January  2019, 39(1): 75-113. doi: 10.3934/dcds.2019004

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)$

1. 

Departament de Matemàtiques, Facultat de Ciències Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

2. 

Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Ⅷ-Región, Chile

3. 

Grupo de Investigación en Sistemas Dinámicos y Aplicaciones-GISDA, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Ⅷ-región, Chile

* Corresponding author: Claudio Vidal

Received  August 2017 Revised  May 2018 Published  October 2018

We study the phase portraits on the Poincaré disc for all the linear type centers of polynomial Hamiltonian systems of degree $5$ with Hamiltonian function $H(x,y) = H_1(x)+H_2(y)$, where $H_1(x) = \frac{1}{2} x^2+\frac{a_3}{3}x^3+ \frac{a_4}{4}x^4+ \frac{a_5}{5}x^5$ and $H_2(y) = \frac{1}{2} y^2+ \frac{b_3}{3}y^3+ \frac{b_4}{4}y^4+ \frac{b_5}{5}y^5$ as function of the six real parameters $a_3, a_4, a_5, b_3, b_4$ and $b_5$ with $a_5 b_5≠ 0$. We characterize the type and multiplicity of the roots of the polynomials $\hat{p}(y) = 1+b_3y + b_4 y^2+b_5y^3$ and $\hat{q}(x) = 1+a_3x+a_4x^2+a_5x^3$ and we prove that the finite equilibria are saddles, centers, cusps or the union of two hyperbolic sectors. For the infinite equilibria we found that there only exist two nodes on the Poincaré disc with opposite stability. We also characterize the separatrices of the equilibria and analyze the possible connections between them. As a complement we use the energy level to complete the global phase portrait.

Citation: 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
References:
[1]

A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators, Dover Publications Inc., New York, 1987. Translated form the Russian by F. Immirzi, Reprint of the 1966 translation.

[2]

V. I. Arnold and Y. S. Ilyashenko, Dynamical Systems I, Ordinary Differential Equation, Encyclopaedia of Mathematical Sciences, Vols 1-2, Springer-Verlag, Heidelberg, 1988.

[3]

J. C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95. doi: 10.1006/jdeq.1994.1004.

[4]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb., 30 (1952), 181-196; Mer. Math. Soc. Transl., 100 (1954), 1-19.

[5]

J. Chavarriga and J. Giné, Integrability of a linear center perturbed by a fourth degree homogeneous polynomial, Publ. Mat., 40 (1996), 21-39. doi: 10.5565/PUBLMAT_40196_03.

[6]

J. Chavarriga and J. Giné, Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat., 41 (1997), 335-356. doi: 10.5565/PUBLMAT_41297_02.

[7]

I. ColakJ. Llibre and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Adv. Math., 259 (2014), 655-687. doi: 10.1016/j.aim.2014.04.002.

[8]

I. ColakJ. Llibre and C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257 (2014), 1623-1661. doi: 10.1016/j.jde.2014.05.024.

[9]

I. ColakJ. Llibre and C. Valls, Bifurcations diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 258 (2015), 846-879. doi: 10.1016/j.jde.2014.10.006.

[10]

I. ColakJ. Llibre and C. Valls, Bifurcations diagrams for nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 262 (2017), 5518-5533. doi: 10.1016/j.jde.2017.02.001.

[11]

H. Dulac, Détermination et integration dâ une certaine classe dâ équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér.(2), 32 (1908), 230-252.

[12]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, 2006.

[13]

J. García-SaldanaA. Gasull and H. Giacomini, Bifurcation values for a family of planar vector fields of degree five, Discrete Contin. Dyn. Syst., 35 (2015), 669-701. doi: 10.3934/dcds.2015.35.669.

[14]

A. GarijoA. Gasull and X. Jarque, Local and global phase portrait of equation $ \dot{z} = f(z)$, Discrete Contin. Dyn. Syst., 17 (2007), 309-329. doi: 10.3934/dcds.2007.17.309.

[15]

H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, Mass, 1951.

