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June  2018, 23(4): 1835-1850. doi: 10.3934/dcdsb.2018094

Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation

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

* Corresponding author

Received  December 2016 Revised  November 2017 Published  March 2018

We consider a non-autonomous ordinary differential equation over a finite time interval $[T_1,T_2]$. The area of exponential attraction consists of solutions such that the distance to adjacent solutions exponentially contracts from $T_1$ to $T_2$. One can use a contraction metric to determine an area of exponential attraction and to provide a bound on the rate of attraction.

In this paper, we will give the first method to algorithmically construct a contraction metric for finite-time systems in one spatial dimension. We will show the existence of a contraction metric, given by a function which satisfies a second-order partial differential equation with boundary conditions. We then use meshless collocation to approximately solve this equation, and show that the resulting approximation itself defines a contraction metric, if the collocation points are sufficiently dense. We give error estimates and apply the method to an example.

Citation: Peter Giesl, James McMichen. Determination of the area of exponential attraction in one-dimensional finite-time systems using meshless collocation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1835-1850. doi: 10.3934/dcdsb.2018094
References:
[1]

A. Berger, On finite-time hyperbolicity, Commun. Pure Appl. Anal., 10 (2011), 963-981. Google Scholar

[2]

A. BergerT. Doan and S. Siegmund, A definition of spectrum for differential equations on finite time, J. Differential Equations, 246 (2009), 1098-1118. doi: 10.1016/j.jde.2008.06.036. Google Scholar

[3]

M. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, 2003. Google Scholar

[4]

C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, New York, 1999. Google Scholar

[5]

T. DoanD. KarraschT. Nguyen and S. Siegmund, A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents, J. Differential Equations, 252 (2012), 5535-5554. doi: 10.1016/j.jde.2012.02.002. Google Scholar

[6]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Mathematics, 1904 Springer, 2007. Google Scholar

[7]

P. Giesl, Construction of a finite-time Lyapunov function by meshless collocation, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2387-2412. doi: 10.3934/dcdsb.2012.17.2387. Google Scholar

[8]

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. Google Scholar

[9]

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. Google Scholar

[10]

P. Giesl and H. Wendland, Kernel-based discretisation for solving matrix-valued PDEs, preprint, arXiv: 1706.09360.Google Scholar

[11]

G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10 (2000), 99-108. doi: 10.1063/1.166479. Google Scholar

[12]

Ph. Hartman, Ordinary Differential Equations, Baltimore, Md., 1973. Google Scholar

[13]

D. Karrasch and G. Haller, Do finite-size Lyapunov exponents detect coherent structures?, Chaos, 23 (2013), 043126, 11 pp. Google Scholar

[14]

J. McMichen, Determination of Areas and Basins of Attraction in Planar Dynamical Systems using Meshless Collocation, Ph. D thesis, University of Sussex, 2016.Google Scholar

[15]

Th. Peacock and J. Dabiri, Introduction to focus issue: Lagrangian coherent structures, Chaos, 20 (2010), 17501-17505. Google Scholar

[16]

M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics 1907 Springer, 2007. Google Scholar

[17]

M. Rasmussen, Finite-time attractivity and bifurcation for nonautonomous differential equations, Differ. Equ. Dyn. Syst., 18 (2010), 57-78. doi: 10.1007/s12591-010-0009-7. Google Scholar

[18]

S. ShaddenF. Lekien and J. 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. Google Scholar

[19]

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. Google Scholar

[20]

H. Wendland, Scattered Data Approximation, Cambridge University Press, 2005. Google Scholar

[21]

J. Wloka, Partial Differential Equations, Cambridge University Press, 1987. Google Scholar

show all references

References:
[1]

A. Berger, On finite-time hyperbolicity, Commun. Pure Appl. Anal., 10 (2011), 963-981. Google Scholar

[2]

A. BergerT. Doan and S. Siegmund, A definition of spectrum for differential equations on finite time, J. Differential Equations, 246 (2009), 1098-1118. doi: 10.1016/j.jde.2008.06.036. Google Scholar

[3]

M. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, 2003. Google Scholar

[4]

C. Chicone, Ordinary Differential Equations with Applications, Springer-Verlag, New York, 1999. Google Scholar

[5]

T. DoanD. KarraschT. Nguyen and S. Siegmund, A unified approach to finite-time hyperbolicity which extends finite-time Lyapunov exponents, J. Differential Equations, 252 (2012), 5535-5554. doi: 10.1016/j.jde.2012.02.002. Google Scholar

[6]

P. Giesl, Construction of Global Lyapunov Functions Using Radial Basis Functions, Lecture Notes in Mathematics, 1904 Springer, 2007. Google Scholar

[7]

P. Giesl, Construction of a finite-time Lyapunov function by meshless collocation, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2387-2412. doi: 10.3934/dcdsb.2012.17.2387. Google Scholar

[8]

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. Google Scholar

[9]

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. Google Scholar

[10]

P. Giesl and H. Wendland, Kernel-based discretisation for solving matrix-valued PDEs, preprint, arXiv: 1706.09360.Google Scholar

[11]

G. Haller, Finding finite-time invariant manifolds in two-dimensional velocity fields, Chaos, 10 (2000), 99-108. doi: 10.1063/1.166479. Google Scholar

[12]

Ph. Hartman, Ordinary Differential Equations, Baltimore, Md., 1973. Google Scholar

[13]

D. Karrasch and G. Haller, Do finite-size Lyapunov exponents detect coherent structures?, Chaos, 23 (2013), 043126, 11 pp. Google Scholar

[14]

J. McMichen, Determination of Areas and Basins of Attraction in Planar Dynamical Systems using Meshless Collocation, Ph. D thesis, University of Sussex, 2016.Google Scholar

[15]

Th. Peacock and J. Dabiri, Introduction to focus issue: Lagrangian coherent structures, Chaos, 20 (2010), 17501-17505. Google Scholar

[16]

M. Rasmussen, Attractivity and Bifurcation for Nonautonomous Dynamical Systems, Lecture Notes in Mathematics 1907 Springer, 2007. Google Scholar

[17]

M. Rasmussen, Finite-time attractivity and bifurcation for nonautonomous differential equations, Differ. Equ. Dyn. Syst., 18 (2010), 57-78. doi: 10.1007/s12591-010-0009-7. Google Scholar

[18]

S. ShaddenF. Lekien and J. 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. Google Scholar

[19]

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. Google Scholar

[20]

H. Wendland, Scattered Data Approximation, Cambridge University Press, 2005. Google Scholar

[21]

J. Wloka, Partial Differential Equations, Cambridge University Press, 1987. Google Scholar

Figure 1.  The collocation points $X_1$ and $X_2$ as well as some numerically computed solutions of the system (21) with $T_2 = 2$.
Figure 2.  The function $L_m(t,x)$, using the approximation $w$ with $T_2 = 2$.
Figure 3.  Some level sets of $L_m(t,x)$. The 0-level set of $L_m$ crosses the $x$-axis at $\pm0.1789$ and is an approximation of the area of exponential attraction.
Figure 4.  Zero level sets of $L_m(t,x)$ for different values of $T_2$, namely $0.4$ (black), $0.8$ (blue), $1.2$ (red), $1.6$ (green), $2$ (magenta) and $2.4$ (cyan). The 0-level set of $L_m$ is an approximation of the area of exponential attraction. The size of the area of exponential attraction in $x$-direction shrinks until $T_2 = 1.5$ and then grows again.
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