American Institute of Mathematical Sciences

Advanced
March  2018, 38(3): 967-988. doi: 10.3934/dcds.2018041

Non-formally integrable centers admitting an algebraic inverse integrating factor

Antonio Algaba , Natalia Fuentes , Cristóbal García and  Manuel Reyes ,

Department of Mathematics, Faculty of Experimental Sciences, Avda. Tres de Marzo s/n, 21071 Huelva, Spain

* Corresponding author: Manuel Reyes

Received  April 2016 Revised  October 2017 Published  December 2017

We study the existence of a class of inverse integrating factor for a family of non-formally integrable systems whose lowest-degree quasi-homogeneous term is a Hamiltonian vector field. Once the existence of an inverse integrating factor is established, we study the systems having a center. Among others, we characterize the centers of the perturbations of the system $ -y^3\partial_x+x^3\partial_y$ having an algebraic inverse integrating factor.

Keywords: Nonlinear differential systems, integrability, inverse integrating factor, centers, normal form.
Mathematics Subject Classification: 34C05, 34C14, 34C20.
Citation: Antonio Algaba, Natalia Fuentes, Cristóbal García, Manuel Reyes. Non-formally integrable centers admitting an algebraic inverse integrating factor. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 967-988. doi: 10.3934/dcds.2018041
References:
[1]

A. Algaba, E. Freire, E. Gamero and C. García, Quasihomogeneous normal forms, J. Comput. Appl. Math., 150 (2003), 193-216. doi: 10.1016/S0377-0427(02)00660-X. Google Scholar

[2]

A. Algaba, N. Fuentes, C. García and M. Reyes, A class of non-integrable systems admitting an inverse integrating factor, J. Math. Anal. Appl., 420 (2014), 1439-1454. doi: 10.1016/j.jmaa.2014.06.047. Google Scholar

[3]

A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420. doi: 10.1088/0951-7715/22/2/009. Google Scholar

[4]

A. Algaba, C. García and J. Giné, Analytic integrability for some degenerate planar vector fields, J. Differential Equations, 257 (2014), 549-565. doi: 10.1016/j.jde.2014.04.010. Google Scholar

[5]

A. Algaba, C. García and M. Reyes, Nilpotent systems admitting an algebraic inverse integrating factor over $ \mathbb{C}((x,y))$, Qualitative Theory of Dynamical Systems, 10 (2011), 303-316. doi: 10.1007/s12346-011-0046-9. Google Scholar

[6]

A. Algaba, C. García and M. Reyes, Characterization of a monodromic singular point of a planar vector field, Nonlinear Analysis, 74 (2011), 5402-5414. doi: 10.1016/j.na.2011.05.023. Google Scholar

[7]

A. Algaba, C. García and M. Reyes, Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems, Chaos Solitons & Fractals, 45 (2012), 869-878. doi: 10.1016/j.chaos.2012.02.016. Google Scholar

[8]

A. García, C. Algaba and M. Reyes, Like-linearizations of vector fields, Bulletin des Sciences Mathématiques, 133 (2009), 806-816. doi: 10.1016/j.bulsci.2009.09.006. Google Scholar

[9]

M. Berthier and R. Moussu, Réversibilité et classification des centres nilpotents, Ann. Inst. Fourier, 44 (1994), 465-494. doi: 10.5802/aif.1406. Google Scholar

[10]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems; Geometry, Topology, Classification, Chapman and Hall, 2004. Google Scholar

[11]

J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. Differential Equations, 157 (1999), 163-182. doi: 10.1006/jdeq.1998.3621. Google Scholar

[12]

J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Darboux integrability and the inverse integrating factor, J. Differential Equations, 194 (2003), 116-139. doi: 10.1016/S0022-0396(03)00190-6. Google Scholar

[13]

C. J. Christopher and J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations, 16 (2000), 5-19. Google Scholar

[14]

C. Christopher, P. Mardesic and C. Rousseau, Normalizable, integrable, and linealizable saddle points for complex quadratic systems in $ \mathbb{C}^2$, J. Dyn. Control Syst., 9 (2003), 311-363. doi: 10.1023/A:1024643521094. Google Scholar

[15]

H. R. Dullin and A. Pelayo, Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Science, 26 (2016), 787-811. doi: 10.1007/s00332-016-9290-0. Google Scholar

[16]

H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting, integrals -elliptic case, Comm. Math. Helv., 65 (1990), 4-35. doi: 10.1007/BF02566590. Google Scholar

[17]