[16]

A. Guillamon and C. Pantazi, Phase portraits of separable Hamiltonian systems, Nonl. Analysis, 74 (2011), 4012-4035. doi: 10.1016/j.na.2011.03.030.

[17]

W. Kapteyn, On the midpoints of integral curves of differential equations of the first Degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl., Nederland, (1911), 1446-1457 (in Dutch).

[18]

W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk, 20 (1912), 1354-1365; Nederl. Akad. Wetensch. Verslag Afd. Natuurk, 21 (1913), 27-33 (in Dutch).

[19]

J. H. KimS. W. LeeH. Massen and H. W. Lee, Relativistic oscillator of constant period, Phys. Rev. A, 53 (1996), 2991-2997.

[20]

K. Lan and Ch. Zhu, Phase portraits of predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst., 32 (2012), 901-933. doi: 10.3934/dcds.2012.32.901.

[21]

K. E. Malkin, Criteria for the center for a certain differential equation, Vols. Mat. Sb. Vyp., 2 (1964), 87-91 (in Russian).

[22]

L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc., 76 (1954), 127-148. doi: 10.1090/S0002-9947-1954-0060657-0.

[23]

Y. P. Martínez and C. Vidal, Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential, J. Differential Equations, 261 (2016), 5923-5948. doi: 10.1016/j.jde.2016.08.024.

[24]

D. A. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81. doi: 10.1090/S0002-9939-1975-0356138-6.

[25]

M. M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, Acad. Press, New York, (1973), 389-420.

[26]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 37 (1881), 375-422; Oeuvres de Henri Poincaré, vol. Ⅰ, Gauthier-Villars, Paris, 1951, 3-84.

[27]

H. Stommel, Trajectories of small bodies sinking slowly through convection cells, J. Mar. Res., 8 (1949), 24-29.

[28]

N. I. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differential Equations, 19 (1983), 273-280.

[29]

N. I. Vulpe and K. S. Sibirskii, Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, Dokl. Akad. Nauk. SSSR, 301 (1988), 1297-1301 (in Russian); translation in: Soviet Math. Dokl., 38 (1989), 198-201.

show all references

References:
[1]

A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators, Dover Publications Inc., New York, 1987. Translated form the Russian by F. Immirzi, Reprint of the 1966 translation.

[2]

V. I. Arnold and Y. S. Ilyashenko, Dynamical Systems I, Ordinary Differential Equation, Encyclopaedia of Mathematical Sciences, Vols 1-2, Springer-Verlag, Heidelberg, 1988.

[3]

J. C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95. doi: 10.1006/jdeq.1994.1004.

[4]

N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb., 30 (1952), 181-196; Mer. Math. Soc. Transl., 100 (1954), 1-19.

[5]

J. Chavarriga and J. Giné, Integrability of a linear center perturbed by a fourth degree homogeneous polynomial, Publ. Mat., 40 (1996), 21-39. doi: 10.5565/PUBLMAT_40196_03.

[6]

J. Chavarriga and J. Giné, Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat., 41 (1997), 335-356. doi: 10.5565/PUBLMAT_41297_02.

[7]

I. ColakJ. Llibre and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Adv. Math., 259 (2014), 655-687. doi: 10.1016/j.aim.2014.04.002.

[8]

I. ColakJ. Llibre and C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257 (2014), 1623-1661. doi: 10.1016/j.jde.2014.05.024.

[9]

I. ColakJ. Llibre and C. Valls, Bifurcations diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 258 (2015), 846-879. doi: 10.1016/j.jde.2014.10.006.

[10]

I. ColakJ. Llibre and C. Valls, Bifurcations diagrams for nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 262 (2017), 5518-5533. doi: 10.1016/j.jde.2017.02.001.

[11]

H. Dulac, Détermination et integration dâ une certaine classe dâ équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér.(2), 32 (1908), 230-252.

[12]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Universitext, Springer-Verlag, 2006.

[13]

J. García-SaldanaA. Gasull and H. Giacomini, Bifurcation values for a family of planar vector fields of degree five, Discrete Contin. Dyn. Syst., 35 (2015), 669-701. doi: 10.3934/dcds.2015.35.669.

