메뉴 건너뛰기




Volumn , Issue , 2009, Pages 221-244

Calculating centre manifolds for delay differential equations using Maple™

Author keywords

Center manifold; Delay differential equation; Symbolic computation

Indexed keywords


EID: 84892271881     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-0-387-85595-0_8     Document Type: Chapter
Times cited : (27)

References (56)
  • 1
    • 0024172778 scopus 로고
    • BIFDE: Software for the investigation of the Hopf bifurcation problem in functional differential equations
    • Aboud N., Sathaye A., Stech H.W. (1988) BIFDE: software for the investigation of the Hopf bifurcation problem in functional differential equations. In: Proceedings of the 27th IEEE Conference on Decision and Control. Vol. 1. pp. 821-824
    • (1988) Proceedings of the 27th IEEE Conference on Decision and Control , vol.1 , pp. 821-824
    • Aboud, N.1    Sathaye, A.2    Stech, H.W.3
  • 3
    • 0028517922 scopus 로고
    • Stability and bifurcations of equilibria in a multiple-delayed differential equation
    • Bélair J., Campbell S.A. (1994) Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM Journal on Applied Mathematics 54(5), 1402-1424
    • (1994) SIAM Journal on Applied Mathematics , vol.54 , Issue.5 , pp. 1402-1424
    • Bélair, J.1    Campbell, S.A.2
  • 5
    • 32844462800 scopus 로고    scopus 로고
    • Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system
    • Campbell S.A., Ncube I., Wu J. (2006) Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system. Physica D 214(2), 101-119
    • (2006) Physica D , vol.214 , Issue.2 , pp. 101-119
    • Campbell, S.A.1    Ncube, I.2    Wu, J.3
  • 6
    • 58149357294 scopus 로고    scopus 로고
    • Zero singularities of codimension two and three in delay differential equations
    • Campbell S.A., Yuan Y. (2008) Zero singularities of codimension two and three in delay differential equations. Nonlinearity 21, 2671-2691
    • (2008) Nonlinearity , vol.21 , pp. 2671-2691
    • Campbell, S.A.1    Yuan, Y.2
  • 7
    • 26944444656 scopus 로고    scopus 로고
    • Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling
    • Campbell S.A., Yuan Y., Bungay S.D. (2005) Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling. Nonlinearity 18, 2827-2846
    • (2005) Nonlinearity , vol.18 , pp. 2827-2846
    • Campbell, S.A.1    Yuan, Y.2    Bungay, S.D.3
  • 9
    • 0036922219 scopus 로고    scopus 로고
    • Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations
    • Das S.L., Chatterjee A. (2002) Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dynamics 30(4), 323-335
    • (2002) Nonlinear Dynamics , vol.30 , Issue.4 , pp. 323-335
    • Das, S.L.1    Chatterjee, A.2
  • 10
    • 39749182700 scopus 로고    scopus 로고
    • Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation
    • Deshmukh V., Butcher E.A., Bueler E. (2008) Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation. Nonlinear Dynamics 52, 137-149
    • (2008) Nonlinear Dynamics , vol.52 , pp. 137-149
    • Deshmukh, V.1    Butcher, E.A.2    Bueler, E.3
  • 11
    • 0000840165 scopus 로고
    • Normal forms for retarded functional differential equations with parameters and applications to Bogdanov-Takens singularity
    • Faria T., Magalh-aes L. (1995a) Normal forms for retarded functional differential equations with parameters and applications to Bogdanov-Takens singularity. Journal of Differential Equations 122, 201-224
    • (1995) Journal of Differential Equations , vol.122 , pp. 201-224
    • Faria, T.1    Magalh-Aes, L.2
  • 12