A. Enciso and D. Peralta-Salas, Existence and vanishing set of inverse integrating factors for analytic vector fields, Bull. London Math. Soc., 41 (2009), 1112-1124. doi: 10.1112/blms/bdp090. Google Scholar

[18]

I. García, H. Giacomini and M. Grau, The inverse integrating factor and the Poincaré map, Trans. Amer. Math. Soc., 362 (2010), 3591-3612. doi: 10.1090/S0002-9947-10-05014-2. Google Scholar

[19]

I. García, H. Giacomini and M. Grau, Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor, J. Dyn. Differ. Equat., 23 (2011), 251-281. doi: 10.1007/s10884-011-9209-2. Google Scholar

[20]

I. García and M. Grau, A survey on the inverse integrating factor, Qual. Theory Dyn. Sist., 9 (2010), 115-166. doi: 10.1007/s12346-010-0023-8. Google Scholar

[21]

I. García and D. Shafer, Integral invariants and limit sets of planar vector fields, J. Differential Equations, 217 (2005), 363-376. doi: 10.1016/j.jde.2005.06.022. Google Scholar

[22]

A. Gasull and J. Torregrosa, Center problem for several differential equations via Cherkas' method, J. Math. Anal. Appl., 228 (1998), 322-343. doi: 10.1006/jmaa.1998.6112. Google Scholar

[23]

H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996), 501-516. doi: 10.1088/0951-7715/9/2/013. Google Scholar

[24]

J. Giné and D. Peralta-Salas, Existence of inverse integrating factors and Lie symmetries for degenerate planar centers, J. Differential Equations, 252 (2012), 344-357. doi: 10.1016/j.jde.2011.08.044. Google Scholar

[25]

R. E. Kooij and C. J. Christopher, Algebraic invariant curves and the integrability of polynomial systems, Appl. Math. Lett., 6 (1993), 51-53. doi: 10.1016/0893-9659(93)90123-5. Google Scholar

[26]

L. Mazzi and M. Sabatini, A characterization of centers via first integrals, J. Differential Equations, 76 (1988), 222-237. doi: 10.1016/0022-0396(88)90072-1. Google Scholar

[27]

R. Moussu, Symétrie et forme normaledes centres et foyers dégénérés, Ergodic Theory Dynam. Sys., 2 (1982), 241-251. Google Scholar

[28]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math., 37 (1881), 375-422. Google Scholar

[29]

M. J. Prelle and M. F. Singer, Elementary first integrals of of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X. Google Scholar

[30]

A. P. Sadovskii, Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, Diff. Urav., 12 (1976), 1238-1246 (in Russian). Google Scholar

[31]

S. Walcher, On the Poincaré problem, J. Differential Equations, 166 (2000), 51-78. doi: 10.1006/jdeq.2000.3801. Google Scholar

[32]

S. Walcher, Local integrating factors, J. Lie Theory, 13 (2003), 279-289. Google Scholar

show all references

References:
[1]

A. Algaba, E. Freire, E. Gamero and C. García, Quasihomogeneous normal forms, J. Comput. Appl. Math., 150 (2003), 193-216. doi: 10.1016/S0377-0427(02)00660-X. Google Scholar

[2]

A. Algaba, N. Fuentes, C. García and M. Reyes, A class of non-integrable systems admitting an inverse integrating factor, J. Math. Anal. Appl., 420 (2014), 1439-1454. doi: 10.1016/j.jmaa.2014.06.047. Google Scholar

[3]

A. Algaba, E. Gamero and C. García, The integrability problem for a class of planar systems, Nonlinearity, 22 (2009), 395-420. doi: 10.1088/0951-7715/22/2/009. Google Scholar

[4]

A. Algaba, C. García and J. Giné, Analytic integrability for some degenerate planar vector fields, J. Differential Equations, 257 (2014), 549-565. doi: 10.1016/j.jde.2014.04.010. Google Scholar

[5]

A. Algaba, C. García and M. Reyes, Nilpotent systems admitting an algebraic inverse integrating factor over $ \mathbb{C}((x,y))$, Qualitative Theory of Dynamical Systems, 10 (2011), 303-316. doi: 10.1007/s12346-011-0046-9. Google Scholar

[6]

A. Algaba, C. García and M. Reyes, Characterization of a monodromic singular point of a planar vector field, Nonlinear Analysis, 74 (2011), 5402-5414. doi: 10.1016/j.na.2011.05.023. Google Scholar

[7]