[14]

A. GarijoA. Gasull and X. Jarque, Local and global phase portrait of equation $ \dot{z} = f(z)$, Discrete Contin. Dyn. Syst., 17 (2007), 309-329. doi: 10.3934/dcds.2007.17.309.

[15]

H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, Mass, 1951.

[16]

A. Guillamon and C. Pantazi, Phase portraits of separable Hamiltonian systems, Nonl. Analysis, 74 (2011), 4012-4035. doi: 10.1016/j.na.2011.03.030.

[17]

W. Kapteyn, On the midpoints of integral curves of differential equations of the first Degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl., Nederland, (1911), 1446-1457 (in Dutch).

[18]

W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk, 20 (1912), 1354-1365; Nederl. Akad. Wetensch. Verslag Afd. Natuurk, 21 (1913), 27-33 (in Dutch).

[19]

J. H. KimS. W. LeeH. Massen and H. W. Lee, Relativistic oscillator of constant period, Phys. Rev. A, 53 (1996), 2991-2997.

[20]

K. Lan and Ch. Zhu, Phase portraits of predator-prey systems with harvesting rates, Discrete Contin. Dyn. Syst., 32 (2012), 901-933. doi: 10.3934/dcds.2012.32.901.

[21]

K. E. Malkin, Criteria for the center for a certain differential equation, Vols. Mat. Sb. Vyp., 2 (1964), 87-91 (in Russian).

[22]

L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc., 76 (1954), 127-148. doi: 10.1090/S0002-9947-1954-0060657-0.

[23]

Y. P. Martínez and C. Vidal, Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential, J. Differential Equations, 261 (2016), 5923-5948. doi: 10.1016/j.jde.2016.08.024.

[24]

D. A. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81. doi: 10.1090/S0002-9939-1975-0356138-6.

[25]

M. M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, Acad. Press, New York, (1973), 389-420.

[26]

H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Journal de Mathématiques, 37 (1881), 375-422; Oeuvres de Henri Poincaré, vol. Ⅰ, Gauthier-Villars, Paris, 1951, 3-84.

[27]

H. Stommel, Trajectories of small bodies sinking slowly through convection cells, J. Mar. Res., 8 (1949), 24-29.

[28]

N. I. Vulpe, Affine-invariant conditions for the topological discrimination of quadratic systems with a center, Differential Equations, 19 (1983), 273-280.

[29]

N. I. Vulpe and K. S. Sibirskii, Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, Dokl. Akad. Nauk. SSSR, 301 (1988), 1297-1301 (in Russian); translation in: Soviet Math. Dokl., 38 (1989), 198-201.

Figure 11.  Phase portraits for the systems associated to the Hamiltonian (5) when the roots of $\hat{p}(y)$ are $r\in \mathbb{R}$ and $a\pm ib$, and the roots of $\hat{q}(x)$ are $\rho$ and $\alpha\pm i\beta$. The separatrices are in bold. (a) $h_2<h_3$. (b) $h_2 = h_3$. (c) $h_2>h_3$
Figure 17.  Global phase portraits for the system (3) in the case Ⅰ-ⅱ for $r>0$ and $0<\sigma<\rho$. (a) $h_2 = h_4$ and $h_3 = h_5$. (b) $h_2 = h_4$ and $h_3<h_5$. (c) $h_2 = h_4$ and $h_5<h_3$. (d) $h_2<h_3<h_4<h_5$. (e) $h_3 = h_4$ and $h_2<h_5$. (f) $h_2<h_4<h_3<h_5$. (g) $h_3 = h_5$ and $h_2<h_4$. (h) $h_2<h_4<h_5<h_3$. (i) $h_4<h_2<h_3<h_5$. (j) $h_3 = h_5$ and $h_2>h_4$. (k) $h_4<h_2<h_5<h_3$
Figure 16.  Global phase portraits for the system (3) in the case Ⅰ-ⅱ for $r>0$ and $0<\rho<\sigma$. (a) $h_2<h_3 = h_4 = h_5$. (b) $h_2 = h_4<h_3 = h_5$. (c) $h_2 = h_4<h_5<h_3$. (d) $h_2<h_4 = h_5<h_3$. (e) $h_2<h_3<h_4<h_5$. (f) $h_2<h_3<h_4 = h_5$. (g) $h_2<h_3<h_5<h_4$. (h) $h_2<h_3 = h_4<h_5$. (i) $h_2<h_4<h_3<h_5$. (j) $h_2<h_4<h_3 = h_5$. (k) $h_2<h_4<h_5<h_3$. (l) $h_2<h_3 = h_5<h_4$. (m) $h_4<h_2<h_3<h_5$. (n) $h_4<h_2<h_3 = h_5$. (o) $h_4<h_2<h_5<h_3$. (p) $h_2<h_5<h_3<h_4$. (q) $h_2<h_5<h_3 = h_4$. (r) $h_2<h_5<h_4<h_3$
Figure 22.  Global Phase portraits for the system (3) in the case Ⅰ-ⅳ. Here consider $r>0$, $\rho<0<\sigma<\tau$. When four (a) or three saddle are in the energy level (a)-(i), $h_5 = h_8$ (j)-(o). (a) $h_2 = h_3 = h_5 = h_8$. (b) $h_8<h_2 = h_3 = h_5$. (c) $h_2 = h_3 = h_5<h_8$. (d) $h_5<h_2 = h_3 = h_8$. (e) $h_2 = h_3 = h_8<h_5$. (f) $h_3<h_2 = h_5 = h_8$. (g) $h_2 = h_5 = h_8<h_3$. (h) $h_2<h_3 = h_5 = h_8$. (i) $h_3 = h_5 = h_8<h_8$. (j) $h_5 = h_8<h_2<h_3$. (k) $h_2<h_5 = h_8<h_3$. (l) $h_2<h_3<h_5 = h_8$. (m) $h_5 = h_8<h_3<h_2$. (n) $h_3<h_5 = h_8<h_2$. (o) $h_3<h_2<h_5 = h_8$
Figure 23.  Global Phase portraits for the system (3) in case Ⅰ-ⅳ. Here consider $r>0$, $\rho<0<\sigma<\tau$ when two pairs of saddle are in the same energy level (a)-(f), $r>0$, $\rho<0<\sigma<\tau$ when $h_2 = h_3$ (g)-(l), and $r>0$, $\rho<0<\sigma<\tau$ when $h_2 = h_5$ (m)-(r). (a) $h_2 = h_3<h_5 = h_8$. (b) $h_2 = h_3>h_5 = h_8$. (c) $h_2 = h_5<h_3 = h_8$. (d) $h_2 = h_5>h_3 = h_8$. (e) $h_2 = h_8<h_3 = h_5$. (f) $h_2 = h_8>h_3 = h_5$. (g) $h_2 = h_3<h_5<h_8$. (h) $h_5<h_2 = h_3<h_8$. (i) $h_5<h_8<h_2 = h_3$. (j) $h_2 = h_3<h_8<h_5$. (k) $h_8<h_2 = h_3<h_5$. (l) $h_8<h_5<h_2 = h_3$. (m) $h_2 = h_5<h_3<h_8$. (n) $h_3<h_2 = h_5<h_8$. (o) $h_3<h_8<h_2 = h_5$. (p) $h_2 = h_5<h_8<h_3$. (q) $h_8<h_2 = h_5<h_3$. (r) $h_8<h_3<h_2 = h_5$
