July  2011, 29(3): 1245-1260. doi: 10.3934/dcds.2011.29.1245

Analytical and numerical dissipativity for nonlinear generalized pantograph equations

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China, China

Received  January 2010 Revised  June 2010 Published  November 2010

This paper is concerned with the analytic and numerical dissipativity of nonlinear neutral differential equations with proportional delay, the so-called generalized pantograph equations. A sufficient condition for the dissipativity of the systems is given. It is shown that the backward Euler method inherits the dissipativity of the underlying system. Numerical examples are given to confirm the theoretical results.
Citation: Wansheng Wang, Chengjian Zhang. Analytical and numerical dissipativity for nonlinear generalized pantograph equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1245-1260. doi: 10.3934/dcds.2011.29.1245
References:
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Z. Cheng and C. M. Huang, Dissipativity for nonlinear neutral delay differential equations,, J. Syst. Simul., 19 (2007), 3184. Google Scholar

[2]

S. Q. Gan, Exact and discretized dissipativity of the pantograph equation,, J. Comput. Math., 25 (2007), 81. Google Scholar

[3]

S. Q. Gan, Dissipativity of $\theta-$methods for nonlinear delay differential equations of neutral type,, Appl. Numer. Math., 59 (2009), 1354. doi: 10.1016/j.apnum.2008.08.003. Google Scholar

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A. T. Hill, Global dissipativity for A-stable methods,, SIAM J. Numer. Anal., 34 (1997), 119. doi: 10.1137/S0036142994270971. Google Scholar

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A. T. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces,, BIT, 37 (1997), 37. doi: 10.1007/BF02510171. Google Scholar

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C. M. Huang, Dissipativity of Runge-Kutta methods for dynamical systems with delays,, IMA J. Numer. Anal., 20 (2000), 153. doi: 10.1093/imanum/20.1.153. Google Scholar

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C. M. Huang, Dissipativity of one-leg methods for dynamical systems with delays,, Appl. Numer. Math., 35 (2000), 11. doi: 10.1016/S0168-9274(99)00048-3. Google Scholar

[8]

C. M. Huang, Dissipativity of multistep Runge-Kutta methods for dynamical systems with delays,, Math. Comp. Model., 40 (2004), 1285. doi: 10.1016/j.mcm.2005.01.019. Google Scholar

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A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems,, SIAM J. Numer. Anal., 31 (1994), 1452. doi: 10.1137/0731075. Google Scholar

[10]

A. Iserles, On the generalized pantograph functional-differential equation,, European J. Appl. Math., 4 (1993), 1. doi: 10.1017/S0956792500000966. Google Scholar

[11]

H. Lehninger and Y. Liu, The functional-differential equation $y^'(t)=Ay(t)+By(\lambda t)+Cy^'(qt)+f(t)$,, European J. Appl. Math., 9 (1998), 81. doi: 10.1017/S0956792597003343. Google Scholar

[12]

M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems,, Science, 197 (1977), 287. doi: 10.1126/science.267326. Google Scholar

[13]

J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive,, Proc. Royal Soc. A, 322 (1971), 447. doi: 10.1098/rspa.1971.0078. Google Scholar

[14]

A. M. Stuart and A. R. Humphries, Model problems in numerical stability theory for initial value problems,, SIAM Review, 36 (1994), 226. doi: 10.1137/1036054. Google Scholar

[15]

A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,", Cambridge University Press, (1996). Google Scholar

[16]

H. J. Tian, Numerical and analytic dissipativity of the $\Theta-$method for delay differential equations with a bounde varible lag,, International J. Bifur. Chaos, 14 (2004), 1839. doi: 10.1142/S0218127404010096. Google Scholar

[17]

H. J. Tian, L. Q. Fan and J. X. Xiang, Numerical dissipativity of multistep methods for delay differential equations,, Appl. Math. Comput., 188 (2007), 934. doi: 10.1016/j.amc.2006.10.048. Google Scholar

[18]

H. J. Tian and N. Guo, Asymptotic stability, contractivity and dissipativity of one-leg $\theta$-method for non-autonomous delay functional differential equations,, Appl. Math. Comput., 203 (2008), 333. doi: 10.1016/j.amc.2008.04.045. Google Scholar

[19]

W. S. Wang and S. F. Li, On the one-leg $\theta$-methods for solving nonlinear neutral functional differential equations,, Appl. Math. Comput., 193 (2007), 285. doi: 10.1016/j.amc.2007.03.064. Google Scholar

[20]

W. S. Wang and S. F. Li, Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay,, Appl. Math. Lett., 21 (2008), 983. doi: 10.1016/j.aml.2007.10.014. Google Scholar

[21]

W. S. Wang and S. F. Li, Stability analysis of $\theta$-methods for nonlinear neutral functional differential equations,, SIAM J. Sci. Comput., 30 (2008), 2181. doi: 10.1137/060654116. Google Scholar

[22]

W. S. Wang, T. T. Qin and S. F. Li, Stability of one-leg $\theta$-methods for nonlinear neutral differential equations with proportional delay,, Appl. Math. Comput., 213 (2009), 177. doi: 10.1016/j.amc.2009.03.010. Google Scholar

[23]

L. P. Wen and S. F. Li, Dissipativity of Volterra functional differential equations,, J. Math. Anal. Appl., 324 (2006), 696. doi: 10.1016/j.jmaa.2005.12.031. Google Scholar

[24]

L. P. Wen, W. S. Wang and Y. X. Yu, Dissipativity of $\theta$-methods for a class of nonlinear neutral differential equations,, Appl. Math. Comput., 202 (2008), 780. doi: 10.1016/j.amc.2008.03.022. Google Scholar

[25]

L. P. Wen, Y. X. Yu and W. S. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations,, J. Math. Anal. Appl., 347 (2008), 169. doi: 10.1016/j.jmaa.2008.05.007. Google Scholar

[26]

A. Xiao, Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces,, Math. Numer. Sin. \textbf{22} (2000), 22 (2000), 429. Google Scholar

show all references

References:
[1]

Z. Cheng and C. M. Huang, Dissipativity for nonlinear neutral delay differential equations,, J. Syst. Simul., 19 (2007), 3184. Google Scholar

[2]

S. Q. Gan, Exact and discretized dissipativity of the pantograph equation,, J. Comput. Math., 25 (2007), 81. Google Scholar

[3]

S. Q. Gan, Dissipativity of $\theta-$methods for nonlinear delay differential equations of neutral type,, Appl. Numer. Math., 59 (2009), 1354. doi: 10.1016/j.apnum.2008.08.003. Google Scholar

[4]

A. T. Hill, Global dissipativity for A-stable methods,, SIAM J. Numer. Anal., 34 (1997), 119. doi: 10.1137/S0036142994270971. Google Scholar

[5]

A. T. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces,, BIT, 37 (1997), 37. doi: 10.1007/BF02510171. Google Scholar

[6]

C. M. Huang, Dissipativity of Runge-Kutta methods for dynamical systems with delays,, IMA J. Numer. Anal., 20 (2000), 153. doi: 10.1093/imanum/20.1.153. Google Scholar

[7]

C. M. Huang, Dissipativity of one-leg methods for dynamical systems with delays,, Appl. Numer. Math., 35 (2000), 11. doi: 10.1016/S0168-9274(99)00048-3. Google Scholar

[8]

C. M. Huang, Dissipativity of multistep Runge-Kutta methods for dynamical systems with delays,, Math. Comp. Model., 40 (2004), 1285. doi: 10.1016/j.mcm.2005.01.019. Google Scholar

[9]

A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems,, SIAM J. Numer. Anal., 31 (1994), 1452. doi: 10.1137/0731075. Google Scholar

[10]

A. Iserles, On the generalized pantograph functional-differential equation,, European J. Appl. Math., 4 (1993), 1. doi: 10.1017/S0956792500000966. Google Scholar

[11]

H. Lehninger and Y. Liu, The functional-differential equation $y^'(t)=Ay(t)+By(\lambda t)+Cy^'(qt)+f(t)$,, European J. Appl. Math., 9 (1998), 81. doi: 10.1017/S0956792597003343. Google Scholar

[12]

M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems,, Science, 197 (1977), 287. doi: 10.1126/science.267326. Google Scholar

[13]

J. R. Ockendon and A. B. Tayler, The dynamics of a current collection system for an electric locomotive,, Proc. Royal Soc. A, 322 (1971), 447. doi: 10.1098/rspa.1971.0078. Google Scholar

[14]

A. M. Stuart and A. R. Humphries, Model problems in numerical stability theory for initial value problems,, SIAM Review, 36 (1994), 226. doi: 10.1137/1036054. Google Scholar

[15]

A. M. Stuart and A. R. Humphries, "Dynamical Systems and Numerical Analysis,", Cambridge University Press, (1996). Google Scholar

[16]

H. J. Tian, Numerical and analytic dissipativity of the $\Theta-$method for delay differential equations with a bounde varible lag,, International J. Bifur. Chaos, 14 (2004), 1839. doi: 10.1142/S0218127404010096. Google Scholar

[17]

H. J. Tian, L. Q. Fan and J. X. Xiang, Numerical dissipativity of multistep methods for delay differential equations,, Appl. Math. Comput., 188 (2007), 934. doi: 10.1016/j.amc.2006.10.048. Google Scholar

[18]

H. J. Tian and N. Guo, Asymptotic stability, contractivity and dissipativity of one-leg $\theta$-method for non-autonomous delay functional differential equations,, Appl. Math. Comput., 203 (2008), 333. doi: 10.1016/j.amc.2008.04.045. Google Scholar

[19]

W. S. Wang and S. F. Li, On the one-leg $\theta$-methods for solving nonlinear neutral functional differential equations,, Appl. Math. Comput., 193 (2007), 285. doi: 10.1016/j.amc.2007.03.064. Google Scholar

[20]

W. S. Wang and S. F. Li, Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay,, Appl. Math. Lett., 21 (2008), 983. doi: 10.1016/j.aml.2007.10.014. Google Scholar

[21]

W. S. Wang and S. F. Li, Stability analysis of $\theta$-methods for nonlinear neutral functional differential equations,, SIAM J. Sci. Comput., 30 (2008), 2181. doi: 10.1137/060654116. Google Scholar

[22]

W. S. Wang, T. T. Qin and S. F. Li, Stability of one-leg $\theta$-methods for nonlinear neutral differential equations with proportional delay,, Appl. Math. Comput., 213 (2009), 177. doi: 10.1016/j.amc.2009.03.010. Google Scholar

[23]

L. P. Wen and S. F. Li, Dissipativity of Volterra functional differential equations,, J. Math. Anal. Appl., 324 (2006), 696. doi: 10.1016/j.jmaa.2005.12.031. Google Scholar

[24]

L. P. Wen, W. S. Wang and Y. X. Yu, Dissipativity of $\theta$-methods for a class of nonlinear neutral differential equations,, Appl. Math. Comput., 202 (2008), 780. doi: 10.1016/j.amc.2008.03.022. Google Scholar

[25]

L. P. Wen, Y. X. Yu and W. S. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations,, J. Math. Anal. Appl., 347 (2008), 169. doi: 10.1016/j.jmaa.2008.05.007. Google Scholar

[26]

A. Xiao, Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces,, Math. Numer. Sin. \textbf{22} (2000), 22 (2000), 429. Google Scholar

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