# American Institute of Mathematical Sciences

October  2012, 17(7): 2387-2412. doi: 10.3934/dcdsb.2012.17.2387

## Construction of a finite-time Lyapunov function by meshless collocation

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

Received  July 2011 Revised  February 2012 Published  July 2012

We consider a nonautonomous ordinary differential equation of the form $\dot{x}=f(t,x)$, $x\in \mathbb{R}^n$ over a finite-time interval $t\in [T_1,T_2]$. The basin of attraction of an attracting solution can be determined using a finite-time Lyapunov function.
In this paper, such a finite-time Lyapunov function is constructed by Meshless Collocation, in particular Radial Basis Functions. Thereto, a finite-time Lyapunov function is characterised as the solution of a second-order linear partial differential equation with boundary values. This problem is approximately solved using Meshless Collocation, and it is shown that the approximate solution can be used to determine the basin of attraction. Error estimates are obtained and verified in examples.
Citation: Peter Giesl. Construction of a finite-time Lyapunov function by meshless collocation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2387-2412. doi: 10.3934/dcdsb.2012.17.2387
##### References:
 [1] A. Berger, On finite-time hyperbolicity, Comm. Pure Applied Anal., 10 (2011), 963-981. doi: 10.3934/cpaa.2011.10.963. [2] A. Berger, D. T. Son and S. Siegmund, Nonautonomous finite-time dynamics, Discrete Cont. Dyn. Syst. Ser. B, 9 (2008), 463-492. [3] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515. doi: 10.1137/S0036142996313002. [4] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in "Handbook of Dynamical Systems,'' Vol. 2, North-Holland, Amsterdam, (2002), 221-264. [5] T. S. Doan, D. Karrasch, N. T. Yet and S. Siegmund, A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents, submitted. [6] T. S. Doan, K. Palmer and S. Siegmund, Transient spectral theory, stable and unstable cones and Gershgorin's theorem for finite-time differential equations, J. Diff. Equations, 250 (2011), 4177-4199. doi: 10.1016/j.jde.2011.01.013. [7] C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput., 93 (1998), 73-82. doi: 10.1016/S0096-3003(97)10104-7. [8] P. Giesl, "Construction of Global Lyapunov Functionsusing Radial Basis Functions,'' Lecture Notes in Mathematics, 1904, Springer, Berlin, 2007. [9] P. Giesl and S. Hafstein, Local Lyapunov Functions for periodic and finite-time ODEs, submitted. [10] P. Giesl and M. Rasmussen, Areas of attraction for nonautonomous differential equations on finite time intervals, J. Math. Anal. Appl., 390 (2012), 27-46. doi: 10.1016/j.jmaa.2011.12.051. [11] 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. [12] P. Giesl and H. Wendland, Approximating the basin of attraction of time-periodic ODEs by meshless collocation, Discrete Contin. Dyn. Syst., 25 (2009), 1249-1274. doi: 10.3934/dcds.2009.25.1249. [13] P. Giesl and H. Wendland, Approximating the Basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem, Discrete Contin. Dyn. Syst., 2009, Dynamical Systems, Differential Equations and Applications, 7$^th$ AIMS Conference, suppl., 259-268. [14] L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337. doi: 10.1007/s002110050241. [15] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6. [16] S. Hafstein, A constructive converse Lyapunov Theoremon Exponential Stability, Discrete Contin. Dyn. Syst., 10 (2004), 657-678. doi: 10.3934/dcds.2004.10.657. [17] G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10 (2000), 99-108. doi: 10.1063/1.166479. [18] G. Haller, A variational theory of hyperbolic Lagrangian coherent structures, Physica D, 240 (2011), 574-598. doi: 10.1016/j.physd.2010.11.010. [19] G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent, Chaos, 21 (2011), 7 pp. [20] G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulance, Physica D, 147 (2000), 352-370. doi: 10.1016/S0167-2789(00)00142-1. [21] C. Hsu, Global analysis by cell mapping, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 727-771. [22] B. Krauskopf and H. Osinga, Computing invariant manifolds via the continuation of orbit segments, in "Numerical Continuation Methods for Dynamical Systems'' (eds. B. Krauskopf, H. Osinga and J. Galán-Vioque), Undert. Complex Syst., Springer, Dordrecht, (2007), 117-154. [23] B. Krauskopf, H. Osinga, E. J. Doedel, M. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791. doi: 10.1142/S0218127405012533. [24] A. M. Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203-474; Translation of the Russian version, published 1893 in Comm. Soc. Math. Kharkow. Newly printed: Ann. of Math. Stud., 17, Princeton, 1949. [25] F. J. Narcowich, J. D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Mathematics of Computation, 74 (2005), 743-763. doi: 10.1090/S0025-5718-04-01708-9. [26] G. Osipenko, "Dynamical Systems, Graphs, and Algorithms,'' Lecture Notes in Mathematics, 1889, Springer-Verlag, Berlin, 2007. [27] M. Rasmussen, Finite-time attractivity and bifurcation for nonautonomous differential equation, Differential Equations Dynam. Systems, 18 (2010), 57-78. doi: 10.1007/s12591-010-0009-7. [28] S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212 (2005), 271-304. doi: 10.1016/j.physd.2005.10.007. [29] H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137. [30] H. Wendland, "Scattered Data Approximation,'' Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005. [31] Z. Wu, Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl., 8 (1992), 1-10.

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##### References:
 [1] A. Berger, On finite-time hyperbolicity, Comm. Pure Applied Anal., 10 (2011), 963-981. doi: 10.3934/cpaa.2011.10.963. [2] A. Berger, D. T. Son and S. Siegmund, Nonautonomous finite-time dynamics, Discrete Cont. Dyn. Syst. Ser. B, 9 (2008), 463-492. [3] M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36 (1999), 491-515. doi: 10.1137/S0036142996313002. [4] M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems, in "Handbook of Dynamical Systems,'' Vol. 2, North-Holland, Amsterdam, (2002), 221-264. [5] T. S. Doan, D. Karrasch, N. T. Yet and S. Siegmund, A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents, submitted. [6] T. S. Doan, K. Palmer and S. Siegmund, Transient spectral theory, stable and unstable cones and Gershgorin's theorem for finite-time differential equations, J. Diff. Equations, 250 (2011), 4177-4199. doi: 10.1016/j.jde.2011.01.013. [7] C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput., 93 (1998), 73-82. doi: 10.1016/S0096-3003(97)10104-7. [8] P. Giesl, "Construction of Global Lyapunov Functionsusing Radial Basis Functions,'' Lecture Notes in Mathematics, 1904, Springer, Berlin, 2007. [9] P. Giesl and S. Hafstein, Local Lyapunov Functions for periodic and finite-time ODEs, submitted. [10] P. Giesl and M. Rasmussen, Areas of attraction for nonautonomous differential equations on finite time intervals, J. Math. Anal. Appl., 390 (2012), 27-46. doi: 10.1016/j.jmaa.2011.12.051. [11] 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. [12] P. Giesl and H. Wendland, Approximating the basin of attraction of time-periodic ODEs by meshless collocation, Discrete Contin. Dyn. Syst., 25 (2009), 1249-1274. doi: 10.3934/dcds.2009.25.1249. [13] P. Giesl and H. Wendland, Approximating the Basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem, Discrete Contin. Dyn. Syst., 2009, Dynamical Systems, Differential Equations and Applications, 7$^th$ AIMS Conference, suppl., 259-268. [14] L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation, Numer. Math., 75 (1997), 319-337. doi: 10.1007/s002110050241. [15] H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6. [16] S. Hafstein, A constructive converse Lyapunov Theoremon Exponential Stability, Discrete Contin. Dyn. Syst., 10 (2004), 657-678. doi: 10.3934/dcds.2004.10.657. [17] G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10 (2000), 99-108. doi: 10.1063/1.166479. [18] G. Haller, A variational theory of hyperbolic Lagrangian coherent structures, Physica D, 240 (2011), 574-598. doi: 10.1016/j.physd.2010.11.010. [19] G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent, Chaos, 21 (2011), 7 pp. [20] G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulance, Physica D, 147 (2000), 352-370. doi: 10.1016/S0167-2789(00)00142-1. [21] C. Hsu, Global analysis by cell mapping, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 727-771. [22] B. Krauskopf and H. Osinga, Computing invariant manifolds via the continuation of orbit segments, in "Numerical Continuation Methods for Dynamical Systems'' (eds. B. Krauskopf, H. Osinga and J. Galán-Vioque), Undert. Complex Syst., Springer, Dordrecht, (2007), 117-154. [23] B. Krauskopf, H. Osinga, E. J. Doedel, M. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz and O. Junge, A survey of methods for computing (un)stable manifolds of vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15 (2005), 763-791. doi: 10.1142/S0218127405012533. [24] A. M. Lyapunov, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse, 9 (1907), 203-474; Translation of the Russian version, published 1893 in Comm. Soc. Math. Kharkow. Newly printed: Ann. of Math. Stud., 17, Princeton, 1949. [25] F. J. Narcowich, J. D. Ward and H. Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Mathematics of Computation, 74 (2005), 743-763. doi: 10.1090/S0025-5718-04-01708-9. [26] G. Osipenko, "Dynamical Systems, Graphs, and Algorithms,'' Lecture Notes in Mathematics, 1889, Springer-Verlag, Berlin, 2007. [27] M. Rasmussen, Finite-time attractivity and bifurcation for nonautonomous differential equation, Differential Equations Dynam. Systems, 18 (2010), 57-78. doi: 10.1007/s12591-010-0009-7. [28] S. C. Shadden, F. Lekien and J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D, 212 (2005), 271-304. doi: 10.1016/j.physd.2005.10.007. [29] H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree, J. Approx. Theory, 93 (1998), 258-272. doi: 10.1006/jath.1997.3137. [30] H. Wendland, "Scattered Data Approximation,'' Cambridge Monographs on Applied and Computational Mathematics, 17, Cambridge University Press, Cambridge, 2005. [31] Z. Wu, Hermite-Birkhoff interpolation of scattered data by radial basis functions, Approx. Theory Appl., 8 (1992), 1-10.
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