. Contr., 47 (2002), 410-421. doi: 10.1109/9.989067. [2] E. Aylward, P. Parrilo and J.-J. Slotine, Stability and robustness analysis of nonlinear systems via contraction metrics and SOS programming, Automatica, 44 (2008), 2163-2170. doi: 10.1016/j.automatica.2007.12.012. [3] V. Boichenko, G. Leonov and V. Reitmann, Dimension Theory for Ordinary Differential Equations, volume 141 of Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 2005. doi: 10.1007/978-3-322-80055-8. [4] G. Borg, A Condition for the Existence of Orbitally Stable Solutions of Dynamical Systems, Kungliga Tekniska Högskolan Handlingar Stockholm 153, 1960. [5] M. Buhmann, Radial Basis Functions: Theory and Implementations, volume 12 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2003. doi: 10.1017/CBO9780511543241. [6] F. Forni and R. Sepulchre, A differential Lyapunov framework for Contraction Analysis, IEEE Trans. Automat. Control, 59 (2014), 614-628. doi: 10.1109/TAC.2013.2285771. [7] P. Giesl, Necessary conditions for a limit cycle and its basin of attraction, Nonlinear Anal., 56 (2004), 643-677. doi: 10.1016/j.na.2003.07.020. [8] P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, volume 1904 of Lecture Notes in Mathematics. Springer, Berlin, 2007. [9] P. Giesl, On the determination of the basin of attraction of a periodic orbit in two-dimensional systems, Journal of Mathematical Analysis and Applications, 335 (2007), 461-479. doi: 10.1016/j.jmaa.2007.01.069. [10] P. Giesl, On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems, J. Math. Anal. Appl., 354 (2009), 606-618. doi: 10.1016/j.jmaa.2009.01.027. [11] P. Giesl and S. Hafstein, Construction of a CPA contraction metric for periodic orbits using semidefinite optimization, Nonlinear Anal., 86 (2013), 114-134. doi: 10.1016/j.na.2013.03.012. [12] P. Giesl and S. Hafstein, Review on computational methods for Lyapunov functions, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2291-2331. doi: 10.3934/dcdsb.2015.20.2291. [13] P. Giesl and H. Wendland, Meshless collocation: Error estimates with application to dynamical systems, SIAM J. Numer. Anal., 45 (2007), 1723-1741. doi: 10.1137/060658813. [14] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964. [15] P. Hartman and C. Olech, On global asymptotic stability of solutions of differential equations, Trans. Amer. Math. Soc., 104 (1962), 154-178. doi: 10.2307/1993939. [16] A. Iske, Perfect Centre Placement for Radial Basis Function Methods, Technical report, Technical report TUM M9809, 1999. [17] G. Leonov, I. Burkin and A. Shepelyavyi, Frequency Methods in Oscillation Theory, Ser. Math. and its Appl., Vol. 357, Kluwer, 1996. doi: 10.1007/978-94-009-0193-3. [18] D. Lewis, Metric properties of differential equations, Amer. J. Math., 71 (1949), 294-312. doi: 10.2307/2372245. [19] W. Lohmiller and J.-J. Slotine, On contraction analysis for non-linear systems, Automatica, 34 (1998), 683-696. doi: 10.1016/S0005-1098(98)00019-3. [20] I. Manchester and J.-J. Slotine, Transverse contraction criteria for existence, stability, and robustness of a limit cycle, Systems Control Lett., 63 (2014), 32-38. doi: 10.1016/j.sysconle.2013.10.005. [21] J. McMichen, Determination of Areas and Basins of Attraction in Planar Dynamical Systems using Meshless Collocation, PhD thesis, University of Sussex, 2016. [22] B. Stenström, Dynamical systems with a certain local contraction property, Math. Scand., 11 (1962), 151-155. doi: 10.7146/math.scand.a-10661. [23] H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, Journal of Approximation Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137. [24] H. Wendland, Scattered Data Approximation, volume 17 of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2005.
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