
Previous Article
Asymptotic behaviour of random tridiagonal Markov chains in biological applications
 DCDSB Home
 This Issue

Next Article
Reduction and identification of dynamic models. Simple example: Generic receptor model
Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay
1.  Voronezh State University, 1 Universitetskaya pl., 394006, Voronezh, Russian Federation, Russian Federation 
References:
[1] 
R. R. Akhmerov, M. I. Kamenskii, V. S. Kozyakin and A. V. Sobolev, Periodic solutions to autonomous functionaldifferrential neutraltype equations with a small delay, Differential Equations, 10 (1974), 19231931 [in Russian]. 
[2] 
R. R.Akhmerov, M. I.Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, "Measures of NonCompactness and Condensing Operators," Nauka, Novosibirsk, 1986 [in Russian]. 
[3] 
P. G. Ayzengendler, Exception theory application to a problem of bifurcation of solutions to nonlinear equations, Scient. notes Mosc. reg. Krupskaya's ped. inst., 166 (1966), 253273 [in Russian]. 
[4] 
P. G . Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to autonomous systems and differential equations in Banach spaces, USSR Academy of science reports, 176 (1967), 912 [in Russian]. 
[5] 
P. G. Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, I, IHE proceedings, 10 (1969), 310 [in Russian]. 
[6] 
P. G. Ayzengendler and M.M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, II, IHE proceedings, 11 (1969), 312 [in Russian]. 
[7] 
P. G. Ayzengendler and M. M. Vainberg, On periodic solutions to nonautonomous systems, USSR Academy of science reports, 165 (1965), 255257 [in Russian]. 
[8] 
P. G. Ayzengendler and M. M. Vainberg, Theory of bifurcation of solutions to nonlinear equations in multidimensional case, USSR Academy of science reports, 163 (1965), 543546 [in Russian]. 
[9] 
A. Buică, J. P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Communication of Pure and Applied Analysis, 6 (2007), 103111. 
[10] 
C. N. Fang and Q. Y. Wang, Existence, uniqueness and stability of periodic solutions to a class of neutral functional differential equations, J. Fuzhou Univ. Nat. Sci. Ed., 37 (2009), 471477 [in Chinese]. 
[11] 
A. F. Filippov, "Differential Equations with Discontinuous RightHand Side," Nauka, Moscow, 1985 [in Russian]. 
[12] 
J. R. Graef and L. Kong, Periodic solutions for functional differential equations with signchanging nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 597616. doi: 10.1017/S0308210509000523. 
[13] 
J. R. Graef and S. H. Saker, New oscillation criteria for generalized secondorder nonlinear neutral functional differential equations, Dynam. Systems Appl., 19 (2010), 455472. 
[14] 
L. X. Guo, S. P. Lu, B. Du and F. Liang, Existence of periodic solutions to a secondorder neutral functional differential equation with deviating arguments, J. Math. (Wuhan), 30 (2010), 839847 [in Chinese]. 
[15] 
M. Kamenskii, O. Makarenkov and P. Nistri, Variables Scaling to Solve a Singular Bifurcation Problem with Application to Periodically perturbed Autonomous Systems, Journal of Dynamic and Differential Equations, 8 (2011), 135153. 
[16] 
M. Kamenskii, O. Makarenkov and P. Nistri, Periodic bifurcation for semilinear differential equations with Lipschitzian perturbations in Banach spaces, Advanced Nonlinear Studies, 8 (2008), 271289. 
[17] 
M. I. Kamenskii and B. A. Mikhaylenko, On a small perturbations of systems with multidimensional degeneracy, Aut. and Rem. Contr., 5 (2011), 148160 [in Russian]. 
[18] 
M. A. Krasnoselskii, "Translation Operator Along the Trajectories of Differential Equations," Nauka, Moscow, 1966 [in Russian]. 
[19] 
W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. Math., 70 (1959), 490529. 
[20] 
L. P. Luo, Oscillation theorems for nonlinear neutral hyperbolic partial functional differential equations, J. Math. (Wuhan), 30 (2010), 10231028 [in Chinese]. 
[21] 
O. Makarenkov and P. Nistri, Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations, J. Math. Anal. Appl., 338 (2008), 14011417. doi: 10.1016/j.jmaa.2007.05.086. 
[22] 
I. G. Malkin, "Some Problems of NonLinear Oscillations Theory," State publishers of technics and theory literature, Moscow, 1956 [in Russian]. 
[23] 
M. B. H. Rhouma and C. Chicone, On the continuation of periodic orbits, Methods Appl. Anal., 7 (2000), 85104. 
[24] 
A. E. Rodkina and B. N. Sadovskiy, On differentiability of translation operator along the trajectories of neutraltype equation, Math. fac. proc., 12 (1974), 3137 [in Russian]. 
[25] 
G. Sansone, "Equazioni Differenziali Nel Campo Reale," p.1., Seconda edizione, Bologna, 1948. 
[26] 
S. N. Shimanov, Oscillations of quasilinear autonomous systems with delay, IHE proceedings. Radiophisics, 3 (1960), 456466 [in Russian]. 
[27] 
S. N. Shimanov, To the oscillation theory of quasilinear systems with delay, AMM., V.XXII (1959), 836844 [in Russian]. 
[28] 
S. L. Wan, J. Yang, C. H. Feng and J. M. Huang, Existence of periodic solutions to higherorder nonlinear neutral functional differential equations with infinite delay, Pure Appl. Math. (Xi'an), 25 (2009), 556562, 594 [in Chinese]. 
[29] 
C. Wang, Y. Li and Y. Fei, Three positive periodic solutions to nonlinear neutral functional differential equations with impulses and parameters on time scales, Math. Comput. Modelling, 52 (2010), 14511462. 
[30] 
C. Wang and J. Wei, Hopf bifurcation for neutral functional differential equations, Nonlinear Anal. Real World Appl., 11 (2010), 12691277. 
[31] 
F. Wei and K. Wang, The periodic solution of functional differential equations with infinite delay, Nonlinear Anal. Real World Appl., 11 (2010), 26692674. doi: 10.1016/j.nonrwa.2009.09.014. 
[32] 
Y. Zhu, Periodic solutions for a higher order nonlinear neutral functional differential equation, Int. J. Comput. Math. Sci., 5 (2011), 812. 
show all references
References:
[1] 
R. R. Akhmerov, M. I. Kamenskii, V. S. Kozyakin and A. V. Sobolev, Periodic solutions to autonomous functionaldifferrential neutraltype equations with a small delay, Differential Equations, 10 (1974), 19231931 [in Russian]. 
[2] 
R. R.Akhmerov, M. I.Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, "Measures of NonCompactness and Condensing Operators," Nauka, Novosibirsk, 1986 [in Russian]. 
[3] 
P. G. Ayzengendler, Exception theory application to a problem of bifurcation of solutions to nonlinear equations, Scient. notes Mosc. reg. Krupskaya's ped. inst., 166 (1966), 253273 [in Russian]. 
[4] 
P. G . Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to autonomous systems and differential equations in Banach spaces, USSR Academy of science reports, 176 (1967), 912 [in Russian]. 
[5] 
P. G. Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, I, IHE proceedings, 10 (1969), 310 [in Russian]. 
[6] 
P. G. Ayzengendler and M.M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, II, IHE proceedings, 11 (1969), 312 [in Russian]. 
[7] 
P. G. Ayzengendler and M. M. Vainberg, On periodic solutions to nonautonomous systems, USSR Academy of science reports, 165 (1965), 255257 [in Russian]. 
[8] 
P. G. Ayzengendler and M. M. Vainberg, Theory of bifurcation of solutions to nonlinear equations in multidimensional case, USSR Academy of science reports, 163 (1965), 543546 [in Russian]. 
[9] 
A. Buică, J. P. Françoise and J. Llibre, Periodic solutions of nonlinear periodic differential systems with a small parameter, Communication of Pure and Applied Analysis, 6 (2007), 103111. 
