American Institute of Mathematical Sciences

Advanced
March  2013, 18(2): 437-452. doi: 10.3934/dcdsb.2013.18.437

Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay

Mikhail Kamenskii 1, and  Boris Mikhaylenko 1,

1. 

Voronezh State University, 1 Universitetskaya pl., 394006, Voronezh, Russian Federation, Russian Federation

Received  October 2011 Revised  April 2012 Published  November 2012

This paper proposes an approach to investigate bifurcation of periodic solutions to functional-differential equations of neutral type with a small delay and a small periodic perturbation from the limit cycle under the assumption that there exists adjoint Floquet solutions to the linearized equation.
Keywords: Equation of neutral type, adjoint Floquet solution, Malkin function., small delay, periodic solutions bifurcation, small perturbations.
Mathematics Subject Classification: Primary: 34K13, 34K18, 34K27, 34K40; Secondary: 46T9.
Citation: Mikhail Kamenskii, Boris Mikhaylenko. Bifurcation of periodic solutions from a degenerated cycle in equations of neutral type with a small delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 437-452. doi: 10.3934/dcdsb.2013.18.437
References:
[1]

R. R. Akhmerov, M. I. Kamenskii, V. S. Kozyakin and A. V. Sobolev, Periodic solutions to autonomous functional-differrential neutral-type equations with a small delay,, Differential Equations, 10 (1974), 1923.   Google Scholar

[2]

R. R.Akhmerov, M. I.Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, "Measures of Non-Compactness and Condensing Operators,", Nauka, (1986).   Google Scholar

[3]

P. G. Ayzengendler, Exception theory application to a problem of bifurcation of solutions to non-linear equations,, Scient. notes Mosc. reg. Krupskaya's ped. inst., 166 (1966), 253.   Google Scholar

[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), 9.   Google Scholar

[5]

P. G. Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, I,, IHE proceedings, 10 (1969), 3.   Google Scholar

[6]

P. G. Ayzengendler and M.M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, II,, IHE proceedings, 11 (1969), 3.   Google Scholar

[7]

P. G. Ayzengendler and M. M. Vainberg, On periodic solutions to non-autonomous systems,, USSR Academy of science reports, 165 (1965), 255.   Google Scholar

[8]

P. G. Ayzengendler and M. M. Vainberg, Theory of bifurcation of solutions to non-linear equations in multidimensional case,, USSR Academy of science reports, 163 (1965), 543.   Google Scholar

[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), 103.   Google Scholar

[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), 471.   Google Scholar

[11]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Side,", Nauka, (1985).   Google Scholar

[12]

J. R. Graef and L. Kong, Periodic solutions for functional differential equations with sign-changing nonlinearities,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 597.  doi: 10.1017/S0308210509000523.  Google Scholar

[13]

J. R. Graef and S. H. Saker, New oscillation criteria for generalized second-order nonlinear neutral functional differential equations,, Dynam. Systems Appl., 19 (2010), 455.   Google Scholar

[14]

L. X. Guo, S. P. Lu, B. Du and F. Liang, Existence of periodic solutions to a second-order neutral functional differential equation with deviating arguments,, J. Math. (Wuhan), 30 (2010), 839.   Google Scholar

[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), 135.   Google Scholar

[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), 271.   Google Scholar

[17]

M. I. Kamenskii and B. A. Mikhaylenko, On a small perturbations of systems with multidimensional degeneracy,, Aut. and Rem. Contr., 5 (2011), 148.   Google Scholar

[18]

M. A. Krasnoselskii, "Translation Operator Along the Trajectories of Differential Equations,", Nauka, (1966).   Google Scholar

[19]

W. S. Loud, Periodic solutions of a perturbed autonomous system,, Ann. Math., 70 (1959), 490.   Google Scholar

[20]

L. P. Luo, Oscillation theorems for nonlinear neutral hyperbolic partial functional differential equations,, J. Math. (Wuhan), 30 (2010), 1023.   Google Scholar

[21]

O. Makarenkov and P. Nistri, Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations,, J. Math. Anal. Appl., 338 (2008), 1401.  doi: 10.1016/j.jmaa.2007.05.086.  Google Scholar

[22]

I. G. Malkin, "Some Problems of Non-Linear Oscillations Theory,", State publishers of technics and theory literature, (1956).   Google Scholar

[23]

M. B. H. Rhouma and C. Chicone, On the continuation of periodic orbits,, Methods Appl. Anal., 7 (2000), 85.   Google Scholar

[24]

A. E. Rodkina and B. N. Sadovskiy, On differentiability of translation operator along the trajectories of neutral-type equation,, Math. fac. proc., 12 (1974), 31.   Google Scholar