    • 0000840164 scopus 로고
    • Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation
    • Faria T., Magalh-aes L. (1995b) Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. Journal of Differential Equations 122, 181-200
    • (1995) Journal of Differential Equations , vol.122 , pp. 181-200
    • Faria, T.1    Magalh-Aes, L.2
  • 13
    • 37949006267 scopus 로고
    • Extensions of an algorithm for the analysis of nongeneric Hopf bifurcations, with applications to delay-difference equations
    • Busenberg S, Martelli M(eds) of Springer Lecture Notes in Mathematics. Springer- Verlag, Berlin
    • Franke J.M., Stech H.W. (1991) Extensions of an algorithm for the analysis of nongeneric Hopf bifurcations, with applications to delay-difference equations. In: Busenberg S, Martelli M(eds), Delay Differential Equations and Dynamical Systems. Vol. 1475 of Springer Lecture Notes in Mathematics. Springer-Verlag, Berlin, pp. 161-175
    • (1991) Delay Differential Equations and Dynamical Systems , vol.1475 , pp. 161-175
    • Franke, J.M.1    Stech, H.W.2
  • 14
    • 0036803765 scopus 로고    scopus 로고
    • Estimating critical Hopf bifurcation parameters for a second order delay differential equation with application to machine tool chatter
    • Gilsinn D.E. (2002) Estimating critical Hopf bifurcation parameters for a second order delay differential equation with application to machine tool chatter. Nonlinear Dynamics 30, 103-154
    • (2002) Nonlinear Dynamics , vol.30 , pp. 103-154
    • Gilsinn, D.E.1
  • 15
    • 84892230211 scopus 로고    scopus 로고
    • Bifurcations, center manifolds, and periodic solutions
    • Balachandran B, Gilsinn DE, Kalmár-Nagy T (eds) Springer Verlag, New York
    • Gilsinn D.E. (2008) Bifurcations, center manifolds, and periodic solutions. In: Balachandran B, Gilsinn DE, Kalmár-Nagy T (eds), Delay Differential Equations: Recent Advances and New Directions. Springer Verlag, New York, pp. 157-204
    • (2008) Delay Differential Equations: Recent Advances and New Directions , pp. 157-204
    • Gilsinn, D.E.1
  • 16
    • 22244473030 scopus 로고    scopus 로고
    • Delay induced periodicity in a neural netlet of excitation and inhibition
    • Gopalsamy K., Leung I. (1996) Delay induced periodicity in a neural netlet of excitation and inhibition. Physica D 89, 395-426
    • (1996) Physica D , vol.89 , pp. 395-426
    • Gopalsamy, K.1    Leung, I.2
  • 18
    • 23444456002 scopus 로고    scopus 로고
    • Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay
    • Guo S. (2005) Spatio-temporal patterns of nonlinear oscillations in an excitatory ring network with delay. Nonlinearity 18, 2391-2407
    • (2005) Nonlinearity , vol.18 , pp. 2391-2407
    • Guo, S.1
  • 19
    • 0041511877 scopus 로고    scopus 로고
    • Hopf bifurcating periodic orbits in a ring of neurons with delays
    • Guo S., Huang L. (2003) Hopf bifurcating periodic orbits in a ring of neurons with delays. Physica D 183, 19-44
    • (2003) Physica D , vol.183 , pp. 19-44
    • Guo, S.1    Huang, L.2
  • 20
    • 4944247888 scopus 로고    scopus 로고
    • Linear stability and Hopf bifurcation in a two neuron network with three delays
    • Guo S., Huang L., Wang L. (2004) Linear stability and Hopf bifurcation in a two neuron network with three delays. International Journal of Bifurcation and Chaos 14, 2799-2810
    • (2004) International Journal of Bifurcation and Chaos , vol.14 , pp. 2799-2810
    • Guo, S.1    Huang, L.2    Wang, L.3
  • 21
    • 84974185478 scopus 로고