[10] 
C. N. Fang and Q. Y. Wang, Existence, uniqueness and stability of periodic solutions to a class of neutral functional differential equations, J. Fuzhou Univ. Nat. Sci. Ed., 37 (2009), 471477 [in Chinese]. 
[11] 
A. F. Filippov, "Differential Equations with Discontinuous RightHand Side," Nauka, Moscow, 1985 [in Russian]. 
[12] 
J. R. Graef and L. Kong, Periodic solutions for functional differential equations with signchanging nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 597616. doi: 10.1017/S0308210509000523. 
[13] 
J. R. Graef and S. H. Saker, New oscillation criteria for generalized secondorder nonlinear neutral functional differential equations, Dynam. Systems Appl., 19 (2010), 455472. 
[14] 
L. X. Guo, S. P. Lu, B. Du and F. Liang, Existence of periodic solutions to a secondorder neutral functional differential equation with deviating arguments, J. Math. (Wuhan), 30 (2010), 839847 [in Chinese]. 
[15] 
M. Kamenskii, O. Makarenkov and P. Nistri, Variables Scaling to Solve a Singular Bifurcation Problem with Application to Periodically perturbed Autonomous Systems, Journal of Dynamic and Differential Equations, 8 (2011), 135153. 
[16] 
M. Kamenskii, O. Makarenkov and P. Nistri, Periodic bifurcation for semilinear differential equations with Lipschitzian perturbations in Banach spaces, Advanced Nonlinear Studies, 8 (2008), 271289. 
[17] 
M. I. Kamenskii and B. A. Mikhaylenko, On a small perturbations of systems with multidimensional degeneracy, Aut. and Rem. Contr., 5 (2011), 148160 [in Russian]. 
[18] 
M. A. Krasnoselskii, "Translation Operator Along the Trajectories of Differential Equations," Nauka, Moscow, 1966 [in Russian]. 
[19] 
W. S. Loud, Periodic solutions of a perturbed autonomous system, Ann. Math., 70 (1959), 490529. 
[20] 
L. P. Luo, Oscillation theorems for nonlinear neutral hyperbolic partial functional differential equations, J. Math. (Wuhan), 30 (2010), 10231028 [in Chinese]. 
[21] 
O. Makarenkov and P. Nistri, Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations, J. Math. Anal. Appl., 338 (2008), 14011417. doi: 10.1016/j.jmaa.2007.05.086. 
[22] 
I. G. Malkin, "Some Problems of NonLinear Oscillations Theory," State publishers of technics and theory literature, Moscow, 1956 [in Russian]. 
[23] 
M. B. H. Rhouma and C. Chicone, On the continuation of periodic orbits, Methods Appl. Anal., 7 (2000), 85104. 
[24] 
A. E. Rodkina and B. N. Sadovskiy, On differentiability of translation operator along the trajectories of neutraltype equation, Math. fac. proc., 12 (1974), 3137 [in Russian]. 
[25] 
G. Sansone, "Equazioni Differenziali Nel Campo Reale," p.1., Seconda edizione, Bologna, 1948. 
[26] 
S. N. Shimanov, Oscillations of quasilinear autonomous systems with delay, IHE proceedings. Radiophisics, 3 (1960), 456466 [in Russian]. 
[27] 
S. N. Shimanov, To the oscillation theory of quasilinear systems with delay, AMM., V.XXII (1959), 836844 [in Russian]. 
[28] 
S. L. Wan, J. Yang, C. H. Feng and J. M. Huang, Existence of periodic solutions to higherorder nonlinear neutral functional differential equations with infinite delay, Pure Appl. Math. (Xi'an), 25 (2009), 556562, 594 [in Chinese]. 
[29] 
C. Wang, Y. Li and Y. Fei, Three positive periodic solutions to nonlinear neutral functional differential equations with impulses and parameters on time scales, Math. Comput. Modelling, 52 (2010), 14511462. 
