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Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets
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 |
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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