2009, 2009(Special): 259-268. doi: 10.3934/proc.2009.2009.259

Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem

1. 

Department of Mathematics, University of Sussex, Brighton, BN1 9RF

Received  July 2008 Revised  April 2009 Published  September 2009

The basin of attraction of an equilibrium or periodic orbit can be determined by sublevel sets of a Lyapunov function. A Lyapunov function is a function with negative orbital derivative, which is defined by $LV(t,x)= {\nabla_x V(t,x),f(t,x)} + \partial_t V(t,x)$. We construct a Lyapunov function by approximately solving a Cauchy problem with a linear PDE for its orbital derivative and boundary conditions on a non-characteristic hypersurface. For the approximation we use meshless collocation. We describe the general approximate reconstruction of multivariate functions, which are periodic in one variable, from discrete data sets and derive error estimates. This method has already been applied to autonomous dynamical systems. In this paper, however, we consider a time-periodic ODE $\dot{x}=f(t,x)$, $x\in \mathbb R^n$, and study the basin of attraction of an exponentially asymptotically stable periodic orbit.
Citation: Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation of a Cauchy problem. Conference Publications, 2009, 2009 (Special) : 259-268. doi: 10.3934/proc.2009.2009.259
[1]

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

[2]

Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1249-1274. doi: 10.3934/dcds.2009.25.1249

[3]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

[4]

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

[5]

Hjörtur Björnsson, Sigurdur Hafstein, Peter Giesl, Enrico Scalas, Skuli Gudmundsson. Computation of the stochastic basin of attraction by rigorous construction of a Lyapunov function. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4247-4269. doi: 10.3934/dcdsb.2019080

[6]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

[7]

Martin D. Buhmann, Slawomir Dinew. Limits of radial basis function interpolants. Communications on Pure and Applied Analysis, 2007, 6 (3) : 569-585. doi: 10.3934/cpaa.2007.6.569

[8]

Peter Giesl. Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 355-373. doi: 10.3934/dcds.2007.18.355

[9]

Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111

[10]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385

[11]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220

[12]

Robert Baier, Lars Grüne, Sigurđur Freyr Hafstein. Linear programming based Lyapunov function computation for differential inclusions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 33-56. doi: 10.3934/dcdsb.2012.17.33

[13]

Imtiaz Ahmad, Siraj-ul-Islam, Mehnaz, Sakhi Zaman. Local meshless differential quadrature collocation method for time-fractional PDEs. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2641-2654. doi: 10.3934/dcdss.2020223

[14]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[15]

Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourth-order differential equation. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1225-1235. doi: 10.3934/cpaa.2008.7.1225

[16]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263

[17]

Sigurdur Hafstein, Skuli Gudmundsson, Peter Giesl, Enrico Scalas. Lyapunov function computation for autonomous linear stochastic differential equations using sum-of-squares programming. Discrete and Continuous Dynamical Systems - B, 2018, 23 (2) : 939-956. doi: 10.3934/dcdsb.2018049

[18]

Avner Friedman, Harsh Vardhan Jain. A partial differential equation model of metastasized prostatic cancer. Mathematical Biosciences & Engineering, 2013, 10 (3) : 591-608. doi: 10.3934/mbe.2013.10.591

[19]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

[20]

Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252

 Impact Factor: 

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]