    • Flows on center manifolds for scalar functional differential equations
    • Hale J.K. (1985) Flows on center manifolds for scalar functional differential equations. Proceedings of the Royal Society of Edinburgh 101A, 193-201
    • (1985) Proceedings of the Royal Society of Edinburgh , vol.101 A , pp. 193-201
    • Hale, J.K.1
  • 23
    • 30944453872 scopus 로고    scopus 로고
    • Stability, bifurcation and a new chaos in the logistic differential equation with delay
    • Jiang M., Shen Y., Jian J., Liao X. (2006) Stability, bifurcation and a new chaos in the logistic differential equation with delay. Physics Letters A 350(3-4), 221-227
    • (2006) Physics Letters A , vol.350 , Issue.3-4 , pp. 221-227
    • Jiang, M.1    Shen, Y.2    Jian, J.3    Liao, X.4
  • 25
    • 0035493311 scopus 로고    scopus 로고
    • Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations
    • Kalmár-Nagy T., Stépán G., Moon F.C. (2001) Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations. Nonlinear Dynamics 26, 121-142
    • (2001) Nonlinear Dynamics , vol.26 , pp. 121-142
    • Kalmár-Nagy, T.1    Stépán, G.2    Moon, F.C.3
  • 26
    • 0033102023 scopus 로고    scopus 로고
    • Theory and applications of Hopf bifurcations in symmetric functional-differential equations
    • Series A: Theory Methods
    • Krawcewicz W., Wu J. (1999) Theory and applications of Hopf bifurcations in symmetric functional-differential equations. Nonlinear Analysis 35 (7, Series A: Theory Methods), 845-870
    • (1999) Nonlinear Analysis , vol.35 , Issue.7 , pp. 845-870
    • Krawcewicz, W.1    Wu, J.2
  • 28
    • 4243051466 scopus 로고    scopus 로고
    • Stability and bifurcation in a harmonic oscillator with delays
    • Liu Z., Yuan R. (2005) Stability and bifurcation in a harmonic oscillator with delays. Chaos, Solitons and Fractals 23, 551-562
    • (2005) Chaos, Solitons and Fractals , vol.23 , pp. 551-562
    • Liu, Z.1    Yuan, R.2
  • 29
    • 29344437674 scopus 로고    scopus 로고
    • Maplesoft, a division of Waterloo Maple Inc., Toronto, Canada
    • Maple 9.5 Getting Started Guide (2004) Maplesoft, a division of Waterloo Maple Inc., Toronto, Canada
    • (2004) Maple 9.5 Getting Started Guide
  • 30
    • 38349094163 scopus 로고    scopus 로고
    • Order reduction of retarded nonlinear systems - The method of multiple scales versus center-manifold reduction
    • Nayfeh A.H. (2008) Order reduction of retarded nonlinear systems - the method of multiple scales versus center-manifold reduction. Nonlinear Dynamics 51, 483-500
    • (2008) Nonlinear Dynamics , vol.51 , pp. 483-500
    • Nayfeh, A.H.1
  • 31
    • 84867929802 scopus 로고    scopus 로고
    • Hopf bifurcation calculations in delayed systems with translational symmetry
    • Orosz G., Stépán G. (2004) Hopf bifurcation calculations in delayed systems with translational symmetry. Journal of Nonlinear Science 14(6), 505-528
    • (2004) Journal of Nonlinear Science , vol.14 , Issue.6 , pp. 505-528
    • Orosz, G.1    Stépán, G.2
  • 32
    • 33748103156 scopus 로고    scopus 로고
    • Subcritical Hopf bifurcations in a car-following model with reaction-time delay
    • Orosz G., Stépán G. (2006) Subcritical Hopf bifurcations in a car-following model with reaction-time delay. Proceedings of the Royal Society of London, series A 462(2073), 2643-2670
    • (2006) Proceedings of the Royal Society of London, Series A , vol.462 , Issue.2073 , pp. 2643-2670
    • Orosz, G.1    Stépán, G.2
  • 34
    • 33749519969 scopus 로고    scopus 로고
    • Center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with Fold-Hopf singularity
