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

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.
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]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[2]

Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266

[3]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[4]

Maoding Zhen, Binlin Zhang, Vicenţiu D. Rădulescu. Normalized solutions for nonlinear coupled fractional systems: Low and high perturbations in the attractive case. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020379

[5]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[6]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[7]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[8]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[9]

Oussama Landoulsi. Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the $ L^2 $-supercritical case. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 701-746. doi: 10.3934/dcds.2020298

[10]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[11]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[12]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454

[13]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[14]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[15]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[16]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[17]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[18]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[19]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[20]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]