Figure 24.  Global Phase portraits for the system (3) in the case Ⅰ-ⅳ. Here consider $r>0$, $\rho<0<\sigma<\tau$ when $h_2 = h_8$ (a)-(f), $h_3 = h_5$ (g)-(l) and $h_3 = h_8$ (m)-(r). (a) $h_2 = h_8<h_3<h_5$. (b) $h_3<h_2 = h_8<h_5$. (c) $h_3<h_5<h_2 = h_8$. (d) $h_2 = h_8<h_5<h_3$. (e) $h_5<h_2 = h_8<h_3$. (f) $h_5<h_3<h_2 = h_8$. (g) $h_3 = h_5<h_2<h_8$. (h) $h_2<h_3 = h_5<h_8$. (i) $h_2<h_8<h_3 = h_5$. (j) $h_3 = h_5<h_8<h_2$. (k) $h_8<h_3 = h_5<h_2$. (l) $h_8<h_2<h_3 = h_5$. (m) $h_3 = h_8<h_2<h_5$. (n) $h_2<h_3 = h_8<h_5$. (o) $h_2<h_5<h_3 = h_8$. (p) $h_3 = h_8<h_5<h_2$. (q) $h_5<h_3 = h_8<h_2$. (r) $h_5<h_2<h_3 = h_8$
Figure 25.  Global Phase portraits for the system (3) in the case Ⅰ-ⅳ. Here consider $r>0$, $\rho<0<\sigma<\tau$ when $h_2<h_3<h_5$ (a)-(d), $h_2<h_5<h_3$ (e)-(h), $h_3<h_2<h_5$ (i)-(l), and $h_3<h_5<h_2$ (m)-(p). (a) $h_2<h_3<h_5<h_8$. (b) $h_2<h_3<h_8<h_5$. (c) $h_2<h_8<h_3<h_5$. (d) $h_8<h_3<h_3<h_5$. (e) $h_2<h_5<h_3<h_8$. (f) $h_2<h_5<h_8<h_3$. (g) $h_2<h_8<h_5<h_3$. (h) $h_8<h_2<h_5<h_3$. (i) $h_3<h_2<h_5<h_8$. (j) $h_3<h_2<h_8<h_5$. (k) $h_3<h_8<h_2<h_5$. (l) $h_8<h_3<h_2<h_5$. (m) $h_3<h_5<h_2<h_8$. (n) $h_3<h_5<h_8<h_2$. (o) $h_3<h_8<h_5<h_2$. (p) $h_8<h_3<h_5<h_2$
Figure 26.  Global phase portraits for the system (3) in the case Ⅰ-ⅳ. Here consider $r>0$, $\rho<0<\sigma<\tau$ when all the saddle are in different energy level and its satisfy $h_5<h_2<h_3$ (a)-(d), $h_5<h_3<h_2$ (e)-(h). (a) $h_5<h_2<h_3<h_8$. (b) $h_5<h_2<h_8<h_3$. (c) $h_5<h_8<h_2<h_3$. (d) $h_8<h_5<h_2<h_3$. (e) $h_5<h_3<h_2<h_8$. (f) $h_5<h_3<h_8<h_2$. (g) $h_5<h_8<h_3<h_2$. (h) $h_8<h_5<h_3<h_2$
Figure 29.  Local phase portraits at the equilibria of system (3) in the case Ⅱ-ⅱ. (a) $0<r<s$ and $0<\rho<\sigma$. (b) $r<0<s$ and $0<\rho<\sigma$. (c) $0<s<r$ and $\sigma<0<\rho$. (d) $r<0<s$ and $\sigma<0<\rho$. (e) $0<s<r$ and $0<\sigma<\rho$
Figure 30.  Local phase portraits at the equilibria of system (3) in the case Ⅱ-ⅳ. (a) if $0<r<s$. (b) if $0<s<r$
Figure 31.  Local phase portraits for system (3) in the case Ⅳ-ⅳ if $0<r<s<t$ and $0<\rho<\sigma<\tau$
Figure 2.  Local phase portrait at the origin of system (9). (a) $\tilde{w}>0$ and $\kappa>0$. (b) $\tilde{w}>0$ and $\kappa<0$. (c) $\tilde{w}<0$ and $\kappa>0$. (d) $\tilde{w}<0$ and $\kappa<0$
Figure 1.  Phase portrait in a neighborhood of the $w$-axis of system (10). (a) $\tilde{w}>0$ and $\kappa>0$. (b) $\tilde{w}>0$ and $\kappa<0$. (c) $\tilde{w}<0$ and $\kappa>0$. (d) $\tilde{w}<0$ and $\kappa<0$
Figure 3.  Blow-up at the origin of system (11). (a) Local phase portrait at the origin of the system (12). (b) Local phase portrait at the origin of the system (11)
Figure 4.  Assume $(\alpha_3-\tilde{y})/(\alpha_3 \tilde{y})>0$. (a) Local phase portrait in a neighbourhood at the $w$-axis of system (14). (b) Local phase portrait at the origin of system (13)