[25]

G. Sansone, "Equazioni Differenziali Nel Campo Reale,", p.1., (1948).   Google Scholar

[26]

S. N. Shimanov, Oscillations of quasi-linear autonomous systems with delay,, IHE proceedings. Radiophisics, 3 (1960), 456.   Google Scholar

[27]

S. N. Shimanov, To the oscillation theory of quasi-linear systems with delay,, AMM., V.XXII (1959), 836.   Google Scholar

[28]

S. L. Wan, J. Yang, C. H. Feng and J. M. Huang, Existence of periodic solutions to higher-order nonlinear neutral functional differential equations with infinite delay,, Pure Appl. Math. (Xi'an), 25 (2009), 556.   Google Scholar

[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), 1451.   Google Scholar

[30]

C. Wang and J. Wei, Hopf bifurcation for neutral functional differential equations,, Nonlinear Anal. Real World Appl., 11 (2010), 1269.   Google Scholar

[31]

F. Wei and K. Wang, The periodic solution of functional differential equations with infinite delay,, Nonlinear Anal. Real World Appl., 11 (2010), 2669.  doi: 10.1016/j.nonrwa.2009.09.014.  Google Scholar

[32]

Y. Zhu, Periodic solutions for a higher order nonlinear neutral functional differential equation,, Int. J. Comput. Math. Sci., 5 (2011), 8.   Google Scholar

show all references

References:
[1]

R. R. Akhmerov, M. I. Kamenskii, V. S. Kozyakin and A. V. Sobolev, Periodic solutions to autonomous functional-differrential neutral-type equations with a small delay,, Differential Equations, 10 (1974), 1923.   Google Scholar

[2]

R. R.Akhmerov, M. I.Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, "Measures of Non-Compactness and Condensing Operators,", Nauka, (1986).   Google Scholar

[3]

P. G. Ayzengendler, Exception theory application to a problem of bifurcation of solutions to non-linear equations,, Scient. notes Mosc. reg. Krupskaya's ped. inst., 166 (1966), 253.   Google Scholar

[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), 9.   Google Scholar

[5]

P. G. Ayzengendler and M. M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, I,, IHE proceedings, 10 (1969), 3.   Google Scholar

[6]

P. G. Ayzengendler and M.M. Vainberg, On bifurcation of periodic solutions to differential equations with delay, II,, IHE proceedings, 11 (1969), 3.   Google Scholar

[7]

P. G. Ayzengendler and M. M. Vainberg, On periodic solutions to non-autonomous systems,, USSR Academy of science reports, 165 (1965), 255.   Google Scholar

[8]

P. G. Ayzengendler and M. M. Vainberg, Theory of bifurcation of solutions to non-linear equations in multidimensional case,, USSR Academy of science reports, 163 (1965), 543.   Google Scholar

[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), 103.   Google Scholar

[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), 471.   Google Scholar

[11]

A. F. Filippov, "Differential Equations with Discontinuous Right-Hand Side,", Nauka, (1985).   Google Scholar

[12]

J. R. Graef and L. Kong, Periodic solutions for functional differential equations with sign-changing nonlinearities,, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 597.  doi: 10.1017/S0308210509000523.  Google Scholar

[13]

J. R. Graef and S. H. Saker, New oscillation criteria for generalized second-order nonlinear neutral functional differential equations,, Dynam. Systems Appl., 19 (2010), 455.   Google Scholar

[14]

L. X. Guo, S. P. Lu, B. Du and F. Liang, Existence of periodic solutions to a second-order neutral functional differential equation with deviating arguments,, J. Math. (Wuhan), 30 (2010), 839.   Google Scholar

[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), 135.   Google Scholar

[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), 271.   Google Scholar

[17]

M. I. Kamenskii and B. A. Mikhaylenko, On a small perturbations of systems with multidimensional degeneracy,, Aut. and Rem. Contr., 5 (2011), 148.   Google Scholar

[18]

M. A. Krasnoselskii, "Translation Operator Along the Trajectories of Differential Equations,", Nauka, (1966).   Google Scholar

[19]

W. S. Loud, Periodic solutions of a perturbed autonomous system,, Ann. Math., 70 (1959), 490.   Google Scholar

[20]

L. P. Luo, Oscillation theorems for nonlinear neutral hyperbolic partial functional differential equations,, J. Math. (Wuhan), 30 (2010), 1023.   Google Scholar

[21]

O. Makarenkov and P. Nistri, Periodic solutions for planar autonomous systems with nonsmooth periodic perturbations,, J. Math. Anal. Appl., 338 (2008), 1401.  doi: 10.1016/j.jmaa.2007.05.086.  Google Scholar