A. Algaba, C. García and M. Reyes, Existence of an inverse integrating factor, center problem and integrability of a class of nilpotent systems, Chaos Solitons & Fractals, 45 (2012), 869-878. doi: 10.1016/j.chaos.2012.02.016. Google Scholar

[8]

A. García, C. Algaba and M. Reyes, Like-linearizations of vector fields, Bulletin des Sciences Mathématiques, 133 (2009), 806-816. doi: 10.1016/j.bulsci.2009.09.006. Google Scholar

[9]

M. Berthier and R. Moussu, Réversibilité et classification des centres nilpotents, Ann. Inst. Fourier, 44 (1994), 465-494. doi: 10.5802/aif.1406. Google Scholar

[10]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems; Geometry, Topology, Classification, Chapman and Hall, 2004. Google Scholar

[11]

J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, On the integrability of two-dimensional flows, J. Differential Equations, 157 (1999), 163-182. doi: 10.1006/jdeq.1998.3621. Google Scholar

[12]

J. Chavarriga, H. Giacomini, J. Giné and J. Llibre, Darboux integrability and the inverse integrating factor, J. Differential Equations, 194 (2003), 116-139. doi: 10.1016/S0022-0396(03)00190-6. Google Scholar

[13]

C. J. Christopher and J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Ann. Differential Equations, 16 (2000), 5-19. Google Scholar

[14]

C. Christopher, P. Mardesic and C. Rousseau, Normalizable, integrable, and linealizable saddle points for complex quadratic systems in $ \mathbb{C}^2$, J. Dyn. Control Syst., 9 (2003), 311-363. doi: 10.1023/A:1024643521094. Google Scholar

[15]

H. R. Dullin and A. Pelayo, Generating hyperbolic singularities in semitoric systems via Hopf bifurcations, J. Nonlinear Science, 26 (2016), 787-811. doi: 10.1007/s00332-016-9290-0. Google Scholar

[16]

H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting, integrals -elliptic case, Comm. Math. Helv., 65 (1990), 4-35. doi: 10.1007/BF02566590. Google Scholar

[17]

A. Enciso and D. Peralta-Salas, Existence and vanishing set of inverse integrating factors for analytic vector fields, Bull. London Math. Soc., 41 (2009), 1112-1124. doi: 10.1112/blms/bdp090. Google Scholar

[18]

I. García, H. Giacomini and M. Grau, The inverse integrating factor and the Poincaré map, Trans. Amer. Math. Soc., 362 (2010), 3591-3612. doi: 10.1090/S0002-9947-10-05014-2. Google Scholar

[19]

I. García, H. Giacomini and M. Grau, Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor, J. Dyn. Differ. Equat., 23 (2011), 251-281. doi: 10.1007/s10884-011-9209-2. Google Scholar

[20]

I. García and M. Grau, A survey on the inverse integrating factor, Qual. Theory Dyn. Sist., 9 (2010), 115-166. doi: 10.1007/s12346-010-0023-8. Google Scholar

[21]

I. García and D. Shafer, Integral invariants and limit sets of planar vector fields, J. Differential Equations, 217 (2005), 363-376. doi: 10.1016/j.jde.2005.06.022. Google Scholar

[22]

A. Gasull and J. Torregrosa, Center problem for several differential equations via Cherkas' method, J. Math. Anal. Appl., 228 (1998), 322-343. doi: 10.1006/jmaa.1998.6112. Google Scholar

[23]

H. Giacomini, J. Llibre and M. Viano, On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity, 9 (1996), 501-516. doi: 10.1088/0951-7715/9/2/013. Google Scholar

[24]

J. Giné and D. Peralta-Salas, Existence of inverse integrating factors and Lie symmetries for degenerate planar centers, J. Differential Equations, 252 (2012), 344-357. doi: 10.1016/j.jde.2011.08.044. Google Scholar

[25]

R. E. Kooij and C. J. Christopher, Algebraic invariant curves and the integrability of polynomial systems, Appl. Math. Lett., 6 (1993), 51-53. doi: 10.1016/0893-9659(93)90123-5. Google Scholar

[26]

L. Mazzi and M. Sabatini, A characterization of centers via first integrals, J. Differential Equations, 76 (1988), 222-237. doi: 10.1016/0022-0396(88)90072-1. Google Scholar

[27]

R. Moussu, Symétrie et forme normaledes centres et foyers dégénérés, Ergodic Theory Dynam. Sys., 2 (1982), 241-251. Google Scholar

[28]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math., 37 (1881), 375-422. Google Scholar

[29]