    • Qesmi R., Ait Babram M., Hbid M.L. (2006a) Center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with Fold-Hopf singularity. Applied Mathematics and Computation 181(1), 220-246
    • (2006) Applied Mathematics and Computation , vol.181 , Issue.1 , pp. 220-246
    • Qesmi, R.1    Ait Babram, M.2    Hbid, M.L.3
  • 35
    • 33645887211 scopus 로고    scopus 로고
    • Computation of terms of center manifolds and normal elements of bifurcations for a class of functional differential equations associated with Hopf singularity
    • Qesmi R., Ait Babram M., Hbid M.L. (2006b) Computation of terms of center manifolds and normal elements of bifurcations for a class of functional differential equations associated with Hopf singularity. Applied Mathematics and Computation 175(2), 932-968
    • (2006) Applied Mathematics and Computation , vol.175 , Issue.2 , pp. 932-968
    • Qesmi, R.1    Ait Babram, M.2    Hbid, M.L.3
  • 36
    • 33947156772 scopus 로고    scopus 로고
    • Symbolic computation for center manifolds and normal forms of Bogdanov bifurcation in retarded functional differential equations
    • Qesmi R., Ait Babram M., Hbid M.L. (2007) Symbolic computation for center manifolds and normal forms of Bogdanov bifurcation in retarded functional differential equations. Nonlinear Analysis 66, 2833-2851
    • (2007) Nonlinear Analysis , vol.66 , pp. 2833-2851
    • Qesmi, R.1    Ait Babram, M.2    Hbid, M.L.3
  • 38
    • 0242573708 scopus 로고    scopus 로고
    • A centre-manifold analysis of variable speed machining
    • Sri Namachchivaya N., van Roessel H.J. (2003) A centre-manifold analysis of variable speed machining. Dynamical Systems 18(3), 245-270
    • (2003) Dynamical Systems , vol.18 , Issue.3 , pp. 245-270
    • Sri Namachchivaya, N.1    Van Roessel, H.J.2
  • 40
    • 84892246119 scopus 로고
    • Hopf bifurcation analysis in a class of scalar functional differential equations
    • Lighthourne J, Rankin S (eds) Marcel Dekker, New York
    • Stech H.W. (1985a) Hopf bifurcation analysis in a class of scalar functional differential equations. In: Lighthourne J, Rankin S (eds), Physical mathematics and nonlinear partial differential equations. Marcel Dekker, New York, pp. 175-186
    • (1985) Physical Mathematics and Nonlinear Partial Differential Equations , pp. 175-186
    • Stech, H.W.1
  • 41
    • 0012278310 scopus 로고
    • Hopf bifurcation calculations for functional differential equations
    • Stech H.W. (1985b) Hopf bifurcation calculations for functional differential equations. Journal of Mathematical Analysis and Applications 109, 472-491
    • (1985) Journal of Mathematical Analysis and Applications , vol.109 , pp. 472-491
    • Stech, H.W.1
  • 42
    • 37949006646 scopus 로고
    • Nongeneric Hopf bifurcations in functional differential equations
    • Stech H.W. (1985c) Nongeneric Hopf bifurcations in functional differential equations. SIAM Journal on Mathematical Analysis 16, 1134-1151
    • (1985) SIAM Journal on Mathematical Analysis , vol.16 , pp. 1134-1151
    • Stech, H.W.1
  • 43
    • 0029486988 scopus 로고
    • Quasiperiodic oscillations in robot dynamics
    • Stépán G., Haller G. (1995) Quasiperiodic oscillations in robot dynamics. Nonlinear Dynamics 8, 513-528
    • (1995) Nonlinear Dynamics , vol.8 , pp. 513-528
    • Stépán, G.1    Haller, G.2
  • 44
    • 0036523478 scopus 로고    scopus 로고
    • Nonlinear models of chatter in drilling processes
    • Stone E., Askari A. (2002) Nonlinear models of chatter in drilling processes. Dynamical Systems 17(1), 65-85