[30] 
C. Wang and J. Wei, Hopf bifurcation for neutral functional differential equations, Nonlinear Anal. Real World Appl., 11 (2010), 12691277. 
[31] 
F. Wei and K. Wang, The periodic solution of functional differential equations with infinite delay, Nonlinear Anal. Real World Appl., 11 (2010), 26692674. doi: 10.1016/j.nonrwa.2009.09.014. 
[32] 
Y. Zhu, Periodic solutions for a higher order nonlinear neutral functional differential equation, Int. J. Comput. Math. Sci., 5 (2011), 812. 
[1] 
Jinjing Jiao, Guanghua Shi. Quasiperiodic solutions for the twodimensional systems with an elliptictype degenerate equilibrium point under small perturbations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 51575180. doi: 10.3934/cpaa.2020231 
[2] 
Bernold Fiedler, Isabelle Schneider. Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay. Discrete and Continuous Dynamical Systems  S, 2020, 13 (4) : 11451185. doi: 10.3934/dcdss.2020068 
[3] 
Meina Gao. Smalldivisor equation with largevariable coefficient and periodic solutions of DNLS equations. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 173204. doi: 10.3934/dcds.2015.35.173 
[4] 
Adriana Buică, Jean–Pierre Françoise, Jaume Llibre. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure and Applied Analysis, 2007, 6 (1) : 103111. doi: 10.3934/cpaa.2007.6.103 
[5] 
Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753772. doi: 10.3934/eect.2020032 
[6] 
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) : 20312068. doi: 10.3934/cpaa.2013.12.2031 
[7] 
Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure and Applied Analysis, 2014, 13 (3) : 11051117. doi: 10.3934/cpaa.2014.13.1105 
[8] 
Xu Zhang, Yuming Shi, Guanrong Chen. Coupledexpanding maps under small perturbations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 12911307. doi: 10.3934/dcds.2011.29.1291 
[9] 
Yingte Sun. Floquet solutions for the Schrödinger equation with fastoscillating quasiperiodic potentials. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 45314543. doi: 10.3934/dcds.2021047 
[10] 
M.I. Gil’. Existence and stability of periodic solutions of semilinear neutral type systems. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 809820. doi: 10.3934/dcds.2001.7.809 
[11] 
LingJun Wang. The dynamics of small amplitude solutions of the SwiftHohenberg equation on a large interval. Communications on Pure and Applied Analysis, 2012, 11 (3) : 11291156. doi: 10.3934/cpaa.2012.11.1129 
[12] 
Roger Grimshaw, Dmitry Pelinovsky. Global existence of smallnorm solutions in the reduced Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 557566. doi: 10.3934/dcds.2014.34.557 
[13] 
Philippe Laurençot, Christoph Walker. The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions. Kinetic and Related Models, 2021, 14 (6) : 961980. doi: 10.3934/krm.2021032 
[14] 
Wei Wang, Kai Liu, Xiulian Wang. Sensitivity to small delays of mean square stability for stochastic neutral evolution equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 24032418. doi: 10.3934/cpaa.2020105 
[15] 
Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 23692387. doi: 10.3934/dcds.2013.33.2369 
[16] 
Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems and Imaging, 2018, 12 (4) : 971992. doi: 10.3934/ipi.2018041 
[17] 
Mickael Chekroun, Michael Ghil, Jean Roux, Ferenc Varadi. Averaging of time  periodic systems without a small parameter. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 753782. doi: 10.3934/dcds.2006.14.753 
[18] 
Hua Chen, LingJun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic and Related Models, 2012, 5 (2) : 261281. doi: 10.3934/krm.2012.5.261 
[19] 
Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 10571067. doi: 10.3934/dcds.2005.13.1057 
[20] 
Kazuyuki Yagasaki. Existence of finite time blowup solutions in a normal form of the subcritical Hopf bifurcation with timedelayed feedback for small initial functions. Discrete and Continuous Dynamical Systems  B, 2022, 27 (5) : 26212634. doi: 10.3934/dcdsb.2021151 
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]