M. J. Prelle and M. F. Singer, Elementary first integrals of of differential equations, Trans. Amer. Math. Soc., 279 (1983), 215-229. doi: 10.1090/S0002-9947-1983-0704611-X. Google Scholar

[30]

A. P. Sadovskii, Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, Diff. Urav., 12 (1976), 1238-1246 (in Russian). Google Scholar

[31]

S. Walcher, On the Poincaré problem, J. Differential Equations, 166 (2000), 51-78. doi: 10.1006/jdeq.2000.3801. Google Scholar

[32]

S. Walcher, Local integrating factors, J. Lie Theory, 13 (2003), 279-289. Google Scholar

Table 1.  Range and co-range of operator $\ell_{j} $ for the system (8).
Range($\ell_{2}$)=span{$-bx^2-3y^2,3ax^2+2bxy$}.
If $a\ne 0,$ Cor($\ell_{2}$)=span{$xy$}. If $a=0$, Cor($\ell_{2}$)=span{$x^2$}.
Range($\ell_{3}$)=span{$-2bx^3-6xy^2,6ax^3+4bx^2y-3h,6ax^2y+4bxy^2$}.
Cor($\ell_{3}$)=span{$h$}.
Range($\ell_{4}$)=span{$3bx^4+9x^2y^2,-9ax^4-6bx^3y+6xh,$ $ -9ax^3y-6bx^2y^2+3yh$}.
Cor($\ell_{4}$)=span{$xh,yh$}.
Table Options

Download as excel

Range($\ell_{2}$)=span{$-bx^2-3y^2,3ax^2+2bxy$}.
If $a\ne 0,$ Cor($\ell_{2}$)=span{$xy$}. If $a=0$, Cor($\ell_{2}$)=span{$x^2$}.
Range($\ell_{3}$)=span{$-2bx^3-6xy^2,6ax^3+4bx^2y-3h,6ax^2y+4bxy^2$}.
Cor($\ell_{3}$)=span{$h$}.
Range($\ell_{4}$)=span{$3bx^4+9x^2y^2,-9ax^4-6bx^3y+6xh,$ $ -9ax^3y-6bx^2y^2+3yh$}.
Cor($\ell_{4}$)=span{$xh,yh$}.
Table 2.  Range and co-range of operator $\ell_{j} $ for the system (15)
Range($\ell_{6}$)=span{$0$}, Cor($\ell_{6}$)=span{$x^2$}
Range($\ell_{7}$)=span{$0$}, Cor($\ell_{7}$)=span{$xy$}
Range($\ell_{8}$)=span{$y^2$}, Cor($\ell_{8}$)=span{$0$}
Range($\ell_{9}$)=span{$x^3$}, Cor($\ell_{9}$)=span{$0$}
Range($\ell_{10}$)=span{$0$}, Cor($\ell_{10}$)=span{$x^2y$}
Range($\ell_{11}$)=span{$xy^2$}, Cor($\ell_{11}$)=span{$0$}
Range($\ell_{12}$)=span{$7x^4-12h$}, Cor($\ell_{12}$)=span{$h$}
Range($\ell_{13}$)=span{$x^3y$}, Cor($\ell_{13}$)={$0$}
Range($\ell_{14}$)=span{$x^2y^2$}, Cor($\ell_{14}$)={$0$}
Range($\ell_{15}$)=span{$x^3-6xh$}, Cor($\ell_{15}$)=span{$xh$}
Range($\ell_{16}$)=span{$11x^4y-12yh$}, Cor($\ell_{16}$)=span{$yh$}
Range($\ell_{17}$)=span{$x^3y^2$}, Cor($\ell_{17}$)={$0$}
Range($\ell_{18}$)=span{$13x^6-36x^2h$}, Cor($\ell_{18}$)=span{$x^2h$}
Range($\ell_{19}$)=span{$7x^5-12xyh$}, Cor($\ell_{19}$)=span{$xyh$}
Range($\ell_{22}$)=span{$17x^6y-9x^2yh$}, Cor($\ell_{22}$)=span{$x^2yh$}
Table Options

Download as excel

Range($\ell_{6}$)=span{$0$}, Cor($\ell_{6}$)=span{$x^2$}
Range($\ell_{7}$)=span{$0$}, Cor($\ell_{7}$)=span{$xy$}
Range($\ell_{8}$)=span{$y^2$}, Cor($\ell_{8}$)=span{$0$}
Range($\ell_{9}$)=span{$x^3$}, Cor($\ell_{9}$)=span{$0$}
Range($\ell_{10}$)=span{$0$}, Cor($\ell_{10}$)=span{$x^2y$}
Range($\ell_{11}$)=span{$xy^2$}, Cor($\ell_{11}$)=span{$0$}
Range($\ell_{12}$)=span{$7x^4-12h$}, Cor($\ell_{12}$)=span{$h$}
Range($\ell_{13}$)=span{$x^3y$}, Cor($\ell_{13}$)={$0$}
Range($\ell_{14}$)=span{$x^2y^2$}, Cor($\ell_{14}$)={$0$}
Range($\ell_{15}$)=span{$x^3-6xh$}, Cor($\ell_{15}$)=span{$xh$}
Range($\ell_{16}$)=span{$11x^4y-12yh$}, Cor($\ell_{16}$)=span{$yh$}
Range($\ell_{17}$)=span{$x^3y^2$}, Cor($\ell_{17}$)={$0$}
Range($\ell_{18}$)=span{$13x^6-36x^2h$}, Cor($\ell_{18}$)=span{$x^2h$}
Range($\ell_{19}$)=span{$7x^5-12xyh$}, Cor($\ell_{19}$)=span{$xyh$}
Range($\ell_{22}$)=span{$17x^6y-9x^2yh$}, Cor($\ell_{22}$)=span{$x^2yh$}
Table 3.  Range and co-range of operator $\ell_{j} $ for the system (21).
Range($\ell_{3}$)=span{$x^3,y^3$}.
Cor($\ell_{3}$)=span{$x^2y,xy^2$}.
Range($\ell_{4}$)=span{$xy^3,x^4+2h,x^3y$}.
Cor($\ell_{4}$)=span{$x^2y^2,h$}.
Range($\ell_{5}$)=span{$x^2y^3,3x^5+8xh,3x^4y+4yh,x^3y^2$}.
Cor($\ell_{5}$)=span{$xh,yh$}.
Range($\ell_{6}$)=span{$x^3y^3,x^6+3x^2h,x^5y+2xyh, x^4y^2+y^2h$}.
Cor($\ell_{6}$)=span{$x^2h,xyh,y^2h$}.
Table Options

Download as excel

Range($\ell_{3}$)=span{$x^3,y^3$}.
Cor($\ell_{3}$)=span{$x^2y,xy^2$}.
Range($\ell_{4}$)=span{$xy^3,x^4+2h,x^3y$}.
Cor($\ell_{4}$)=span{$x^2y^2,h$}.
Range($\ell_{5}$)=span{$x^2y^3,3x^5+8xh,3x^4y+4yh,x^3y^2$}.
Cor($\ell_{5}$)=span{$xh,yh$}.
Range($\ell_{6}$)=span{$x^3y^3,x^6+3x^2h,x^5y+2xyh, x^4y^2+y^2h$}.
Cor($\ell_{6}$)=span{$x^2h,xyh,y^2h$}.
[1]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[2]

Jackson Itikawa, Jaume Llibre, Ana Cristina Mereu, Regilene Oliveira. Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3259-3272. doi: 10.3934/dcdsb.2017136
[3]

Hebai Chen, Jaume Llibre, Yilei Tang. Centers of discontinuous piecewise smooth quasi–homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6495-6509. doi: 10.3934/dcdsb.2019150
[4]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557
[5]

Jaume Llibre, Claudia Valls. Analytic integrability of a class of planar polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2657-2661. doi: 10.3934/dcdsb.2015.20.2657
[6]

Antoni Ferragut, Jaume Llibre, Adam Mahdi. Polynomial inverse integrating factors for polynomial vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 387-395. doi: 10.3934/dcds.2007.17.387
[7]

Ali Hamidoǧlu. On general form of the Tanh method and its application to nonlinear partial differential equations. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 175-181. doi: 10.3934/naco.2016007
[8]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345
[9]

Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $ H = H_1(x)+H_2(y)$. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 75-113. doi: 10.3934/dcds.2019004
[10]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[11]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[12]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[13]

Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362
[14]

Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
[15]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[16]

Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667
[17]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010
[18]

Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612
[19]

Dan Endres, Martin Kummer. Nonlinear normal modes for the isosceles DST. Conference Publications, 1998, 1998 (Special) : 231-241. doi: 10.3934/proc.1998.1998.231
[20]

Nguyen Dinh Cong, Doan Thai Son, Stefan Siegmund, Hoang The Tuan. An instability theorem for nonlinear fractional differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3079-3090. doi: 10.3934/dcdsb.2017164

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (109)
  • HTML views (146)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences