Dirichlet series for dynamical systems of first-order ordinary differential equations

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January 2014, 19(1): 281-298. doi: 10.3934/dcdsb.2014.19.281

Dirichlet series for dynamical systems of first-order ordinary differential equations

1.	School of Mathematics & Physics, Qingdao University of Science & Technology, Qingdao 266061, China
2.	Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Rd, Cambridge CB3 0WA, United Kingdom

Received August 2012 Revised October 2013 Published December 2013

Abstract
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In this paper, inspired by the work by A. Iserles and G. Söderlind [Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), pp. 97-112], we present comprehensive discussion on Dirichlet series for dynamical systems of first-order ordinary differential equations (ODEs). We first derive the scheme of Dirichlet approximation for scalar dynamical systems and present the bounds on the terms of Dirichlet series. The global error and the right choice of a term in Dirichlet series are analysed and two numerical experiments are carried out to demonstrate the efficiency of Dirichlet approximation. Then we consider applying Dirichlet series to multivariate dynamical systems and present a new scheme of Dirichlet approximation for such systems. Some discussion and a numerical experiment are accordingly carried out for the new Dirichlet approximation. Compared with routine time-stepping algorithms, Dirichlet series does not need time stepping and yields a continuous solution that is equally valid along an interval, which is significant for obtaining long-time numerical solution. As a result of the special nature of Dirichlet series, the Dirichlet approximation delivers considerable information on dynamical systems of first-order ODEs and provides a novel and effective approach to numerical solutions of these dynamical systems.

Keywords: dynamical systems, stationary points, global bounds., Dirichlet series, first-order ordinary differential equations.

Mathematics Subject Classification: Primary: 37M10, 37N30; Secondary: 65L05, 65L7.

Citation: Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

References:

[1]	J. C. Butcher, Numerical Methods for Ordinary Differential Equations,, 2nd ed., (2008). doi: 10.1002/9780470753767.
[2]	Y. Fang, Y. Song and X. Wu, New embedded pairs of explicit Runge-Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems,, Phys. Lett. A, 372 (2008), 6551. doi: 10.1016/j.physleta.2008.09.014.
[3]	A. B. González, P. Martín and J. M. Farto, A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators,, Numer. Math., 82 (1999), 635. doi: 10.1007/s002110050434.
[4]	E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method,, Acta Numer., 12 (2003), 399. doi: 10.1017/S0962492902000144.
[5]	E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems,, Springer-Verlag, (1993).
[6]	G. H. Hardy and M. Riesz, The general theory of Dirichlet series,, Cambridge Tracts in Mathematics and Mathematical Physics, 18 (1964).
[7]	M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilineal parabolic problems,, SIAM J. Numer. Anal., 43 (2005), 1069. doi: 10.1137/040611434.
[8]	A. Iserles, A First Course in the Numerical Analysis of Differential Equations,, 2nd ed., (2008).
[9]	A. Iserles, G. P. Ramaswami and M. Sofroniou, Runge-Kutta methods for quadratic ordinary differential equations,, BIT, 38 (1998), 315. doi: 10.1007/BF02512370.
[10]	A. Iserles and G. Söderlind, Global bounds on numerical error for ordinary differential equations,, J. Complexity, 9 (1993), 97. doi: 10.1006/jcom.1993.1007.
[11]	A. Iserles and A. Zanna, Preserving algebraic invariants with Runge-Kutta methods,, J. Comp. Appl. Maths., 125 (2000), 69. doi: 10.1016/S0377-0427(00)00459-3.
[12]	S. Mandelbrojt, Dirichlet Series: Principles and Methods,, D. Reidel Publishing Company, (1972).
[13]	L. Perko, Differential Equations and Dynamical Systems,, 3rd ed., (2001).
[14]	R. C. Robinson, An Introduction to Dynamical Systems: Countinuous and Discrete,, Pearson Prentice Hall, (2004).
[15]	J. M. Sanz-Serna, Runge-Kutta schems for Hamiltonian systems,, BIT, 28 (1988), 877. doi: 10.1007/BF01954907.
[16]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Springer-Verlag, (1990). doi: 10.1007/978-3-642-97149-5.

show all references

References:

[1]	J. C. Butcher, Numerical Methods for Ordinary Differential Equations,, 2nd ed., (2008). doi: 10.1002/9780470753767.
[2]	Y. Fang, Y. Song and X. Wu, New embedded pairs of explicit Runge-Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems,, Phys. Lett. A, 372 (2008), 6551. doi: 10.1016/j.physleta.2008.09.014.
[3]	A. B. González, P. Martín and J. M. Farto, A new family of Runge-Kutta type methods for the numerical integration of perturbed oscillators,, Numer. Math., 82 (1999), 635. doi: 10.1007/s002110050434.
[4]	E. Hairer, C. Lubich and G. Wanner, Geometric numerical integration illustrated by the Störmer-Verlet method,, Acta Numer., 12 (2003), 399. doi: 10.1017/S0962492902000144.
[5]	E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems,, Springer-Verlag, (1993).
[6]	G. H. Hardy and M. Riesz, The general theory of Dirichlet series,, Cambridge Tracts in Mathematics and Mathematical Physics, 18 (1964).
[7]	M. Hochbruck and A. Ostermann, Explicit exponential Runge-Kutta methods for semilineal parabolic problems,, SIAM J. Numer. Anal., 43 (2005), 1069. doi: 10.1137/040611434.
[8]	A. Iserles, A First Course in the Numerical Analysis of Differential Equations,, 2nd ed., (2008).
[9]	A. Iserles, G. P. Ramaswami and M. Sofroniou, Runge-Kutta methods for quadratic ordinary differential equations,, BIT, 38 (1998), 315. doi: 10.1007/BF02512370.
[10]	A. Iserles and G. Söderlind, Global bounds on numerical error for ordinary differential equations,, J. Complexity, 9 (1993), 97. doi: 10.1006/jcom.1993.1007.
[11]	A. Iserles and A. Zanna, Preserving algebraic invariants with Runge-Kutta methods,, J. Comp. Appl. Maths., 125 (2000), 69. doi: 10.1016/S0377-0427(00)00459-3.
[12]	S. Mandelbrojt, Dirichlet Series: Principles and Methods,, D. Reidel Publishing Company, (1972).
[13]	L. Perko, Differential Equations and Dynamical Systems,, 3rd ed., (2001).
[14]	R. C. Robinson, An Introduction to Dynamical Systems: Countinuous and Discrete,, Pearson Prentice Hall, (2004).
[15]	J. M. Sanz-Serna, Runge-Kutta schems for Hamiltonian systems,, BIT, 28 (1988), 877. doi: 10.1007/BF01954907.
[16]	F. Verhulst, Nonlinear Differential Equations and Dynamical Systems,, Springer-Verlag, (1990). doi: 10.1007/978-3-642-97149-5.

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