    • (2002) Dynamical Systems , vol.17 , Issue.1 , pp. 65-85
    • Stone, E.1    Askari, A.2
  • 45
    • 84867969858 scopus 로고    scopus 로고
    • Stability and bifurcation analysis of a nonlinear DDE model for drilling
    • Stone E., Campbell S.A. (2004) Stability and bifurcation analysis of a nonlinear DDE model for drilling. Journal of Nonlinear Science 14(1), 27-57
    • (2004) Journal of Nonlinear Science , vol.14 , Issue.1 , pp. 27-57
    • Stone, E.1    Campbell, S.A.2
  • 48
    • 27644482807 scopus 로고    scopus 로고
    • Synchronized Hopf bifurcation analysis in a neural network model with delays
    • Wei J.J., Yuan Y. (2005) Synchronized Hopf bifurcation analysis in a neural network model with delays. Journal of Mathematical Analysis and Applications 312(1), 205-229
    • (2005) Journal of Mathematical Analysis and Applications , vol.312 , Issue.1 , pp. 205-229
    • Wei, J.J.1    Yuan, Y.2
  • 50
    • 0036845207 scopus 로고    scopus 로고
    • The dynamics of two coupled van der Pol oscillators with delay coupling
    • Wirkus S., Rand R. (2004) The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dynamics 30(3), 205-221
    • (2004) Nonlinear Dynamics , vol.30 , Issue.3 , pp. 205-221
    • Wirkus, S.1    Rand, R.2
  • 52
    • 33646864830 scopus 로고    scopus 로고
    • Symmetric functional-differential equations and neural networks with memory
    • Wu J. (1998) Symmetric functional-differential equations and neural networks with memory. Transactions of the American Mathematical Society 350(12), 4799-4838
    • (1998) Transactions of the American Mathematical Society , vol.350 , Issue.12 , pp. 4799-4838
    • Wu, J.1
  • 53
    • 0033166162 scopus 로고    scopus 로고
    • Synchronization and stable phase-locking in a network of neurons with memory
    • Wu J., Faria T., Huang Y.S. (1999) Synchronization and stable phase-locking in a network of neurons with memory. Mathematical and Computer Modelling 30(1-2), 117-138
    • (1999) Mathematical and Computer Modelling , vol.30 , Issue.1-2 , pp. 117-138
    • Wu, J.1    Faria, T.2    Huang, Y.S.3
  • 54
    • 26944442186 scopus 로고    scopus 로고
    • Stability and synchronization of a ring of identical cells with delayed coupling
    • Yuan Y., Campbell S.A. (2004) Stability and synchronization of a ring of identical cells with delayed coupling. Journal of Dynamics and Differential Equations 16(1), 709-744
    • (2004) Journal of Dynamics and Differential Equations , vol.16 , Issue.1 , pp. 709-744
    • Yuan, Y.1    Campbell, S.A.2
  • 55
    • 33845533069 scopus 로고    scopus 로고
    • Multiple bifurcation analysis in a neural network model with delays
    • Yuan Y., Wei J.J. (2005) Multiple bifurcation analysis in a neural network model with delays. International Journal of Bifurcation and Chaos 16(10), 2903-2913
    • (2005) International Journal of Bifurcation and Chaos , vol.16 , Issue.10 , pp. 2903-2913
    • Yuan, Y.1    Wei, J.J.2
  • 56
    • 4944238379 scopus 로고    scopus 로고
    • Aeroelastiticy of time-delayed feedback control of two-dimensional supersonic lifting surfaces
    • Yuan Y., Yu P., Librescu L., Marzocca P. (2004) Aeroelastiticy of time-delayed feedback control of two-dimensional supersonic lifting surfaces. Journal of Guidance, Control, and Dynamics 27(5), 795-803
    • (2004) Journal of Guidance, Control, and Dynamics , vol.27 , Issue.5 , pp. 795-803
    • Yuan, Y.1    Yu, P.2    Librescu, L.3    Marzocca, P.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.