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 and Continuous Dynamical Systems, 2011, 29 (3) : 1245-1260. doi: 10.3934/dcds.2011.29.1245
References:
[1]

Z. Cheng and C. M. Huang, Dissipativity for nonlinear neutral delay differential equations, J. Syst. Simul., 19 (2007), 3184-3187.

[2]

S. Q. Gan, Exact and discretized dissipativity of the pantograph equation, J. Comput. Math., 25 (2007), 81-88.

[3]

S. Q. Gan, Dissipativity of $\theta-$methods for nonlinear delay differential equations of neutral type, Appl. Numer. Math., 59 (2009), 1354-1365. doi: 10.1016/j.apnum.2008.08.003.

[4]

A. T. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.

[5]

A. T. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.

[6]

C. M. Huang, Dissipativity of Runge-Kutta methods for dynamical systems with delays, IMA J. Numer. Anal., 20 (2000), 153-166. doi: 10.1093/imanum/20.1.153.

[7]

C. M. Huang, Dissipativity of one-leg methods for dynamical systems with delays, Appl. Numer. Math., 35 (2000), 11-22. doi: 10.1016/S0168-9274(99)00048-3.

[8]

C. M. Huang, Dissipativity of multistep Runge-Kutta methods for dynamical systems with delays, Math. Comp. Model., 40 (2004), 1285-1296. doi: 10.1016/j.mcm.2005.01.019.

[9]

A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal., 31 (1994), 1452-1485. doi: 10.1137/0731075.

[10]

A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. doi: 10.1017/S0956792500000966.

[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), 81-91. doi: 10.1017/S0956792597003343.

[12]

M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.

[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), 447-468. doi: 10.1098/rspa.1971.0078.

[14]

A. M. Stuart and A. R. Humphries, Model problems in numerical stability theory for initial value problems, SIAM Review, 36 (1994), 226-257. doi: 10.1137/1036054.

[15]

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

[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), 1839-1845. doi: 10.1142/S0218127404010096.

[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), 934-941. doi: 10.1016/j.amc.2006.10.048.

[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), 333-342. doi: 10.1016/j.amc.2008.04.045.

[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), 285-301. doi: 10.1016/j.amc.2007.03.064.

[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), 983-991. doi: 10.1016/j.aml.2007.10.014.

[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), 2181-2205. doi: 10.1137/060654116.

[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), 177-183. doi: 10.1016/j.amc.2009.03.010.

[23]

L. P. Wen and S. F. Li, Dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 324 (2006), 696-706. doi: 10.1016/j.jmaa.2005.12.031.

[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), 780-786. doi: 10.1016/j.amc.2008.03.022.

[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), 169-178. doi: 10.1016/j.jmaa.2008.05.007.

[26]

A. Xiao, Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces, Math. Numer. Sin. 22 (2000), 429-436.

show all references

References:
[1]

Z. Cheng and C. M. Huang, Dissipativity for nonlinear neutral delay differential equations, J. Syst. Simul., 19 (2007), 3184-3187.

[2]

S. Q. Gan, Exact and discretized dissipativity of the pantograph equation, J. Comput. Math., 25 (2007), 81-88.

[3]

S. Q. Gan, Dissipativity of $\theta-$methods for nonlinear delay differential equations of neutral type, Appl. Numer. Math., 59 (2009), 1354-1365. doi: 10.1016/j.apnum.2008.08.003.

[4]

A. T. Hill, Global dissipativity for A-stable methods, SIAM J. Numer. Anal., 34 (1997), 119-142. doi: 10.1137/S0036142994270971.

[5]

A. T. Hill, Dissipativity of Runge-Kutta methods in Hilbert spaces, BIT, 37 (1997), 37-42. doi: 10.1007/BF02510171.

[6]

C. M. Huang, Dissipativity of Runge-Kutta methods for dynamical systems with delays, IMA J. Numer. Anal., 20 (2000), 153-166. doi: 10.1093/imanum/20.1.153.

[7]

C. M. Huang, Dissipativity of one-leg methods for dynamical systems with delays, Appl. Numer. Math., 35 (2000), 11-22. doi: 10.1016/S0168-9274(99)00048-3.

[8]

C. M. Huang, Dissipativity of multistep Runge-Kutta methods for dynamical systems with delays, Math. Comp. Model., 40 (2004), 1285-1296. doi: 10.1016/j.mcm.2005.01.019.

[9]

A. R. Humphries and A. M. Stuart, Runge-Kutta methods for dissipative and gradient dynamical systems, SIAM J. Numer. Anal., 31 (1994), 1452-1485. doi: 10.1137/0731075.

[10]

A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math., 4 (1993), 1-38. doi: 10.1017/S0956792500000966.

[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), 81-91. doi: 10.1017/S0956792597003343.

[12]

M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.

[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), 447-468. doi: 10.1098/rspa.1971.0078.

[14]

A. M. Stuart and A. R. Humphries, Model problems in numerical stability theory for initial value problems, SIAM Review, 36 (1994), 226-257. doi: 10.1137/1036054.

[15]

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

[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), 1839-1845. doi: 10.1142/S0218127404010096.

[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), 934-941. doi: 10.1016/j.amc.2006.10.048.

[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), 333-342. doi: 10.1016/j.amc.2008.04.045.

[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), 285-301. doi: 10.1016/j.amc.2007.03.064.

[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), 983-991. doi: 10.1016/j.aml.2007.10.014.

[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), 2181-2205. doi: 10.1137/060654116.

[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), 177-183. doi: 10.1016/j.amc.2009.03.010.

[23]

L. P. Wen and S. F. Li, Dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 324 (2006), 696-706. doi: 10.1016/j.jmaa.2005.12.031.

[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), 780-786. doi: 10.1016/j.amc.2008.03.022.

[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), 169-178. doi: 10.1016/j.jmaa.2008.05.007.

[26]

A. Xiao, Dissipativity of general linear methods for dissipative dynamical systems in Hilbert spaces, Math. Numer. Sin. 22 (2000), 429-436.

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