[22]

I. G. Malkin, "Some Problems of Non-Linear Oscillations Theory,", State publishers of technics and theory literature, (1956).   Google Scholar

[23]

M. B. H. Rhouma and C. Chicone, On the continuation of periodic orbits,, Methods Appl. Anal., 7 (2000), 85.   Google Scholar

[24]

A. E. Rodkina and B. N. Sadovskiy, On differentiability of translation operator along the trajectories of neutral-type equation,, Math. fac. proc., 12 (1974), 31.   Google Scholar

[25]

G. Sansone, "Equazioni Differenziali Nel Campo Reale,", p.1., (1948).   Google Scholar

[26]

S. N. Shimanov, Oscillations of quasi-linear autonomous systems with delay,, IHE proceedings. Radiophisics, 3 (1960), 456.   Google Scholar

[27]

S. N. Shimanov, To the oscillation theory of quasi-linear systems with delay,, AMM., V.XXII (1959), 836.   Google Scholar

[28]

S. L. Wan, J. Yang, C. H. Feng and J. M. Huang, Existence of periodic solutions to higher-order nonlinear neutral functional differential equations with infinite delay,, Pure Appl. Math. (Xi'an), 25 (2009), 556.   Google Scholar

[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), 1451.   Google Scholar

[30]

C. Wang and J. Wei, Hopf bifurcation for neutral functional differential equations,, Nonlinear Anal. Real World Appl., 11 (2010), 1269.   Google Scholar

[31]

F. Wei and K. Wang, The periodic solution of functional differential equations with infinite delay,, Nonlinear Anal. Real World Appl., 11 (2010), 2669.  doi: 10.1016/j.nonrwa.2009.09.014.  Google Scholar

[32]

Y. Zhu, Periodic solutions for a higher order nonlinear neutral functional differential equation,, Int. J. Comput. Math. Sci., 5 (2011), 8.   Google Scholar

[1]

Bernold Fiedler, Isabelle Schneider. Stabilized rapid oscillations in a delay equation: Feedback control by a small resonant delay. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1145-1185. doi: 10.3934/dcdss.2020068
[2]

Meina Gao. Small-divisor equation with large-variable coefficient and periodic solutions of DNLS equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 173-204. doi: 10.3934/dcds.2015.35.173
[3]

Adriana Buică, Jean–Pierre Françoise, Jaume Llibre. Periodic solutions of nonlinear periodic differential systems with a small parameter. Communications on Pure & Applied Analysis, 2007, 6 (1) : 103-111. doi: 10.3934/cpaa.2007.6.103
[4]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations & Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
[5]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[6]

Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105
[7]

Xu Zhang, Yuming Shi, Guanrong Chen. Coupled-expanding maps under small perturbations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1291-1307. doi: 10.3934/dcds.2011.29.1291
[8]

M.I. Gil’. Existence and stability of periodic solutions of semilinear neutral type systems. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 809-820. doi: 10.3934/dcds.2001.7.809
[9]

Ling-Jun Wang. The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1129-1156. doi: 10.3934/cpaa.2012.11.1129
[10]

Roger Grimshaw, Dmitry Pelinovsky. Global existence of small-norm solutions in the reduced Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 557-566. doi: 10.3934/dcds.2014.34.557
[11]

Wei Wang, Kai Liu, Xiulian Wang. Sensitivity to small delays of mean square stability for stochastic neutral evolution equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2403-2418. doi: 10.3934/cpaa.2020105
[12]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[13]

Fioralba Cakoni, Shari Moskow, Scott Rome. Asymptotic expansions of transmission eigenvalues for small perturbations of media with generally signed contrast. Inverse Problems & Imaging, 2018, 12 (4) : 971-992. doi: 10.3934/ipi.2018041
[14]

Hua Chen, Ling-Jun Wang. A perturbation approach for the transverse spectral stability of small periodic traveling waves of the ZK equation. Kinetic & Related Models, 2012, 5 (2) : 261-281. doi: 10.3934/krm.2012.5.261
[15]

Mickael Chekroun, Michael Ghil, Jean Roux, Ferenc Varadi. Averaging of time - periodic systems without a small parameter. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 753-782. doi: 10.3934/dcds.2006.14.753
[16]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057
[17]

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
[18]

Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885
[19]

Yang Cao, Jingxue Yin. Small perturbation of a semilinear pseudo-parabolic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 631-642. doi: 10.3934/dcds.2016.36.631
[20]

Boyan Jonov, Thomas C. Sideris. Global and almost global existence of small solutions to a dissipative wave equation in 3D with nearly null nonlinear terms. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1407-1442. doi: 10.3934/cpaa.2015.14.1407

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences