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 & 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. doi: 10.3934/cpaa.2011.10.963. Google Scholar

[2]

A. Berger, D. T. Son and S. Siegmund, Nonautonomous finite-time dynamics,, Discrete Cont. Dyn. Syst. Ser. B, 9 (2008), 463. Google Scholar

[3]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM J. Numer. Anal., 36 (1999), 491. doi: 10.1137/S0036142996313002. Google Scholar

[4]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in, (2002), 221. Google Scholar

[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., (). Google Scholar

[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. doi: 10.1016/j.jde.2011.01.013. Google Scholar

[7]

C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions,, Appl.\ Math.\ Comput., 93 (1998), 73. doi: 10.1016/S0096-3003(97)10104-7. Google Scholar

[8]

P. Giesl, "Construction of Global Lyapunov Functionsusing Radial Basis Functions,'', Lecture Notes in Mathematics, 1904 (2007). Google Scholar

[9]

P. Giesl and S. Hafstein, Local Lyapunov Functions for periodic and finite-time ODEs,, submitted., (). Google Scholar

[10]

P. Giesl and M. Rasmussen, Areas of attraction for nonautonomous differential equations on finite time intervals,, J. Math. Anal. Appl., 390 (2012), 27. doi: 10.1016/j.jmaa.2011.12.051. Google Scholar

[11]

P. Giesl and H. Wendland, Meshless Collocation: Error Estimates with Application to Dynamical Systems,, SIAM J. Numer. Anal., 45 (2007), 1723. doi: 10.1137/060658813. Google Scholar

[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. doi: 10.3934/dcds.2009.25.1249. Google Scholar

[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 (): 259. Google Scholar

[14]

L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation,, Numer. Math., 75 (1997), 319. doi: 10.1007/s002110050241. Google Scholar

[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. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar

[16]

S. Hafstein, A constructive converse Lyapunov Theoremon Exponential Stability,, Discrete Contin. Dyn. Syst., 10 (2004), 657. doi: 10.3934/dcds.2004.10.657. Google Scholar

[17]

G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields,, Chaos, 10 (2000), 99. doi: 10.1063/1.166479. Google Scholar

[18]

G. Haller, A variational theory of hyperbolic Lagrangian coherent structures,, Physica D, 240 (2011), 574. doi: 10.1016/j.physd.2010.11.010. Google Scholar

[19]

G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent,, Chaos, 21 (2011). Google Scholar

[20]

G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulance,, Physica D, 147 (2000), 352. doi: 10.1016/S0167-2789(00)00142-1. Google Scholar

[21]

C. Hsu, Global analysis by cell mapping,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 727. Google Scholar

[22]

B. Krauskopf and H. Osinga, Computing invariant manifolds via the continuation of orbit segments,, in, (2007), 117. Google Scholar

[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. doi: 10.1142/S0218127405012533. Google Scholar

[24]

A. M. Lyapunov, Problème général de la stabilité du mouvement,, Ann. Fac. Sci. Toulouse, 9 (1907), 203. Google Scholar

[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. doi: 10.1090/S0025-5718-04-01708-9. Google Scholar

[26]

G. Osipenko, "Dynamical Systems, Graphs, and Algorithms,'', Lecture Notes in Mathematics, 1889 (1889). Google Scholar

[27]

M. Rasmussen, Finite-time attractivity and bifurcation for nonautonomous differential equation,, Differential Equations Dynam. Systems, 18 (2010), 57. doi: 10.1007/s12591-010-0009-7. Google Scholar

[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. doi: 10.1016/j.physd.2005.10.007. Google Scholar

[29]

H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree,, J. Approx. Theory, 93 (1998), 258. doi: 10.1006/jath.1997.3137. Google Scholar

[30]

H. Wendland, "Scattered Data Approximation,'', Cambridge Monographs on Applied and Computational Mathematics, 17 (2005). Google Scholar

[31]

Z. Wu, Hermite-Birkhoff interpolation of scattered data by radial basis functions,, Approx. Theory Appl., 8 (1992), 1. Google Scholar

show all references

References:
[1]

A. Berger, On finite-time hyperbolicity,, Comm. Pure Applied Anal., 10 (2011), 963. doi: 10.3934/cpaa.2011.10.963. Google Scholar

[2]

A. Berger, D. T. Son and S. Siegmund, Nonautonomous finite-time dynamics,, Discrete Cont. Dyn. Syst. Ser. B, 9 (2008), 463. Google Scholar

[3]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behavior,, SIAM J. Numer. Anal., 36 (1999), 491. doi: 10.1137/S0036142996313002. Google Scholar

[4]

M. Dellnitz and O. Junge, Set oriented numerical methods for dynamical systems,, in, (2002), 221. Google Scholar

[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., (). Google Scholar

[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. doi: 10.1016/j.jde.2011.01.013. Google Scholar

[7]

C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions,, Appl.\ Math.\ Comput., 93 (1998), 73. doi: 10.1016/S0096-3003(97)10104-7. Google Scholar

[8]

P. Giesl, "Construction of Global Lyapunov Functionsusing Radial Basis Functions,'', Lecture Notes in Mathematics, 1904 (2007). Google Scholar

[9]

P. Giesl and S. Hafstein, Local Lyapunov Functions for periodic and finite-time ODEs,, submitted., (). Google Scholar

[10]

P. Giesl and M. Rasmussen, Areas of attraction for nonautonomous differential equations on finite time intervals,, J. Math. Anal. Appl., 390 (2012), 27. doi: 10.1016/j.jmaa.2011.12.051. Google Scholar

[11]

P. Giesl and H. Wendland, Meshless Collocation: Error Estimates with Application to Dynamical Systems,, SIAM J. Numer. Anal., 45 (2007), 1723. doi: 10.1137/060658813. Google Scholar

[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. doi: 10.3934/dcds.2009.25.1249. Google Scholar

[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 (): 259. Google Scholar

[14]

L. Grüne, An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation,, Numer. Math., 75 (1997), 319. doi: 10.1007/s002110050241. Google Scholar

[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. doi: 10.1090/S0002-9939-08-09341-6. Google Scholar

[16]

S. Hafstein, A constructive converse Lyapunov Theoremon Exponential Stability,, Discrete Contin. Dyn. Syst., 10 (2004), 657. doi: 10.3934/dcds.2004.10.657. Google Scholar

[17]

G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields,, Chaos, 10 (2000), 99. doi: 10.1063/1.166479. Google Scholar

[18]

G. Haller, A variational theory of hyperbolic Lagrangian coherent structures,, Physica D, 240 (2011), 574. doi: 10.1016/j.physd.2010.11.010. Google Scholar

[19]

G. Haller and T. Sapsis, Lagrangian coherent structures and the smallest finite-time Lyapunov exponent,, Chaos, 21 (2011). Google Scholar

[20]

G. Haller and G. Yuan, Lagrangian coherent structures and mixing in two-dimensional turbulance,, Physica D, 147 (2000), 352. doi: 10.1016/S0167-2789(00)00142-1. Google Scholar

[21]

C. Hsu, Global analysis by cell mapping,, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2 (1992), 727. Google Scholar

[22]

B. Krauskopf and H. Osinga, Computing invariant manifolds via the continuation of orbit segments,, in, (2007), 117. Google Scholar

[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. doi: 10.1142/S0218127405012533. Google Scholar

[24]

A. M. Lyapunov, Problème général de la stabilité du mouvement,, Ann. Fac. Sci. Toulouse, 9 (1907), 203. Google Scholar

[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. doi: 10.1090/S0025-5718-04-01708-9. Google Scholar

[26]

G. Osipenko, "Dynamical Systems, Graphs, and Algorithms,'', Lecture Notes in Mathematics, 1889 (1889). Google Scholar

[27]

M. Rasmussen, Finite-time attractivity and bifurcation for nonautonomous differential equation,, Differential Equations Dynam. Systems, 18 (2010), 57. doi: 10.1007/s12591-010-0009-7. Google Scholar

[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. doi: 10.1016/j.physd.2005.10.007. Google Scholar

[29]

H. Wendland, Error estimates for interpolation by compactly supported radial basis functions of minimal degree,, J. Approx. Theory, 93 (1998), 258. doi: 10.1006/jath.1997.3137. Google Scholar

[30]

H. Wendland, "Scattered Data Approximation,'', Cambridge Monographs on Applied and Computational Mathematics, 17 (2005). Google Scholar

[31]

Z. Wu, Hermite-Birkhoff interpolation of scattered data by radial basis functions,, Approx. Theory Appl., 8 (1992), 1. Google Scholar

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