Figure 5.  Assume $(\alpha_3-\tilde{y})/(\alpha_3 \tilde{y})<0$. (a) Local phase portrait in a neighbourhood at the $w$-axis of system (14). (b) Local phase portrait at the origin of system (13)
Figure 6.  Blow-up of the origin of system (15). (a) Local phase portrait at in a neighborhood of the $w$-axis of system (16). (b) Local phase portrait at the origin of system (15)
Figure 7.  Local phase portrait of the equilibria of for the system (3) in the case Ⅰ-ⅰ
Figure 8.  Flow of system (3) over the straight lines $x = 0$, $x = \rho$, $y = 0$ and $y = r$ in the case Ⅰ-ⅰ
Figure 9.  The graphic of the auxiliary function $\nu_2(y)$
Figure 10.  The graphic of the auxiliary function $\mu_2(x)$. (a) $h_2<h_3$. (b) $h_2>h_3$
Figure 12.  Local phase portraits of the cusp $e_2 = (\sigma,0)$ and $e_5 = (\sigma,r)$ of system (20) according sign of $\sigma-\rho$ (assuming $\sigma,\rho>0$)
Figure 13.  Local phase portraits at the equilibria of system (3) in the case Ⅰ-ⅱ for $r>0$. (a) $0<\rho <\sigma$. (b) $0<\sigma <\rho$
Figure 14.  Flow of the vector field (3) over the straight lines $x = 0$, $x = \rho$, $x = \sigma$, $y = 0$ and $y = r$ in the case Ⅰ-ⅱ. (a) $0<\rho< \sigma$. (b) $0<\sigma<\rho$
Figure 15.  Phase portraits of separatrices in system (3) in the case Ⅰ-ⅱ with $r>0$. (a) $0<\rho< \sigma$. (b) $0<\sigma<\rho$
Figure 18.  Local phase portrait at the equilibrium $e_6$ of system (21) after translate it to the origin. (a) $(r-s)/(rs)>0$. (b) $(r-s)/(rs)<0$
Figure 19.  Local phase portraits at the equilibria of system 21 in the case Ⅱ-ⅲ with $\rho>0$. (a) if $0<r<s$. (b) if $0<s<r$
Figure 20.  Local phase portraits at the equilibria of system (3) in the case Ⅰ-ⅳ. (a) $r>0$ and $0<\rho<\sigma<\tau$. (b) $r>0$ and $\rho<0<\sigma<\tau$
Figure 21.  The vector field (22) over the straights lines $x = 0$, $x = \rho$, $x = \sigma$, $x = \sigma$, $y = 0$ and $y = r$ if $r>0$ and $\rho<0<\sigma<\tau$. (a) Vector field. (b) General analysis of the separatrices
Figure 27.  Local phase portraits for the nilpotent equilibria $e_3$ and $e_6$ in the case Ⅰ-ⅳ for $r>0$ shifted to the origin. (a) $e_3$ when $-(r-s)/(rs)>0$. (b) $e_3$ when $-(r-s)/(rs)<0$. (c) $e_6$ when $(r-s)/(rs)>0$. (d) $e_6$ when $(r-s)/(rs)<0$
Figure 28.  Local phase portrait at the equilibrium $e_9 = (\sigma,s)$ of system (24) after translation to the origin. (a) $(r-s)/(rs)>0$ and $(\rho-\sigma)/(\rho\sigma)>0$. (b) $(r-s)/(rs)>0$ and $(\rho-\sigma)/(\rho\sigma)<0$. (c) $(r-s)/(rs)<0$ and $(\rho-\sigma)/(\rho\sigma)>0$. (d) $(r-s)/(rs)<0$ and $(\rho-\sigma)/(\rho\sigma)<0$
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